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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)).
1
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
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.
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
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