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.

A285636 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, 4, 8, 14, 22, 36, 64, 114, 198, 340, 586, 1018, 1772, 3076, 5332, 9248, 16054, 27872, 48376, 83952, 145700, 252888, 438938, 761846, 1322286, 2295022, 3983384, 6913822, 12000054, 20828006, 36150354, 62744812, 108903838, 189020310, 328075444, 569428264, 988335418, 1715417004
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2017

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 14*x^6 + 22*x^7 + 36*x^8 + 64*x^9 + ...

References

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

Links

Crossrefs

Cf. A003823, A005169, A007325, A055101, A285635, A285637, A285638.

Programs

  • Maple
    A10:= [1, seq([x^i,1],i=1..10)]: B10:= [1, seq([-x^i,1],i=1..10)]:
S:= series(numtheory:-nthconver(A10,10)/numtheory:-nthconver(B10,10),x,51):
A:= [seq(coeff(S,x,i),i=0..50)]; # Robert Israel, Dec 15 2024
  • Mathematica
    nmax = 40; 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 = 40; CoefficientList[Series[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}]  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) = P(x)/(R(x)*Q(x)), where P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / 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 Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).
a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.452642356466453742995961374156022446123012... - Vaclav Kotesovec, Aug 26 2017, updated Sep 24 2020

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).
Showing 1-3 of 3 results.

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