A103626 - OEIS

%I #20 Dec 11 2022 19:31:32

%S 1,1,1,1,1,2,2,2,1,3,4,4,1,5,6,7,1,8,10,11,1,12,16,18,1,19,24,28,1,29,

%T 38,43,1,44,58,67,1,68,88,102,1,103,136,156,1,157,206,239,1,240,314,

%U 363,1,364,480,554,1,555,728,844,1,845,1110,1283,1,1284,1690,1955,1,1956

%N Expansion of (1 + x + x^2 + x^3 + x^5 + x^6 + x^7 - x^8 + x^10 + x^11 - 2*x^12 - x^13 - x^14)/(1 - x^4 - x^8 - x^12 + 2*x^16).

%C Four interleaved sequences (1,1,1,1,1,1....), (1,2,3,5,8,12,...), (1,2,4,6,10,16,..) and (1,2,4,7,11,18,..) each with recurrence b(n) = b(n-1) + b(n-2) + b(n-3) - 2*b(n-4).

%D J. J. P. Veerman, Hausdorff Dimension of Boundaries of Self-Affine Tiles in R^n, Bol. Soc. Mex. Mat. 3, Vol. 4, No 2, 1998, 159 - 182

%H G. C. Greubel, <a href="/A103626/b103626.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,-2).

%F Using the matrix M = {{1,0,0,0}, {1,0,0,1}, {0,2,0,0}, {0,1,1,0}} and vector v(0) = (1,1,1,1), then v(n) = M.v(n-1) gives v(n) = (a(4n), a(4n+1), a(4n+2), a(4n+3)).

%F From _R. J. Mathar_, Jul 10 2012: (Start)

%F a(n) = +a(n-4) +a(n-8) +a(n-12) -2*a(n-16).

%F a(4*n) = 1.

%F G.f.: (1+x+x^2+x^3+x^5+x^6+x^7-x^8+x^10+x^11-2*x^12-x^13-x^14) / ( (1-x)*(1+x)*(1+x^2)*(1-x^8-2*x^12) ). (End)

%t M= {{1,0,0,0}, {1,0,0,1}, {0,2,0,0}, {0,1,1,0}};

%t v[0]= {1,1,1,1}; v[n_]:= v[n]= M.v[n-1];

%t Flatten[Table[v[n], {n,0,40}]]

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 164); Coefficients(R!( (1+x+ x^2+x^3+x^5+x^6+x^7-x^8+x^10+x^11-2*x^12-x^13-x^14)/(1-x^4-x^8-x^12+ 2*x^16) )); // _G. C. Greubel_, Dec 10 2022

%o (SageMath)

%o def A103626_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1+x+x^2+x^3+x^5+x^6+x^7-x^8 + x^10+x^11-2*x^12-x^13- x^14)/(1-x^4-x^8-x^12+2*x^16) ).list()

%o A103626_list(164) # _G. C. Greubel_, Dec 10 2022

%K nonn,easy

%O 0,6

%A _Roger L. Bagula_, Mar 25 2005

%E Edited by _G. C. Greubel_, Dec 10 2022