OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Paul S. Bruckman, Problem H-704, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 49, No. 3 (2011), p. 281; A Sum Yielding Pell Numbers, Solution to Problem H-704 by Zbigniew Jakubczyk, ibid., Vol. 51, No. 1 (2013), pp. 92-93.
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).
FORMULA
From R. J. Mathar, Mar 27 2012: (Start)
G.f.: 2*x*(-1+2*x) / ( (x^2+2*x-1)*(x-1)^2 ).
a(n) = 2*A100131(n-1). (End)
From Colin Barker, Nov 06 2017: (Start)
a(n) = (-(1-sqrt(2))^n + (1+sqrt(2))^n) / (2*sqrt(2)) + n.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4) for n>3. (End)
a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-2*k-1, 2*k)*2^(n-4*k) (Bruckman, 2011). - Amiram Eldar, Jan 16 2026
MATHEMATICA
a[n_] := Fibonacci[n, 2] + n; Array[a, 32, 0] (* Amiram Eldar, Jan 16 2026 *)
PROG
(PARI) concat(0, Vec( 2*x*(1 - 2*x) / ((1 - x)^2*(1 - 2*x - x^2)) + O(x^50))) \\ Colin Barker, Nov 06 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 25 2012
STATUS
approved