A129080 - OEIS

%I #27 Apr 14 2021 22:26:15

%S 4,8,14,25,48,99,215,482,1100,2534,5865,13606,31599,73425,170656,

%T 396688,922146,2143685,4983416,11584987,26931775,62608726,145547572,

%U 338356994,786584517,1828587086,4250949167,9882257793,22973462076,53406819752,124155792838

%N 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)).

%H G. C. Greubel, <a href="/A129080/b129080.txt">Table of n, a(n) for n = 1..1000</a>

%H Shigeki Akiyama, <a href="http://math.tsukuba.ac.jp/~akiyama/papers/cargese_note.pdf">Pisot number system and its dual tiling</a>, in: "Physics and Theoretical Computer Science", ed. by J. P. Gazeau et al., IOS Press (2007) 133-154.

%H Petr Ambroz, Christiane Frougny, Zuzana Masakova and Edita Pelantova, <a href="http://arxiv.org/abs/math/0603608">Palindromic complexity of infinite words associated with simple Parry numbers</a>, arXiv:math/0603608 [math.CO], 2006.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,8,-4,1)

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

%F 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]

%F a(n) = A095263(n) + 2*n + 1. - _G. C. Greubel_, Apr 12 2021

%p 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):

%p seq(coeff(S, x, j), j=1..m); # _G. C. Greubel_, Apr 12 2021

%t (* b = A095263 *)

%t b[n_]:= b[n]= If[n<4, 2^n -1, 3*b[n-1] -2*b[n-2] +b[n-3]];

%t a[n_]:= a[n]= If[n==1, 4, a[n-1] +b[n] -b[n-1] +2];

%t Table[a[n], {n, 40}] (*modified by _G. C. Greubel_, Apr 12 2021 *)

%t LinearRecurrence[{5,-9,8,-4,1},{4,8,14,25,48},40] (* _Harvey P. Dale_, Feb 14 2015 *)

%o (Magma)

%o b:= func< n | n lt 4 select 2^n -1 else 3*Self(n-1) -2*Self(n-2) +Self(n-3) >;

%o [2*n+1+b(n): n in [1..40]]; // _G. C. Greubel_, Apr 12 2021

%o (Sage)

%o @CachedFunction

%o def b(n): return 2^n -1 if n < 4 else 3*b(n-1) -2*b(n-2) +b(n-3)

%o [2*n+1 +b(n) for n in (1..40)] # _G. C. Greubel_, Apr 12 2021

%Y Cf. A095263.

%K nonn,easy

%O 1,1

%A _Roger L. Bagula_, May 11 2007

%E Edited by _G. C. Greubel_, Apr 12 2021

%E New name using Maksym Voznyy's g.f., _Joerg Arndt_, Apr 13 2021