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

1,1

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)

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A095263.

Sequence in context: A105143 A020185 A008029 * A138643 A183977 A153364

Adjacent sequences: A129077 A129078 A129079 * A129081 A129082 A129083

Keyword

nonn,easy

Author

Roger L. Bagula, May 11 2007

Extensions

Edited by G. C. Greubel, Apr 12 2021

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

Status

approved