A142245 Expansion of 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).

0, 12, 34, 68, 124, 214, 360, 596, 978, 1596, 2596, 4214, 6832, 11068, 17922, 29012, 46956, 75990, 122968, 198980, 321970, 520972, 842964, 1363958, 2206944, 3570924, 5777890, 9348836, 15126748, 24475606, 39602376, 64078004, 103680402, 167758428, 271438852, 439197302

Offset: 0

Author

Paul Curtz, Sep 18 2008

Keywords

• nonn
• easy

Comments

The generic a(n) = 2*a(n-1)-a(n-3) for this family of recurrences (see the link to the OEIS index) leads directly to a common symmetry of the form a(n+1)-2a(n) = 12, 10, 0, -12, -34, -68, -124,... = 12, 10, -a(n).

Links

• Todd Silvestri, Table of n, a(n) for n = 0..999
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A094707, A022095, A000045, A000032.

Programs

Mathematica

a[n_Integer/;n>=0]:=23 Fibonacci[n]+11 LucasL[n]-22 (* Todd Silvestri, Dec 16 2014 *)
LinearRecurrence[{2,0,-1},{0,12,34},40] (* Harvey P. Dale, May 12 2015 *)

PARI

concat(0, Vec(2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)) + O(x^50))) \\ Colin Barker, Nov 13 2017

Formula

G.f.: 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).

a(n) = 10*A094707(2*n) + A094707(2*n+1).

a(n) = 2*A022095(n+3) - 22. - R. J. Mathar, Jul 07 2011

a(n) = 23*F(n)+11*L(n)-22 = 23*A000045(n)+11*A000032(n)-22, where F(n) and L(n) are the n-th Fibonacci and Lucas numbers, respectively. - Todd Silvestri, Dec 16 2014

a(n) = (1/5)*(-110 + (55-23*sqrt(5))*((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n*(55+23*sqrt(5))). - Colin Barker, Nov 13 2017