A127980 a(n) = (n + 2/3)*2^(n-1) - 1/2 - (-1)^(n-1)*(1/6).

1, 5, 14, 37, 90, 213, 490, 1109, 2474, 5461, 11946, 25941, 55978, 120149, 256682, 546133, 1157802, 2446677, 5155498, 10835285, 22719146, 47535445, 99265194, 206918997, 430615210, 894784853, 1856678570, 3847574869, 7963585194

Offset: 1

Author

Artur Jasinski, Feb 09 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• W. Bosma, Signed bits and fast exponentiation, Journal de Théorie des Nombres de Bordeaux, Vol. 13, Fasc. 1 (2001), p. 38 (Proposition 7).
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Crossrefs

Cf. A073371, A127976, A127978, A127979, A073371.

Programs

Magma

I:=[1,5,14,37]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)-4*Self(n-3)+4*Self(n-4): n in [1..30]]; // G. C. Greubel, May 08 2018

Mathematica

Table[(n+2/3)2^(n-1) - 1/2 -(-1)^(n-1)*(1/6), {n, 1, 50}]
LinearRecurrence[{4,-3,-4,4}, {1,5,14,37}, 50] (* G. C. Greubel, May 08 2018 *)

PARI

x='x+O('x^30); Vec(x*(1+x-3*x^2)/((1-x)*(1+x)*(1-2*x)^2)) \\ G. C. Greubel, May 08 2018

Formula

G.f.: x*(1+x-3*x^2)/((1-x)*(1+x)*(1-2*x)^2). - Colin Barker, Apr 02 2012

E.g.f.: ((1 + 3*x)*cosh(2*x) - 2*sinh(x) + cosh(x)*((2 + 6*x)*sinh(x) - 1))/3. - Stefano Spezia, May 25 2023