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A089926 a(n)=12a(n-1)+a(n-2), a(0)=1,a(1)=6. 1
1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The family of recurrences a(n)=2ka(n-1)+a(n-2), a(0)=1,a(1)=k has solution a(n)=((k+sqrt(k^2+1))^n+(k-sqrt(k^2+1))^n)/2; a(n)=sum{j=0..floor(n/2), C(n,2k)(k^2+1)^jk^(n-2j)}; a(n)=T(n,ki)(-i)^n; e.g.f. exp(kx)cosh(sqrt(k^2+1)x).

LINKS

Table of n, a(n) for n=0..17.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,1).

FORMULA

E.g.f. : exp(6x)cosh(sqrt(37)x); a(n)=((6+sqrt(37))^n+(6-sqrt(37))^n)/2; a(n)=sum{k=0..floor(n/2), C(n, 2k)37^k6^(n-2k)}. a(n)=T(n, 6i)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1.

G.f.: (1-6x)/(1-12*x-x^2). [From Philippe Deléham, Nov 21 2008]

CROSSREFS

Cf. A088320, A088317, A005667, A001077.

Sequence in context: A179568 A202557 A041060 * A135594 A168603 A244689

Adjacent sequences:  A089923 A089924 A089925 * A089927 A089928 A089929

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 15 2003

STATUS

approved

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Last modified August 23 01:14 EDT 2017. Contains 290953 sequences.