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.

A206741 G.f.: 1/(1 - x/(1 - x/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^8/(1 -...- x^Fibonacci(n)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 102, 231, 524, 1189, 2698, 6124, 13900, 31551, 71618, 162566, 369013, 837633, 1901368, 4315978, 9796979, 22238489, 50479892, 114585999, 260102617, 590415686, 1340204451, 3042175244, 6905536091, 15675109089, 35581458383, 80767551510
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 102*x^7 +...

Links

Crossrefs

Cf. A206742, A206743.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Fibonacci[Range[nmax + 1]])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
  • PARI
    {a(n)=local(CF=1+x*O(x^n),M=ceil(log(n+1)/log(1.6))); for(k=0, M, CF=1/(1-x^fibonacci(M-k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,50,print1(a(n),", "))

Formula

a(n) ~ c * d^n, where d = 2.2699337019511296354569330617166782764872939098477919669570757033487700138... and c = 0.3272015736512679060779796519077970622372291004190408455581585307453... - Vaclav Kotesovec, Aug 25 2017

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 9, 11, 15, 23, 28, 43, 57, 78, 113, 149, 214, 293, 403, 569, 774, 1086, 1502, 2072, 2896, 3986, 5548, 7691, 10636, 14797, 20459, 28400, 39386, 54542, 75724, 104886, 145468, 201733, 279545, 387786, 537472, 745233, 1033383, 1432415, 1986394
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2017

Keywords

Examples

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

Links

Crossrefs

Cf. A000040, A206739, A206741, A206742, A206743, A227310, A269353.

Programs

  • Maple
    R:= 1:
for i from numtheory:-pi(50) to 1 by -1 do
  R:= series(1-x^ithprime(i)/R, x, 51);
od:
R:= series(1/R, x, 51):
seq(coeff(R,x,j),j=0..50); # Robert Israel, Apr 20 2017
  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * d^n, where d = 1.3864622092472465020397266918102624708859968795203700659786636158522760956... and c = 0.15945087310540003725148530084775272562567007586487061850065597143186... - Vaclav Kotesovec, Aug 25 2017

A307083 Expansion of 1/(1 - x/(1 - 3*x/(1 - 4*x/(1 - 7*x/(1 - 11*x/(1 - 18*x/(1 - ... - Lucas(k)*x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 4, 28, 292, 4408, 97432, 3231256, 164789104, 13170099856, 1670220282544, 338692348412320, 110327835695333920, 57892877044109184160, 49019180876700301391680, 67044425508546158335526080, 148216012413625321252632612160, 529829556146109541834263919553920
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

Links

Crossrefs

Cf. A000204, A001622, A206742, A307082.

Programs

  • Maple
    L:= proc(n) option remember; (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1] end:
b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,
     `if`(x>0, b(x-1, y)*L(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 12 2023
  • Mathematica
    nmax = 17; CoefficientList[Series[1/(1 + ContinuedFractionK[-LucasL[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * phi^(n*(n+1)/2), where phi = A001622 is the golden ratio and c = 5.62026823201787715079864730026619553810473701484813959397175006212578036... - Vaclav Kotesovec, Sep 18 2021

A294922 Expansion of 1/(1 + x/(1 + x^3/(1 + x^4/(1 + x^7/(1 + x^11/(1 + ... + x^Lucas(k)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -6, 8, -11, 16, -22, 31, -44, 61, -85, 119, -166, 232, -325, 454, -634, 886, -1237, 1728, -2415, 3373, -4712, 6583, -9194, 12843, -17941, 25060, -35006, 48899, -68303, 95409, -133272, 186159, -260036, 363230, -507373, 708720, -989969, 1382827, -1931590
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

Crossrefs

Cf. A000204, A007325, A206741, A206742, A206743, A279586.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[1/(1 + ContinuedFractionK[x^LucasL[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 + x/(1 + x^3/(1 + x^4/(1 + x^7/(1 + x^11/(1 + ... + x^A000204(k)/(1 + ...))))))), a continued fraction.
Showing 1-4 of 4 results.

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