A123943 The (1,5)-entry in the 5 X 5 matrix M^n, where 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}}.

0, 1, 5, 40, 315, 2490, 19681, 155563, 1229604, 9719061, 76821600, 607214857, 4799560053, 37936780428, 299860673343, 2370164848026, 18734305316497, 148080078051971, 1170457572108040, 9251554605638681, 73126326541645648, 578006601205833441

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

Cf. A122099, A122100.

Sequence in context: A280158 A073505 A145841 * A067412 A355355 A078846

Adjacent sequences: A123940 A123941 A123942 * A123944 A123945 A123946

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