OFFSET
0,6
COMMENTS
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).
REFERENCES
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,-2).
FORMULA
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)).
From R. J. Mathar, Jul 10 2012: (Start)
a(n) = +a(n-4) +a(n-8) +a(n-12) -2*a(n-16).
a(4*n) = 1.
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)
MATHEMATICA
M= {{1, 0, 0, 0}, {1, 0, 0, 1}, {0, 2, 0, 0}, {0, 1, 1, 0}};
v[0]= {1, 1, 1, 1}; v[n_]:= v[n]= M.v[n-1];
Flatten[Table[v[n], {n, 0, 40}]]
PROG
(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
(SageMath)
def A103626_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
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()
A103626_list(164) # G. C. Greubel, Dec 10 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Mar 25 2005
EXTENSIONS
Edited by G. C. Greubel, Dec 10 2022
STATUS
approved