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 1-5 of 5 results.

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 28 2015

Keywords

Links

Crossrefs

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A266686 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 4, 6, 8, 9, 10, 11, 14, 16, 18, 21, 25, 28, 31, 36, 41, 48, 52, 59, 69, 77, 85, 96, 109, 121, 133, 151, 172, 189, 208, 231, 260, 287, 316, 350, 390, 432, 471, 521, 578, 636, 695, 764, 842, 924, 1009, 1107, 1218, 1330, 1449, 1584
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A264905, A266647, A266650, A275820.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[1+x^k-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A275821 Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)).

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 1, -1, 1, -1, 3, -2, 3, -3, 2, -1, 4, -3, 4, -4, 7, -7, 7, -7, 9, -6, 11, -10, 10, -11, 15, -14, 18, -19, 21, -17, 24, -22, 26, -29, 35, -34, 42, -43, 43, -39, 52, -52, 59, -59, 74, -72, 79, -87, 93, -87, 107, -108, 118, -126, 149, -146
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 15 2016

Keywords

Links

Crossrefs

Cf. A000726, A263401, A264905, A266686, A275820.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Product[1+x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
RootReduce[QPochhammer[Root[-1 + # + #^3 &, 1], x] QPochhammer[Root[-1 + # + #^3 &, 2], x] QPochhammer[Root[-1 + # + #^3 &, 3], x] + O[x]^70][[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = -1; Do[Do[p[[j + 1]] = p[[j + 1]] + If[j < 2 k, 0, p[[j - 2 k + 1]]] - If[j < 3 k, 0, p[[j - 3 k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 06 2018 *)

Formula

a(n) ~ (-1)^n * c^(1/4) * exp(sqrt(c*n)) / (2^(3/2)*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + 2*exp(-x) + exp(-2*x) - exp(-3*x)) dx = 1.522848148277623680909526566...

A276519 Expansion of Product_{k>=1} 1/(1 - x^(2*k) - x^(3*k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 17, 19, 34, 37, 61, 75, 112, 138, 209, 256, 376, 478, 675, 866, 1222, 1566, 2175, 2830, 3873, 5055, 6900, 9011, 12213, 16045, 21599, 28429, 38191, 50290, 67341, 88884, 118669, 156751, 209018, 276200, 367734, 486376, 646688
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 15 2016

Keywords

Links

Crossrefs

Cf. A264905, A266686, A275820, A275821, A276526.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[1/Product[1-x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * p / r^n, where r = A075778 = 1/A060006 = 0.7548776662466927600495... is the real root of the equation r^3 + r^2 - 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) - r^(3*n)) = 3.820450591662541853... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.

A276526 Expansion of Product_{k>=1} 1/(1 - x^(2*k) + x^(3*k)).

Original entry on oeis.org

1, 0, 1, -1, 2, -2, 3, -4, 7, -8, 11, -15, 22, -27, 37, -51, 70, -90, 121, -162, 220, -288, 381, -512, 688, -902, 1197, -1598, 2127, -2809, 3722, -4949, 6581, -8699, 11519, -15301, 20305, -26862, 35581, -47208, 62591, -82859, 109756, -145506, 192856, -255388
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 16 2016

Keywords

Links

Crossrefs

Cf. A264905, A266686, A275820, A275821, A276519.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[1-x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * p / r^n, where r = -A075778 = -0.7548776662466927600495... is the real root of the equation r^3 - r^2 + 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) + r^(3*n)) = 1.9844809074648434... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.
Showing 1-5 of 5 results.

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