A129080 Expansion of g.f. x*(x^4 - 5*x^3 + 10*x^2 - 12*x + 4)/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)).

4, 8, 14, 25, 48, 99, 215, 482, 1100, 2534, 5865, 13606, 31599, 73425, 170656, 396688, 922146, 2143685, 4983416, 11584987, 26931775, 62608726, 145547572, 338356994, 786584517, 1828587086, 4250949167, 9882257793, 22973462076, 53406819752, 124155792838

Offset: 1

Author

Roger L. Bagula, May 11 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Shigeki Akiyama, Pisot number system and its dual tiling, in: "Physics and Theoretical Computer Science", ed. by J. P. Gazeau et al., IOS Press (2007) 133-154.
• Petr Ambroz, Christiane Frougny, Zuzana Masakova and Edita Pelantova, Palindromic complexity of infinite words associated with simple Parry numbers, arXiv:math/0603608 [math.CO], 2006.
• Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4,1)

Crossrefs

Cf. A095263.

Programs

Magma

b:= func< n | n lt 4 select 2^n -1 else 3*Self(n-1) -2*Self(n-2) +Self(n-3) >;
[2*n+1+b(n): n in [1..40]]; // G. C. Greubel, Apr 12 2021

Maple

m:=40; S:=series( x*(x^4-5*x^3+10*x^2-12*x+4)/((1-x)^2*(1-3*x+2*x^2-x^3)), x, m+1):
seq(coeff(S, x, j), j=1..m); # G. C. Greubel, Apr 12 2021

Mathematica

(* b = A095263 *)
b[n_]:= b[n]= If[n<4, 2^n -1, 3*b[n-1] -2*b[n-2] +b[n-3]];
a[n_]:= a[n]= If[n==1, 4, a[n-1] +b[n] -b[n-1] +2];
Table[a[n], {n, 40}] (*modified by G. C. Greubel, Apr 12 2021 *)
LinearRecurrence[{5,-9,8,-4,1},{4,8,14,25,48},40] (* Harvey P. Dale, Feb 14 2015 *)

Sage

@CachedFunction
def b(n): return 2^n -1 if n < 4 else 3*b(n-1) -2*b(n-2) +b(n-3)
[2*n+1 +b(n) for n in (1..40)] # G. C. Greubel, Apr 12 2021

Formula

a(n) = a(n-1) + A095263(n) - A095263(n-1) + 2.

G.f.: x*(x^4 - 5*x^3 + 10*x^2 - 12*x + 4)/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009; corrected by R. J. Mathar, Sep 16 2009]

a(n) = A095263(n) + 2*n + 1. - G. C. Greubel, Apr 12 2021

Extensions

Edited by G. C. Greubel, Apr 12 2021

New name using Maksym Voznyy's g.f., Joerg Arndt, Apr 13 2021