(MAGMAMagma) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +11*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

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From _Peter Bala, _, Jan 07 2022: (Start)

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From Peter Bala, Jan 07 2022: (Start)

a(n) = [x^n] (x + sqrt(1 + 12*x^2))^n.

The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.

O.g.f. A(x) = 1 + x*d/dx(log(B(x)), where B(x) = 1/sqrt(1 - 2*x - 11*x^2) is the o.g.f. of A084603. (End)

Cf. A049310, A015520, A084603.

easy,nonn

nonn,easy

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1, 1, 13, 37, 217, 841, 4069, 17389, 79537, 350353, 1575613, 7005109, 31341961, 139740121, 624241813, 2785624957, 12437909857, 55517694241, 247852396909, 1106399430469, 4939175226937, 22048744189033, 98428415874373, 439393017828109, 1961498610274321, 8756320416657841

<a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,11).

<a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,11).

E.g.f.: exp(x)*cosh(2sqrt2*sqrt(3)*x);.

a(n) = ((1 + 2sqrt2*sqrt(3))^n/2 + (1 - 2sqrt2*sqrt(3))^n)/2.

a[n_] := Simplify[((1 + Sqrt[12])^n + (1 - Sqrt[12])^n)/2]; Array[a, 25, 30, 0] (* or *)

CoefficientList[Series[(x - 1)/(11 x11x^2 + 2 x 2x- 1), {x, 0, 2330}], x] (* or *)

Table[ MatrixPower[{{1, 2}, {6, 1}}, n][[1, 1]], {n, 0, 2530}] (* Robert G. Wilson v, Sep 18 2013 and modified per Wolfdieter Lang Feb 17 2018 *)

LinearRecurrence[{2, 11}, {1, 1}, 2530] (* Ray Chandler, Aug 01 2015 *)

(PARI) x='x+O('x^9930); Vec((1-x)/(1-2*x-11*x^2)) \\ Altug Alkan, Feb 17 2018

(GAP) a := [1, 1];; for n in [3..2530] do a[n] := 2*a[n-1]+ 11*a[n-2]; od; a; # Muniru A Asiru, Feb 18 2018

(MAGMA) I:=[1, 1]; [n le 2 select I[n] else 2*Self(n-1) +11*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

(Sage) ((1-x)/(1-2*x-11*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

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