A122365 The (1,6)-entry of the matrix M^n, where M is the 6 X 6 matrix {{1, 1, 1, 1, 1, 1},{1, 0, 0, 0, 1, 0},{1, 0, 0, 1, 0, 0},{1, 0, 1, 0, 0, 0},{1, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}.

0, 1, 1, 6, 15, 53, 160, 517, 1621, 5150, 16267, 51513, 162944, 515673, 1631609, 5162966, 16336695, 51693645, 163571104, 517580093, 1637750957, 5182251182, 16397926099, 51887105969, 164183665152, 519517828081, 1643883210801, 5201654068774, 16459323152671

Offset

0,4

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,4,-1).

Formula

a(n) = 2*a(n-1) + 5*a(n-2) - 3*a(n-3) - 4*a(n-4) + a(n-5); a(0)=0, a(1)=1, a(2)=1,a(3)=6,a(4)=15 (follows from the minimal polynomial of M).

From Colin Barker, Mar 03 2017: (Start)

G.f.: x*(1 - x) / (1 - 2*x - 4*x^2 + x^3).

a(n) = 2*a(n-1) + 4*a(n-2) - a(n-3) for n>2.

(End)

Maple

a[0]:=0: a[1]:=1: a[2]:=1: a[3]:=6: a[4]:=15: for n from 5 to 26 do a[n]:=2*a[n-1]+5*a[n-2]-3*a[n-3]-4*a[n-4]+a[n-5] od: seq(a[n], n=0..26);

Mathematica

M = {{1, 1, 1, 1, 1, 1}, {1, 0, 0, 0, 1, 0}, {1, 0, 0, 1, 0, 0}, {1, 0, 1, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}; v[1] = {0, 0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

Prog

(PARI) concat(0, Vec(x*(1 - x) / (1 - 2*x - 4*x^2 + x^3) + O(x^40))) \\ Colin Barker, Mar 03 2017

Crossrefs

Sequence in context: A056347 A271332 A244024 * A262912 A119132 A073065

Adjacent sequences: A122362 A122363 A122364 * A122366 A122367 A122368

Keyword

nonn,easy

Author

Gary W. Adamson and Roger L. Bagula, Oct 19 2006

Extensions

Edited by N. J. A. Sloane, Oct 29 2006

Status

approved