OFFSET
0,3
REFERENCES
See A123942 for references.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,0,-6,0,1).
FORMULA
a(n) = 8*a(n-1) - 6*a(n-3) + a(n-5) for n>=5 (follows from the minimal polynomial of the matrix M).
G.f.: x*(1 - 3*x + x^3) / (1 - 8*x + 6*x^3 - x^5). - Colin Barker, Mar 03 2017
MAPLE
with(linalg): M[1]:=matrix(5, 5, [5, 3, 2, 1, 1, 3, 2, 1, 1, 0, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0]): for n from 2 to 30 do M[n]:=multiply(M[1], M[n-1]) od: 0, seq(M[n][1, 5], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=5: a[3]:=40: a[4]:=315: for n from 5 to 30 do a[n]:=8*a[n-1]-6*a[n-3]+a[n-5] od: seq(a[n], n=0..30);
# third Maple program:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,
<1|0|-6|0|8>>^n. <<0, 1, 5, 40, 315>>)[1, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2019
MATHEMATICA
M = {{5, 3, 2, 1, 1}, {3, 2, 1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}; v[1] = {0, 0, 0, 0, 1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
LinearRecurrence[{8, 0, -6, 0, 1}, {0, 1, 5, 40, 315}, 30] (* G. C. Greubel, Aug 05 2019 *)
PROG
(PARI) concat(0, Vec(x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) + O(x^30))) \\ Colin Barker, Mar 03 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) )); // G. C. Greubel, Aug 05 2019
(Sage) (x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019
(GAP) a:=[0, 1, 5, 40, 315];; for n in [6..30] do a[n]:=8*a[n-1]-6*a[n-3] +a[n-5]; od; a; # G. C. Greubel, Aug 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 25 2006
EXTENSIONS
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved