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 1-6 of 6 results.

A376621 Decimal expansion of a constant related to the asymptotics of A369557 and A376580.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Examples

			2.75108509088891993943420496204894703641817860263717...

Crossrefs

Cf. A333198, A369557, A376542, A376580, A376623, A376658, A376659, A376660.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2/2 + 4*PolyLog[2, r^(1/2)] - Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A369557(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376580(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376542(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376623(n)^(1/sqrt(n)).
Equals exp(sqrt(3*log(r)^2/2 + 4*polylog(2, r^(1/2)) - Pi^2/3)), where r = A088559 = 0.465571231876768026656731... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A216222 Counting a set of restricted partitions.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 3, 3, 6, 7, 8, 10, 9, 9, 9, 9, 11, 13, 16, 20, 22, 25, 28, 27, 28, 29, 30, 32, 35, 40, 45, 53, 60, 67, 73, 79, 85, 87, 92, 95, 98, 105, 111, 120, 132, 145, 160, 178, 196, 212, 231, 247, 263, 280, 291, 305, 319, 334, 352, 371, 393
Offset: 0

Views

Author

David S. Newman, Mar 13 2013

Keywords

Links

Crossrefs

Cf. A001622, A306734, A376542, A376580, A376812, A376813.

Programs

  • Mathematica
    Take[CoefficientList[Sum[x^(k^2)*Product[1 + x^i, {i, k}]^2, {k, 0, 7}], x], 63] (* Giovanni Resta, Mar 13 2013 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Oct 09 2024 *)

Formula

G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^2 = 1 +x^1*(1+x)^2 +x^4*(1+x)^2*(1+x^2)^2 +...+ x^k^2*(1+x)^2*(1+x^2)^2*(1+x^3)^2*...*(1+x^k)^2+...
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (4*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 29 2024

Extensions

a(14)-a(62) from Giovanni Resta, Mar 13 2013

A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*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} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A376581 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 13, 17, 22, 30, 38, 48, 62, 78, 97, 122, 151, 184, 228, 278, 335, 408, 491, 588, 707, 843, 1000, 1189, 1407, 1658, 1955, 2295, 2686, 3145, 3670, 4270, 4968, 5763, 6671, 7720, 8909, 10263, 11816, 13577, 15574, 17850, 20424, 23333, 26638, 30365
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 29 2024

Keywords

Links

Crossrefs

Cf. A053264, A306734, A340647, A376542, A376580.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k^2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n/5)) / (4*5^(1/4)*sqrt(n)).

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...

A376626 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 3, 6, 3, 3, 7, 7, 6, 6, 9, 13, 12, 11, 16, 18, 17, 20, 22, 26, 28, 31, 36, 36, 42, 46, 50, 57, 61, 69, 72, 75, 87, 97, 100, 108, 126, 136, 141, 151, 167, 188, 195, 207, 233, 254, 265, 279, 315, 339, 355, 380, 417, 455, 473, 503, 551, 600, 627, 667, 730
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A376580, A376623, A376630, A376659.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[2*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^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^(2*j-1))^2 * x^j.
a(n) ~ c * A376659^sqrt(n) / sqrt(n), where c = sqrt(1/14 + sinh(arcsinh(75*sqrt(69)/2)/3)/(7*sqrt(69))) = 0.3792934340515155206194952273079851598271882968396...
Showing 1-6 of 6 results.

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