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.

A111262 a(n) = (1/n)*Sum_{k=1..n} F(4*k)*B(2*n-2*k)*binomial(2*n,2*k), where F are Fibonacci numbers and B are Bernoulli numbers.

Original entry on oeis.org

3, 12, 65, 403, 2652, 17889, 121859, 833260, 5706081, 39096531, 267936188, 1836369217, 12586419075, 86267964108, 591287758337, 4052742230419, 27777897084444, 190392509164065, 1304969593244291, 8944394450283436
Offset: 1

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Author

Benoit Cloitre, Nov 12 2005, corrected Feb 24 2008

Keywords

Comments

Values are always integers.

Links

Crossrefs

Cf. A001519.

Programs

  • Mathematica
    Table[(1/n)*Sum[Fibonacci[4k]BernoulliB[2n-2k]Binomial[2n,2k],{k,1,n}],{n,1,20}] (* or *) Table[Fibonacci[4n-2]+2Fibonacci[2n-1],{n,1,20}] (* or *) LinearRecurrence[{10,-23,10,-1},{3,12,65,403},20] (* Indranil Ghosh, Feb 26 2017 *)
  • PARI
    a(n)=fibonacci(4*n-2)+2*fibonacci(2*n-1)

Formula

a(n) = F(4*n-2) + 2*F(2*n-1).
Recurrence: a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4).
O.g.f.: -x*(-3+18*x-14*x^2+x^3)/((x^2-3*x+1)*(x^2-7*x+1)) = -1+(2-4*x)/(x^2-3*x+1)+(-1+8*x)/(x^2-7*x+1). - R. J. Mathar, Nov 23 2007
a(n) = (Lucas(2*n-1)+2)*Fibonacci(2*n-1) = A162483(n-1)*A001519(n). - Ehren Metcalfe, Jun 04 2019

A351635 a(n) is the number of perfect matchings of an edge-labeled 2 X n Klein bottle grid graph, or equivalently the number of domino tilings of a 2 X n Klein bottle grid. (The twist is on the length-n side.)

Original entry on oeis.org

2, 6, 10, 16, 38, 54, 142, 196, 530, 726, 1978, 2704, 7382, 10086, 27550, 37636, 102818, 140454, 383722, 524176, 1432070, 1956246, 5344558, 7300804, 19946162, 27246966, 74440090, 101687056, 277814198, 379501254, 1036816702, 1416317956, 3869452610, 5285770566, 14440993738, 19726764304, 53894522342, 73621286646
Offset: 1

Views

Author

Sarah-Marie Belcastro, Feb 15 2022

Keywords

Comments

Output of Lu and Wu's formula for the number of perfect matchings of an m X n Klein bottle grid specializes to this sequence for m=2 and the twist on the length-n side.

Examples

			a(1) = 2 because this is the number of perfect matchings of a 2 X 1 Klein bottle grid graph (one for each choice of the two non-loop edges).

Links

Crossrefs

Cf. A000129, A005248, A001835, A003499, A068397, A103999, A162484, A162483.

Programs

  • Mathematica
    RecurrenceTable[{a[n] ==
   a[n - 1] + a[n - 2] + Mod[n, 2] a[n - 1] - 4 Mod[n, 2], a[1] == 2,
  a[2] == 6}, a, {n, 1, 50}]

Formula

a(n) = a(n-1) + a(n-2) + (n mod 2)*a(n-1) - 4*(n mod 2).
From Stefano Spezia, Feb 15 2022: (Start)
G.f.: 2*x*(1 + 3*x - 7*x^3 - x^4 + 2*x^5)/(1 - 5*x^2 + 5*x^4 - x^6).
a(n) = 5*a(n-2) - 5*a(n-4) + a(n-6) for n > 6. (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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