A053808 Partial sums of A001891.

1, 5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, 24156724, 39087009, 63244757, 102332855, 165578768, 267912848

Offset

0,2

Comments

Antidiagonal sums of the convolution array A213579 and row 1 of the convolution array A213590. - Clark Kimberling, Jun 18 2012

Also number CG(n,2) of complete games with n players of 2 types. - N. J. A. Sloane, Dec 29 2012

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, 2012. - From N. J. A. Sloane, Dec 29 2012

W. Lang, Problem B-858, Fibonacci Quarterly, 36,3 (1998) 373-374; Solution, ibid. 37,2 (1999) 183-184.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

a(n) = a(n-1) + a(n-2) + (n+1)^2, a(-n)=0.

G.f.: (1+x)/((1-x-x^2)*(1-x)^3).

a(n) = Fibonacci(n+6) - (n^2 + 4*n + 8), n >= 2 (see p. 184 of FQ reference).

a(n-2) = Sum_{i=0..n} Fibonacci(i)*(n-i)^2. - Benoit Cloitre, Mar 06 2004

Mathematica

Table[Fibonacci[n+8] -(n^2 +8*n+20), {n, 0, 40}] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{4, -5, 1, 2, -1}, {1, 5, 15, 36, 76}, 40] (* Harvey P. Dale, Apr 14 2022 *)

Prog

(PARI) vector(40, n, n--; fibonacci(n+8) - (n^2 +8*n+20)) \\ G. C. Greubel, Jul 06 2019
(Magma) [Fibonacci(n+8) - (n^2+8*n+20): n in [0..40]]; // G. C. Greubel, Jul 06 2019
(Sage) [fibonacci(n+8) - (n^2 +8*n+20) for n in (0..20)] # G. C. Greubel, Jul 06 2019
(GAP) List([0..40], n-> Fibonacci(n+8) - (n^2 +8*n+20)) # G. C. Greubel, Jul 06 2019

Crossrefs

Convolution of A000290 (squares) with A000045, n >= 1. (Fibonacci) - Wolfdieter Lang, Apr 10 2000

Right-hand column 7 of triangle A011794.

Sequence in context: A360486 A144898 A163250 * A111926 A369804 A137609

Adjacent sequences: A053805 A053806 A053807 * A053809 A053810 A053811

Keyword

easy,nonn

Author

Barry E. Williams, Mar 27 2000

Status

approved