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
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Mar 27 2000
STATUS
approved