|
%I #24 Nov 13 2017 11:43:15
%S 0,12,34,68,124,214,360,596,978,1596,2596,4214,6832,11068,17922,29012,
%T 46956,75990,122968,198980,321970,520972,842964,1363958,2206944,
%U 3570924,5777890,9348836,15126748,24475606,39602376,64078004,103680402,167758428,271438852,439197302
%N Expansion of 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).
%C 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).
%H Todd Silvestri, <a href="/A142245/b142245.txt">Table of n, a(n) for n = 0..999</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F G.f.: 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).
%F a(n) = 10*A094707(2*n) + A094707(2*n+1).
%F a(n) = 2*A022095(n+3) - 22. - _R. J. Mathar_, Jul 07 2011
%F 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
%F 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
%t a[n_Integer/;n>=0]:=23 Fibonacci[n]+11 LucasL[n]-22 (* _Todd Silvestri_, Dec 16 2014 *)
%t LinearRecurrence[{2,0,-1},{0,12,34},40] (* _Harvey P. Dale_, May 12 2015 *)
%o (PARI) concat(0, Vec(2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)) + O(x^50))) \\ _Colin Barker_, Nov 13 2017
%Y Cf. A094707, A022095, A000045, A000032.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Sep 18 2008
|
