cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A376659 Decimal expansion of a constant related to the asymptotics of A376626 and A376627.

Original entry on oeis.org

3, 3, 3, 5, 2, 6, 0, 2, 0, 7, 0, 3, 7, 0, 8, 0, 8, 6, 0, 2, 9, 1, 2, 2, 4, 4, 8, 1, 5, 6, 3, 3, 5, 2, 4, 6, 7, 3, 0, 8, 8, 4, 9, 8, 7, 0, 9, 9, 2, 7, 7, 9, 6, 8, 2, 0, 6, 1, 3, 7, 0, 4, 6, 5, 3, 8, 3, 8, 2, 8, 8, 8, 1, 9, 4, 3, 7, 2, 1, 2, 0, 1, 2, 2, 7, 4, 2, 2, 8, 0, 3, 2, 7, 5, 6, 4, 1, 8, 2, 1, 6, 4, 3, 7, 3
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			3.33526020703708086029122448156335246730884987099277968...

Crossrefs

Cf. A075778, A333198, A376621, A376626, A376627, A376658, A376660.

Programs

  • Mathematica
    RealDigits[E^Sqrt[6*Log[r]^2 + 2*PolyLog[2, 1 - r^2]] /. r -> (-1 + ((25 - 3*Sqrt[69])/2)^(1/3) + ((25 + 3*Sqrt[69])/2)^(1/3))/3, 10, 105][[1]]

Formula

Equals exp(sqrt(2*(3*log(r)^2 + polylog(2, 1 - r^2)))), where r = A075778 = 0.7548776662466927600495088963585286918946... is the real root of the equation r^2*(1+r) = 1.
Equals limit_{n->infinity} A376626(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376627(n)^(1/sqrt(n)).

A376812 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 8, 10, 13, 14, 16, 19, 21, 25, 29, 33, 40, 45, 50, 57, 64, 72, 81, 93, 104, 117, 134, 148, 165, 185, 204, 227, 253, 280, 310, 345, 381, 422, 469, 514, 567, 625, 685, 753, 825, 903, 990, 1086, 1186, 1297, 1419, 1548, 1692, 1845, 2007
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 05 2024

Keywords

Links

Crossrefs

Cf. A053261, A376626, A376627, A376813, A216222.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (1 + x^j)^2 * x^j.
a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 1/(4*sqrt(3/2 - 2*sinh(arcsinh(3^(3/2)/2)/3)/sqrt(3))) = 0.27647151570071656262813536...

A376623 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 4, 2, 2, 4, 2, 2, 4, 4, 5, 4, 6, 10, 6, 6, 9, 8, 9, 8, 10, 12, 11, 14, 14, 16, 17, 18, 23, 20, 22, 26, 26, 26, 27, 34, 31, 32, 39, 40, 43, 42, 48, 54, 55, 56, 63, 72, 68, 74, 80, 84, 88, 90, 101, 104, 109, 112, 121, 130, 132, 144, 152, 160
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A376542, A376580, A376621, A376626, A376629.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1))*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*(1 + x^(2*k - 1))*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1))^2.
a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = sqrt(sinh(arcsinh(3*sqrt(93)/2)/3)) / (sqrt(2)*93^(1/4)) = 0.26678318911398751342...
Showing 1-3 of 3 results.

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