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A091914 a(n) = 2*a(n-1) + 12*a(n-2). 18
1, 2, 16, 56, 304, 1280, 6208, 27776, 130048, 593408, 2747392, 12615680, 58200064, 267788288, 1233977344, 5681414144, 26170556416, 120518082560, 555082842112, 2556382674944, 11773759455232, 54224111009792, 249733335482368 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of 1,1,13,13,169,169,....

The inverse binomial transform of 2^n*c(n), where c(n) is the solution to c(n)=c(n-1)+kc(n-2), a(0)=1,a(1)=1 is 1,1,4k+1,4k+1,(4k+1)^2,...

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,12).

FORMULA

a(n) = A000079(n)*A006130(n).

G.f.: 1/(1-2*x-12*x^2).

a(n) = ((1+sqrt(13))*(1+sqrt(13))^n - (1-sqrt(13))*(1-sqrt(13))^n) /(2*sqrt(13)).

a(n) = Sum_{k=0..floor(n/2)} C(n+1,2*k+1) * 13^k. - Paul Barry, Jan 15 2007

MAPLE

a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then 2*procname(n-1) + 12*procname(n-2) fi; end: # Muniru A Asiru, Jan 31 2018

MATHEMATICA

Join[{a=1, b=2}, Table[c=2*b+12*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)

LinearRecurrence[{2, 12}, {1, 2}, 30] (* or *) With[{s=Sqrt[13]}, Table[ Simplify[ -(((13+s)((1-s)^n-(1+s)^n))/(26(1+s)))], {n, 30}]] (* Harvey P. Dale, May 25 2013 *)

PROG

(Sage) [lucas_number1(n, 2, -12) for n in xrange(1, 24)] # Zerinvary Lajos, Apr 22 2009

(PARI) x='x+O('x^30); Vec(1/(1-2*x-12*x^2)) \\ G. C. Greubel, Jan 30 2018

(MAGMA) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1-2*x-12*x^2))) // G. C. Greubel, Jan 30 2018

(GAP) a := [1, 2];; for n in [3..350] do a[n] := 2*a[n-1] + 12*a[n-2]; od; a; # Muniru A Asiru, Jan 31 2018

CROSSREFS

Cf. A003683, A063727.

Sequence in context: A225051 A033431 A107610 * A123791 A293620 A206980

Adjacent sequences:  A091911 A091912 A091913 * A091915 A091916 A091917

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 12 2004

STATUS

approved

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Last modified January 15 18:45 EST 2019. Contains 319165 sequences. (Running on oeis4.)