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 1-2 of 2 results.

A291942 Row sums of A291940.

Original entry on oeis.org

1, 1, -2, 5, -9, 14, -24, 47, -87, 151, -272, 505, -918, 1656, -3020, 5512, -10020, 18221, -33172, 60388, -109899, 199992, -364002, 662529, -1205777, 2194488, -3994057, 7269275, -13230124, 24078998, -43824226, 79760817, -145165804, 264203948, -480855437, 875164374
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2017

Keywords

Links

Crossrefs

Cf. A291896, A291905, A291930.

A275762 G.f.: 2 - x/(1+2*x - x^3/(1+2*x^2 - x^5/(1+2*x^3 - x^7/(1+2*x^4 - x^9/(1+2*x^5 - x^11/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

2, -1, 2, -4, 7, -12, 22, -41, 74, -133, 243, -444, 806, -1465, 2669, -4859, 8840, -16087, 29282, -53296, 96994, -176527, 321290, -584755, 1064251, -1936952, 3525296, -6416092, 11677369, -21252993, 38680798, -70399646, 128128414, -233195704, 424419826, -772450633, 1405872057, -2558708924, 4656889892, -8475611623, 15425744240, -28075093283, 51097104306, -92997520459, 169256926243, -308050225082, 560656176744, -1020402917484, 1857149100126, -3380040101304, 6151725289638
Offset: 0

Views

Author

Paul D. Hanna, Aug 10 2016

Keywords

Comments

a(n) ~ c/r^n, where r = -0.54944587773859960333406076695895194626366374257497442830... and c = 0.6098779103867259353642411483841966048261178594794555738...
The g.f. of related triangle A275760 satisfies: G(x,y) = x*y + 1/G(x,x*y) with G(0,y) = 1.

Examples

			G.f.: A(x) = 2 - x + 2*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 22*x^6 - 41*x^7 + 74*x^8 - 133*x^9 + 243*x^10 - 444*x^11 + 806*x^12 - 1465*x^13 + 2669*x^14 - 4859*x^15 +...

Links

Crossrefs

Cf. A275760, A275761, A006958, A227309, A291940 (column 1).

Programs

  • Mathematica
    m = 51;
2 + ContinuedFractionK[-x^(2i-1), 1+2x^i, {i, 1, Sqrt[m]//Floor}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)
  • PARI
    {a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = 1/(A + y*x^(n+1-k))); polcoeff(1 + subst(A,y,1), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

Equals the diagonal sums of the irregular triangle A275760.
G.f.: 1/F(x) + 1, where F(x) is the g.f. of A275761, the row sums of triangle A275760.
G.f.: G(x,1/x), where G(x,y) = x*y + 1/G(x,x*y) with G(0,y) = 1, where G(x,y) is the g.f. of A275760.
G.f.: 2 - x/(1+x + x/(1+x^2 - x^4/(1+x^3 + x^2/(1+x^4 - x^7/(1+x^5 + x^3/(1+x^6 - x^10/(1+x^7 + x^4/(1+x^8 - x^13/(1+x^9 + x^5/(1+x^10 - x^16/(1 + ...))))))))))), a continued fraction.
G.f.: 1/(1 - 1/(1 + (1+x) - x^2/(1 + x*(1+x) - x^4/(1 + x^2*(1+x) - x^6/(1 + x^3*(1+x) - x^8/(1 + x^4*(1+x) - x^10/(1 + x^5*(1+x) - x^12/(1 - ...)))))))), a continued fraction.
G.f.: 1/(1 - 1/(1+x + 1/(1+x^2 - x^3/(1+x^3 + x/(1+x^4 - x^6/(1+x^5 + x^2/(1+x^6 - x^9/(1+x^7 + x^3/(1+x^8 - x^12/(1+x^9 + x^4/(1+x^10 - x^15/(1 + ...)))))))))))), a continued fraction.
G.f.: 1 + 1/(1 + x/(1 + x/(1 + x^2/(1 + x^2/(1 + x^3/(1 + x^3/(1 + ...))))))) since the odd part of this continued fraction equals the defining continued fraction given above. Cf. A006958 and A227309. - Peter Bala, Oct 29 2017
Showing 1-2 of 2 results.

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