A353580 a(n) = 2*a(n-1) + a(n-2) - 1, with a(0) = 0 and a(1) = 2.
0, 2, 3, 7, 16, 38, 91, 219, 528, 1274, 3075, 7423, 17920, 43262, 104443, 252147, 608736, 1469618, 3547971, 8565559, 20679088, 49923734, 120526555, 290976843, 702480240, 1695937322, 4094354883, 9884647087, 23863649056, 57611945198, 139087539451, 335787024099, 810661587648
Offset: 0
Links
- Paul K. Stockmeyer, Table of n, a(n) for n = 0..1000
- Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
- Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
Programs
-
Mathematica
nxt[{a_,b_}]:={b,2b+a-1}; NestList[nxt,{0,2},40][[;;,1]] (* or *) LinearRecurrence[{3,-1,-1},{0,2,3},40] (* Harvey P. Dale, Mar 14 2025 *)
Formula
a(n) = (1/4)((2*s - 1)*(1 + s)^n - (2*s + 1)*(1 - s)^n + 2) where s = sqrt(2).
G.f.: -x*(3*x-2)/((x-1)*(x^2+2*x-1)). - Alois P. Heinz, May 03 2022
E.g.f.: exp(x)*(1 - cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, May 03 2022