A117917 a(n) = 3*a(n-1) + a(n-2) + n.

1, 4, 15, 52, 175, 582, 1927, 6370, 21045, 69514, 229597, 758316, 2504557, 8272000, 27320571, 90233728, 298021771, 984299058, 3250918963, 10737055966, 35462086881, 117123316630, 386832036793, 1277619427032, 4219690317913

Offset: 0

Author

Gary W. Adamson, Apr 02 2006

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to 3.30277563... = exp(ArcSinh(3/2)) (A098316).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-6,1,1).

Crossrefs

Cf. A006190.

Programs

Magma

I:=[1,4,15,52]; [n le 4 select I[n] else 5*Self(n-1) -6*Self(n-2) +Self(n-3) +Self(n-4): n in [1..31]]; // G. C. Greubel, Oct 22 2021

Mathematica

LinearRecurrence[{5,-6,1,1},{1,4,15,52},30] (* Harvey P. Dale, Mar 21 2018 *)

Sage

[(2*lucas_number1(n+2, 3, -1) + 8*lucas_number1(n+1, 3, -1) - 3*n -5)/9 for n in (0..30)] # G. C. Greubel, Oct 22 2021

Formula

O.g.f.: (1 - x + x^2)/((1-x)^2*(1 - 3*x - x^2)). - R. J. Mathar, Mar 17 2008

a(n) = ( -3*n - 5 + 2*A006190(n) + 14*A006190(n+1) )/9. - R. J. Mathar, Oct 21 2012

Extensions

More terms from R. J. Mathar, Mar 17 2008