A084136 Binomial transform of cosh(sqrt(2)*x)^2.
1, 1, 5, 13, 57, 201, 797, 2997, 11569, 44113, 169205, 647197, 2478825, 9488025, 36327821, 139071813, 532438369, 2038379425, 7803827429, 29876310829, 114379413657, 437893003113, 1676441901821, 6418134825429, 24571362963601
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,5,-7).
Programs
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Magma
I:=[1,1,5]; [n le 3 select I[n] else 3*Self(n-1) +5*Self(n-2) - 7*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 11 2018
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Mathematica
LinearRecurrence[{3,5,-7},{1,1,5},30] (* Harvey P. Dale, Nov 08 2017 *) -
PARI
my(x='x+O('x^30)); round(Vec(serlaplace(exp(x)*cosh(sqrt(2)*x)^2))) \\ G. C. Greubel, Sep 11 2018 -
SageMath
A084058=BinaryRecurrenceSequence(2,7,1,1) def A084136(n): return (1+A084058(n))/2 [A084136(n) for n in range(41)] # G. C. Greubel, Oct 13 2022
Formula
a(n) = ((1+sqrt(8))^n + (1-sqrt(8))^n + 2)/4.
a(n) = (A084058(n) + 1)/2.
E.g.f.: exp(x)*cosh(sqrt(2)*x)^2.
G.f.: (1+x)*(1-3*x) / ( (1-x)*(1-2*x-7*x^2) ). - R. J. Mathar, Nov 09 2012