A100296 Sequence generated from a symmetric matrix.

1, 6, 25, 111, 488, 2149, 9461, 41654, 183389, 807403, 3554736, 15650361, 68903513, 303360038, 1335596817, 5880203831, 25888648920, 113979406525, 501814720109, 2209329044566, 9726966211957, 42824708216851, 188543436246752, 830096195208753, 3654646945111665

Offset

1,2

Comments

A100295 is generated from M^n * [1, 0, 0].

Limit_{n -> oo} a(n)/a(n-1) tends to 4.4026788295...a root of the characteristic polynomial of M, x^3 - 4*x^2 - 2*x + 1 and an eigenvalue of M.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,2,-1).

Formula

a(n) = rightmost term in M^n * [1,1,1], where M = [3,2,1; 2,1,0; 1,0,0].

a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3).

G.f.: x*(1+2*x-x^2)/(1-4*x-2*x^2+x^3). - Colin Barker, May 25 2013

Example

a(5) = 488 since M^5 * [1, 1, 1] = [2149, 1263, 488]. 2149 = a(6).
a(8) = 4*a(7) + 2*a(6) - a(5) =41654 = 4*9461 + 2*2149 - 488.

Maple

a:= n-> (<<3|2|1>, <2|1|0>, <1|0|0>>^n. <<1, 1, 1>>)[3, 1]:
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 06 2023

Mathematica

LinearRecurrence[{4, 2, -1}, {1, 6, 25}, 40] (* G. C. Greubel, Feb 05 2023 *)

Prog

(Magma) I:=[1, 6, 25]; [n le 3 select I[n] else 4*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 05 2023
(SageMath)
@CachedFunction
def a(n): # a = A100296
    if (n<3): return (1, 1, 6)[n]
    else: return 4*a(n-1) + 2*a(n-2) - a(n-3)
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 05 2023

Crossrefs

Cf. A100295, A100297.

Sequence in context: A295202 A346894 A094669 * A346818 A120758 A227914

Adjacent sequences: A100293 A100294 A100295 * A100297 A100298 A100299

Keyword

nonn,easy

Author

Gary W. Adamson, Nov 11 2004

Extensions

More terms from Colin Barker, May 25 2013

Status

approved