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

Author

Barry E. Williams, Mar 27 2000

Keywords

• easy
• nonn

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).

Crossrefs

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

Right-hand column 7 of triangle A011794.

Programs

GAP

List([0..40], 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

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 *)

PARI

vector(40, n, n--; fibonacci(n+8) - (n^2 +8*n+20)) \\ 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

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