#binomial — Public Fediverse posts

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

I'm doing some symmetric monoidal algebra that involves careful counting of some signs of certain permutations. The work depends on a couple of combinatorial identities that I haven't seen anywhere else—do you recognize these (below)?!

The identities involve the "choose two" binomial coefficients.
For ease of typing, I'll use this notation:

[a;2] = binomial(a,2) = a·(a-1)/2
(read "a choose 2")

The two identities are

(I1):
[a+b;2] = [a;2] + [b;2] + ab

and

(I2):
[ab;2] = a[b;2] + b[a;2] + 2[a;2][b;2]

In particular, (I2) means there is a mod 2 congruence
[ab;2] ≡ a[b;2] + b[a;2]
and that's the form that has been particularly useful for me.

Neither of these identities are hard to prove directly from the definition, and they hold for positive *and negative* integers a and b. (That extension to all integers is important for my applications too.)

I've done some internet searching (wikipedia [1,2] and other general references), but I haven't found mention of these particular identities. So, I'm wondering if anyone here recognizes them. (Boosts appreciated!)

Note: These particular binomial coefficients [a;2], for positive a, are also called *triangular numbers*. I'll rewrite (I1) and (I2) in terms of triangular numbers in the next post, in case people will recognize that alternate form (but I doubt it).

[1] https://en.wikipedia.org/wiki/Binomial_coefficient
[2] https://en.wikipedia.org/wiki/Triangular_number

(1/2)

#binomial #triangular #PascalsTriangle

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

Bitnomial sues SEC over claim that XRP is a security - The XRP token is already regulated as a commodity and the SEC “duplicate... - https://cointelegraph.com/news/bitnomial-sues-sec-over-xrp-security-claim #regulations #xrpfutures #securities #binomial #lawsuit #ripple #sec

In the process I also thought of a #binomial #coefficient interpretation for choosing k things out of a bag of n objects when the objects can be put back and picked again. In probability problems, this is referred to as picking "with replacement".

Usually, when we say, n choose k, we do not allow repeated choices, every chosen object has to be chosen once. In this case, I want to allow replacements but at the same time, I want to keep using my trusted n choose k idea.

Here's my way around this: We'll still be picking k things, but we'll pick them not from 1 to n but from the set {1, 2, ..., 𝑛, 𝑟₁, 𝑟₂, ..., 𝑟ₖ₋₁}. That is, in addition to the n objects, we add what I'm calling "replacement tokens" 𝑟ₓ. If any of the 𝑟ₓ gets picked, then it is interpreted as the 𝑥th choice was put back and you are now choosing to pick that again. Since the 𝑘th choice is not put back, we only need replacement tokens for choices 1 to k-1.

With these replacement tokens, the problem becomes a standard choose k things out of this set, which we can resolve using the binomial coefficient to get: \({ n+k-1 \choose k }\).

I believe the standard approach to this is via #StarsAndBars but I liked the idea of this "replacement token". Admittedly, I didn't want to use stars and bars here and made up some stuff which I happened to like. :)

#math #counting #proof

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@[email protected]

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

Le fun est total.

(c'est un vrai jeu)
(enfin c'est un logiciel, que j'ai acheté avec mon argent, dans le but de me détendre, sur steam)

#infiniteturtles #zachtronics #zachlike #binomial

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial:

https://youtu.be/g0OQrAVwsTw

In this short video you'll learn how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #ProbabilityDistribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood #SciDAViS

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@[email protected]

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@[email protected]

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@[email protected]

New LabPlot tutorial

https://tube.kockatoo.org/w/9Kmqeefqw35EpEZj914N5p

In this short video you'll learn how to how to fit a distribution to data.

