A103626 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).

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, 38, 43, 1, 44, 58, 67, 1, 68, 88, 102, 1, 103, 136, 156, 1, 157, 206, 239, 1, 240, 314, 363, 1, 364, 480, 554, 1, 555, 728, 844, 1, 845, 1110, 1283, 1, 1284, 1690, 1955, 1, 1956

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

Sequence in context: A352682 A361639 A055253 * A238224 A026268 A089258

Adjacent sequences: A103623 A103624 A103625 * A103627 A103628 A103629

Keyword

nonn,easy

Author

Roger L. Bagula, Mar 25 2005

Extensions

Edited by G. C. Greubel, Dec 10 2022

Status

approved