A098790 a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074, 7422, 17919, 43261, 104442, 252146, 608735, 1469617, 3547970, 8565558, 20679087, 49923733, 120526554, 290976842, 702480239, 1695937321, 4094354882, 9884647086, 23863649055, 57611945197

Offset: 0

Author

Creighton Dement, Oct 30 2004

Keywords

• nonn

Comments

Previous name was: a(n) = A048739(n) - A000129(n).

Partial sums of Pell numbers A000129 except omit next-to-last Pell number. E.g., 37 = 0+1+2+5+12+29 - 12.

References

• M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
• A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
• Hermann Stamm-Wilbrandt, 4 interlaced bisections
• Index entries for linear recurrences with constant coefficients, signature (3, -1, -1).

Crossrefs

Cf. A006451, A000129, A048739, A124124, A024537, A074323, A052542.

Programs

Mathematica

a[0] = 1; a[1] = 2; a[n_] := a[n] = 2a[n - 1] + a[n - 2] + 1; Table[ a[n], {n, 0, 28}] (* Robert G. Wilson v, Nov 04 2004 *)
LinearRecurrence[{3,-1,-1},{1,2,6},31] (* Harvey P. Dale, Oct 15 2011 *)
CoefficientList[Series[(x^2 - x + 1)/((1 - x) (1 - 2 x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 14 2014 *)

Formula

a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

G.f.: (x^2-x+1)/((1-x)(1-2x-x^2)).

a(n+1) = - A024537(n+1) + 2*A048739(n+1) - 2*A048739(n).

a(n) = - A024537(n) + A052542(n+1).

Partial sums of A074323. - Paul Barry, Mar 11 2007

a(n) = (sqrt(2)+1)^n*(3/4+sqrt(2)/4)+(sqrt(2)-1)^n*(3/4-sqrt(2)/4)*(-1)^n-1/2; - Paul Barry, Mar 11 2007

a(0)=1, a(1)=2, a(2)=6, a(n)=3*a(n-1)-a(n-2)-a(n-3). [Harvey P. Dale, Oct 15 2011]

a(2*n) = A124124(2*n+1). - Hermann Stamm-Wilbrandt, Aug 03 2014

a(2*n+1) = A006451(2*n+1). - Hermann Stamm-Wilbrandt, Aug 26 2014

a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - Hermann Stamm-Wilbrandt, Aug 26 2014

2*a(n) = A135532(n+1)-1. - R. J. Mathar, Jan 13 2023

Extensions

More terms from Robert G. Wilson v, Nov 04 2004

Definition edited by N. J. A. Sloane, Aug 03 2014

New name from existing formula by Joerg Arndt, Aug 13 2014