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

0,2

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).

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

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 *)

Crossrefs

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

Sequence in context: A061261 A335903 A291414 * A300344 A018019 A331347

Adjacent sequences: A098787 A098788 A098789 * A098791 A098792 A098793

Keyword

nonn

Author

Creighton Dement, Oct 30 2004

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

Status

approved