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.

A175712 The third column of the Lucas Fibonacci sum of binomials A175685.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 33, 54, 79, 133, 176, 309, 365, 674, 709, 1383, 1300, 2683, 2267, 4950, 3785, 8735, 6085, 14820, 9465, 24285, 14302, 38587
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Crossrefs

Cf. A000045, A175685, A175686.

Programs

  • Mathematica
    Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 3, Floor[(
    n - 1)/2]}], {n, 0, 30}]

Formula

a(n) = A175685(n,3), n>=1 .

A175686 a(n) = binomial(n-j-1,j) + binomial(n-j,j-1) with j= floor((n-1)/2).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 7, 7, 14, 11, 25, 16, 41, 22, 63, 29, 92, 37, 129, 46, 175, 56, 231, 67, 298, 79, 377, 92, 469, 106, 575, 121, 696, 137, 833, 154, 987, 172, 1159, 191, 1350, 211, 1561, 232, 1793, 254, 2047, 277, 2324, 301, 2625, 326, 2951, 352, 3303, 379
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

The column m=1 in the array A175685, where the sum over the binomials reduces to only two terms.

Links

Crossrefs

Cf. A175685, A057979, A152271.

Programs

  • Mathematica
    Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 1, Floor[(
    n - 1)/2]}], {n, 0, 30}]
CoefficientList[Series[-(x^3-x^2-x)(x^4-x^2+1)/(x^2-1)^4, {x, 0, 30}],x] (* Benedict W. J. Irwin, Oct 31 2016 *)
Table[(42+20n+6n^2+n^3+(-1)^n(-42+20n-6n^2+n^3))/96, {n, 0, 30}] (* Benedict W. J. Irwin, Oct 31 2016 *)
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,1,1,2,3,4,7,7},60] (* Harvey P. Dale, Jul 29 2018 *)
  • PARI
    concat(0, Vec(x*(1+x-x^2)*(1-x^2+x^4)/((1-x)^4*(1+x)^4) + O(x^100))) \\ Colin Barker, Oct 31 2016

Formula

a(n) = A057979(n+1) + binomial(n-j,j-1) with j = A004526(n-1), n>0.
From Benedict W. J. Irwin, Oct 31 2016: (Start)
G.f.: -(x^3 - x^2 - x)*(x^4 - x^2 + 1)/(x^2 - 1)^4.
E.g.f.: ((6*x + 3*x^2)*cosh(x) + (42 + 21*x + 6*x^2 + x^3)*sinh(x))/48.
a(n) = (42 + 20*n + 6*n^2 + n^3 + (-1)^n*(-42 + 20*n - 6*n^2 + n^3))/96. (End)
a(n) = 4*a(n-2)-6*a(n-4)+4*a(n-6)-a(n-8) for n>7. - Colin Barker, Oct 31 2016

Extensions

More terms from Colin Barker, Oct 31 2016
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