%I #17 Oct 13 2022 17:32:52

%S 1,1,5,13,57,201,797,2997,11569,44113,169205,647197,2478825,9488025,

%T 36327821,139071813,532438369,2038379425,7803827429,29876310829,

%U 114379413657,437893003113,1676441901821,6418134825429,24571362963601

%N Binomial transform of cosh(sqrt(2)*x)^2.

%H G. C. Greubel, <a href="/A084136/b084136.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,-7).

%F a(n) = ((1+sqrt(8))^n + (1-sqrt(8))^n + 2)/4.

%F a(n) = (A084058(n) + 1)/2.

%F E.g.f.: exp(x)*cosh(sqrt(2)*x)^2.

%F G.f.: (1+x)*(1-3*x) / ( (1-x)*(1-2*x-7*x^2) ). - _R. J. Mathar_, Nov 09 2012

%t LinearRecurrence[{3,5,-7},{1,1,5},30] (* _Harvey P. Dale_, Nov 08 2017 *)

%o (PARI) my(x='x+O('x^30)); round(Vec(serlaplace(exp(x)*cosh(sqrt(2)*x)^2))) \\ _G. C. Greubel_, Sep 11 2018

%o (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

%o (SageMath)

%o A084058=BinaryRecurrenceSequence(2,7,1,1)

%o def A084136(n): return (1+A084058(n))/2

%o [A084136(n) for n in range(41)] # _G. C. Greubel_, Oct 13 2022

%Y Cf. A084058, A084137.

%K easy,nonn

%O 0,3

%A _Paul Barry_, May 16 2003