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.

A291148 Expansion of 1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - x^6/(1 - ... - x^n/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 0, -1, 0, -1, -1, -1, -2, -2, -4, -4, -7, -9, -13, -19, -25, -38, -51, -75, -104, -149, -211, -298, -426, -600, -857, -1211, -1724, -2444, -3471, -4930, -6995, -9940, -14104, -20038, -28444, -40397, -57362, -81453, -115675, -164250, -233262, -331227
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2017

Keywords

Comments

The sequence b(n>=1) = 1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, ... of absolute values counts fountains of n coins that cannot be separated into two or more fountains by cutting vertically through the fountain without splitting a coin. (This separation requires that the fountain is a left-right sequence of more elementary fountains counted by b(n).) A005169(n) = Sum_{compositions n=n1+n2+n3+...} Product b(n1)*b(n2)*.... - R. J. Mathar, Aug 22 2018

Examples

			G.f. = 1 - x - x^3 - x^5 - x^6 - x^7 - 2*x^8 - 2*x^9 - ...

Links

Crossrefs

Cf. A005169, A290975, A291147.

Formula

Convolution inverse of A005169.
a(n) ~ c * d^n, where d = 1.42009048763893649946106129818306075366296460727614... and c = -0.093433697175825717154301151812109730023054876584907211486145769... - Vaclav Kotesovec, Oct 16 2017

A206739 G.f.: 1/(1 - x/(1 - x^4/(1 - x^9/(1 - x^16/(1 - x^25/(1 - x^36/(1 -...- x^(n^2)/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 37, 52, 72, 99, 138, 193, 269, 373, 518, 722, 1006, 1399, 1944, 2705, 3766, 5241, 7290, 10141, 14112, 19638, 27323, 38012, 52889, 73593, 102398, 142470, 198225, 275809, 383760, 533954, 742923, 1033685, 1438254
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Examples

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

Crossrefs

Cf. A206740, A290975.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1]^2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)
  • Maxima
    T(n, m):=if n=m then 1 else  sum(binomial(m, i)*T((n-m)/3, i), i, 1, (n-m)/3);
makelist(sum(T(n,k),k,0,n),n,0,20); /* Vladimir Kruchinin, Mar 21 2015 */
  • PARI
    {a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^((M-k+1)^2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))
  • PARI
    N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^((k+1)^2) / G(k+1) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013

Formula

a(n) = sum(k=0..n, T(n,k)), where T(n, m)=sum(i=1..(n-m)/3, binomial(m, i)*T((n-m)/3,i)), T(n,n)=1. - Vladimir Kruchinin, Mar 21 2015
G.f.: A(x)=1/B(x), where B(x) is g.f. of A290975. - Seiichi Manyama, Aug 18 2017
a(n) ~ c * d^n, where d = 1.391377080590304271048017099353... and c = 0.3625537262803710555422183139... - Vaclav Kotesovec, Aug 24 2017

A291147 Expansion of 1 - x/(1 - x^8/(1 - x^27/(1 - x^64/(1 - x^125/(1 - x^216/(1 - ... - x^(n^3)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, -3, 0, 0, 0, 0, -1, 0, 0, -4, 0, 0, -1, 0, -1, 0, 0, -5, 0, 0, -3, 0, -1, 0, 0, -6, 0, 0, -6, 0, -1
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2017

Keywords

Crossrefs

Cf. A290975, A291148.

Formula

Convolution inverse of A291146.
Showing 1-3 of 3 results.

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