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 1-8 of 8 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

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			2.045390691852050639893704244342601252265948793467833187994662870934455617...

Crossrefs

Cf. A333198, A376621, A376630, A376631, A376658, A376659, A376815.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2/4 + 2*PolyLog[2, r^(1/2)] - Pi^2/6] /. 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} A376630(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376631(n)^(1/sqrt(n)).
Equals A376815^(1/2). - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2/4 + 2*polylog(2, r^(1/2)) - Pi^2/6)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

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)).

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 05 2024

Keywords

Examples

			4.18362308231501037592434207471436289895638697707035887857832710...

Crossrefs

Cf. A333198, A376621, A376658, A376659, A376660, A376812.
Cf. A088559.

Programs

  • Mathematica
    RealDigits[E^Sqrt[3*Log[r]^2 + 8*PolyLog[2, r^(1/2)] - 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} A376812(n)^(1/sqrt(n)).
Equals A376660^2. - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2 + 8*polylog(2, r^(1/2)) - 2*Pi^2/3)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A376624 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, 4, 6, 8, 13, 18, 23, 33, 44, 57, 77, 99, 125, 163, 207, 259, 328, 407, 503, 626, 769, 938, 1149, 1397, 1687, 2044, 2458, 2943, 3531, 4213, 5011, 5957, 7055, 8334, 9838, 11580, 13594, 15948, 18661, 21790, 25425, 29593, 34386, 39918, 46250, 53501, 61824, 71325
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A053282, A072223, A376542, A376622, A376581, A376658.

Programs

  • Mathematica
    nmax=60; 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]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j-1))^2.
a(n) ~ (r^(3/4)/sqrt(8*(1 + 3*r^2))) * A376658^sqrt(n) / sqrt(n), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

A377075 G.f.: Sum_{k>=0} x^(8*k^2) / Product_{j=1..8*k-1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 734, 861, 1011, 1178, 1372, 1586, 1835, 2108, 2422, 2768, 3162, 3595, 4088, 4627, 5237, 5907, 6660, 7485, 8414, 9429, 10568, 11817, 13213
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 15 2024

Keywords

Comments

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m.

Links

Crossrefs

Column 8 of A350889.
Cf. A376658.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(8*k^2)/Product[1-x^j, {j, 1, 8*k-1}], {k, 1, Sqrt[nmax/8]}], {x, 0, nmax}], x]

Formula

Limit_{n->oo} a(n)^(1/sqrt(n)) = A376658.
a(n) ~ r^2 * (8*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((8*log(r)^2 + polylog(2, r^2))*n)) / (8*sqrt(Pi*(4 - 3*r^2)) * n^(3/4)), where r = 0.8511709340670154789... is the positive real root of the equation r^2 = 1 - r^8.

A356032 Decimal expansion of the positive real root of x^4 + x - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Aug 27 2022

Keywords

Comments

The other real (negative) root is -A060007.
One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

Examples

			r = 0.724491959000515611588372282187036565786494481350011017270...

Crossrefs

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

Programs

  • Mathematica
    First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)
  • PARI
    solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022
  • PARI
    polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

Formula

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).
Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025

A376625 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 3, 0, 5, 1, 9, 2, 13, 6, 20, 12, 27, 23, 39, 40, 51, 69, 70, 108, 92, 169, 125, 252, 166, 370, 227, 527, 307, 743, 425, 1021, 586, 1393, 816, 1867, 1132, 2481, 1577, 3256, 2184, 4247, 3019, 5479, 4149, 7036, 5670, 8966, 7698, 11377, 10386, 14356, 13915, 18060
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000700, A072223, A179049, A340647, A376542, A376658.

Programs

  • Mathematica
    nmax=80; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j))^2.
a(n) ~ (r^(1/4) * sqrt(log(r)^2 + 2*polylog(2, sqrt(r))) / (2*Pi*sqrt(1 + 3*r^2))) * A376658^sqrt(n) / n, where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.
Showing 1-8 of 8 results.

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