#DistributionFitting #GaussianDistribution, #Log-normalDistribution #Probability Distribution #PoissonDistribution #Binomial #Distribution #ExponentialDistribution #MaximumLikelihood

@[email protected]

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

One day, one decomposition
A138389: Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1)

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Binomial_primes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Binomial_primes.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #primes #PrimeNumbers #coprime #graph #threejs #webGL

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

One day, one decomposition
A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A121943.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A121943.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #central #binomial #coefficient #divisible #graph #threejs #webGL

Bitnomial sues SEC over claim that XRP is a security - The XRP token is already regulated as a commodity and the SEC “duplicate... - https://cointelegraph.com/news/bitnomial-sues-sec-over-xrp-security-claim #regulations #xrpfutures #securities #binomial #lawsuit #ripple #sec

Bitnomial sues SEC over claim that XRP is a security - The XRP token is already regulated as a commodity and the SEC “duplicate... - https://cointelegraph.com/news/bitnomial-sues-sec-over-xrp-security-claim #regulations #xrpfutures #securities #binomial #lawsuit #ripple #sec

Bitnomial sues SEC over claim that XRP is a security - The XRP token is already regulated as a commodity and the SEC “duplicate... - https://cointelegraph.com/news/bitnomial-sues-sec-over-xrp-security-claim #regulations #xrpfutures #securities #binomial #lawsuit #ripple #sec

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]

\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]

where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.

#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge

In the process I also thought of a #binomial #coefficient interpretation for choosing k things out of a bag of n objects when the objects can be put back and picked again. In probability problems, this is referred to as picking "with replacement".

Usually, when we say, n choose k, we do not allow repeated choices, every chosen object has to be chosen once. In this case, I want to allow replacements but at the same time, I want to keep using my trusted n choose k idea.

Here's my way around this: We'll still be picking k things, but we'll pick them not from 1 to n but from the set {1, 2, ..., 𝑛, 𝑟₁, 𝑟₂, ..., 𝑟ₖ₋₁}. That is, in addition to the n objects, we add what I'm calling "replacement tokens" 𝑟ₓ. If any of the 𝑟ₓ gets picked, then it is interpreted as the 𝑥th choice was put back and you are now choosing to pick that again. Since the 𝑘th choice is not put back, we only need replacement tokens for choices 1 to k-1.

With these replacement tokens, the problem becomes a standard choose k things out of this set, which we can resolve using the binomial coefficient to get: \({ n+k-1 \choose k }\).

I believe the standard approach to this is via #StarsAndBars but I liked the idea of this "replacement token". Admittedly, I didn't want to use stars and bars here and made up some stuff which I happened to like. :)

#math #counting #proof

In this video, we look at binomial expressions using synonyms (words with same or similar meanings).

This is the second video in a short series on binomial expressions.

#synonyms #LearnEnglish #binomial #expressions #EFL #ESL #daybreakenglish

https://youtu.be/3FRLM5-I9pY

In this video, we look at binomial expressions using synonyms (words with same or similar meanings).

This is the second video in a short series on binomial expressions.

#synonyms #LearnEnglish #binomial #expressions #EFL #ESL #daybreakenglish

https://youtu.be/3FRLM5-I9pY

In this video, we look at binomial expressions using antonyms (words that are opposites).

But wait, what's a binomial expression?

#antonyms #LearnEnglish #binomial #expressions #EFL #ESL #DaybreakEnglish

https://youtu.be/kjknk1UpHAE

In this video, we look at binomial expressions using antonyms (words that are opposites).

But wait, what's a binomial expression?

#antonyms #LearnEnglish #binomial #expressions #EFL #ESL #DaybreakEnglish

https://youtu.be/kjknk1UpHAE

One day, one decomposition
A034856: a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A034856.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A034856.html

#decompwlj #maths #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #graph #threejs #webGL

One day, one decomposition
A034856: a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A034856.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A034856.html

#decompwlj #maths #mathematics #sequence #OEIS #javascript #php #3D #numbers #binomial #graph #threejs #webGL

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.

https://www.youtube.com/watch?v=g0OQrAVwsTw

#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot

Le fun est total.

(c'est un vrai jeu)
(enfin c'est un logiciel, que j'ai acheté avec mon argent, dans le but de me détendre, sur steam)

#infiniteturtles #zachtronics #zachlike #binomial

Le fun est total.

(c'est un vrai jeu)
(enfin c'est un logiciel, que j'ai acheté avec mon argent, dans le but de me détendre, sur steam)

#infiniteturtles #zachtronics #zachlike #binomial