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 1-4 of 4 results.

A347901 Decimal expansion of a constant related to the asymptotics of A005169.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 18 2021

Keywords

Examples

			0.576148769142756602297868573719938782354724663118974468685156534319...

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Links

Crossrefs

Cf. A005169, A168445, A226999, A285636, A285903, A285637, A305840.

Programs

  • Mathematica
    FindRoot[Sum[(-1)^k*r^(k^2)/QPochhammer[r, r, k], {k, 0, 1000}] == 0, {r, 1/2}, WorkingPrecision -> 120]

Formula

Lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0.

A285637 G.f.: 1/( (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) * (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) ), a continued fraction.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 6, 10, 19, 30, 55, 92, 161, 282, 483, 846, 1462, 2538, 4409, 7642, 13276, 23032, 39977, 69394, 120426, 209036, 362800, 629698, 1092952, 1896968, 3292522, 5714678, 9918752, 17215620, 29880461, 51862438, 90015657, 156236814, 271174435, 470667300, 816919764, 1417897172, 2460991365
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2017

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 + 4*x^5 + 6*x^6 + 10*x^7 + 19*x^8 + 30*x^9 + 55*x^10 + ...

Links

Crossrefs

Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285638.

Programs

  • Mathematica
    nmax = 42; CoefficientList[Series[(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: A(x) = R(x)*P(x)/Q(x), where R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))), P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).
a(n) ~ c * d^n, where d = 1/A347901 = 1.7356628245303474256582607497196685302546528472903927546099... and c = 0.215558365582078354136603033062960103377669... - Vaclav Kotesovec, Aug 26 2017

A285635 G.f.: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) / (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction.

Original entry on oeis.org

1, -2, 2, -2, 0, 0, -2, 2, -4, 0, -2, -6, 0, -10, -6, -12, -20, -20, -40, -46, -68, -104, -132, -204, -280, -394, -578, -790, -1154, -1616, -2294, -3286, -4614, -6610, -9340, -13278, -18878, -26748, -38060, -53978, -76684, -108912, -154600, -219622, -311812, -442818, -628866, -892962, -1268168
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2017

Keywords

Examples

			G.f.: A(x) = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^6 + 2*x^7 - 4*x^8 - 2*x^10 - 6*x^11 - ...

Links

Crossrefs

Cf. A003823, A005169, A007325, A055101, A285636, A285637, A285638.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k^2)/Product[(1 -
x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: A(x) = R(x)*Q(x)/P(x), where R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))), Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m) and P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m).

A285638 G.f.: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) * (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))), a continued fraction.

Original entry on oeis.org

1, 0, -1, -2, 0, 0, -1, -4, -4, -2, -3, -6, -13, -16, -19, -24, -38, -60, -82, -110, -150, -224, -324, -458, -637, -898, -1289, -1838, -2609, -3680, -5223, -7430, -10571, -15004, -21272, -30202, -42903, -60960, -86543, -122860, -174450, -247762, -351883, -499668, -709521, -1007532
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2017

Keywords

Examples

			G.f.: A(x) = 1 - x^2 - 2*x^3 - x^6 - 4*x^7 - 4*x^8 - 2*x^9 - 3*x^10 - 6*x^11 - 13*x^12 - ...

Links

Crossrefs

Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285637.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[1/((1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0,
nmax}] Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: A(x) = Q(x)/(R(x)*P(x)), where Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m), R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))) and P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m).
Showing 1-4 of 4 results.

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