A084150 A Pell related sequence.

0, 0, 1, 3, 14, 50, 199, 749, 2892, 11028, 42301, 161799, 619706, 2372006, 9081955, 34767953, 133109592, 509594856, 1950956857, 7469077707, 28594853414, 109473250778, 419110475455, 1604533706357, 6142840740900, 23517417426300

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Comments

Binomial transform of expansion of (sinh(sqrt(2)x))^2/4 = (0, 0, 1, 0, 8, 0, 64, ...). Inverse binomial transform of A006668.

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).

Crossrefs

Cf. A006646, A006668, A015519, A084058.

Programs

Magma

[n le 3 select Floor((n-1)/2) else 3*Self(n-1) +5*Self(n-2) -7*Self(n-3): n in [1..41]]; // G. C. Greubel, Oct 11 2022

Mathematica

LinearRecurrence[{3,5,-7}, {0,0,1}, 41] (* G. C. Greubel, Oct 11 2022 *)

SageMath

A084058 = BinaryRecurrenceSequence(2,7,1,1)
def A084150(n): return (A084058(n) - 1)/8
[A084150(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = ( (1+sqrt(8))^n + (1-sqrt(8))^n - 2 )/16.

E.g.f.: (1/4)*exp(x)*( sinh(sqrt(2)*x) )^2.

G.f.: x^2 / ( (1-x)*(1-2*x-7*x^2) ). - R. J. Mathar, Feb 05 2011

a(n) = (A015519(n) - A015519(n-1) - 1)/8 = (A084058(n) - 1)/8. - G. C. Greubel, Oct 11 2022