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A122186 First row sum of the 4 X 4 matrix M^n, where M={{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}. 0
1, 30, 707, 16886, 403104, 9623140, 229729153, 5484227157, 130922641160, 3125460977225, 74612811302754, 1781200165693270, 42521840081752984, 1015105948653689061, 24233196047277585233, 578508865448619225434 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

These matrices resemble Hankel matrices.

LINKS

Table of n, a(n) for n=0..15.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Index entries for linear recurrences with constant coefficients, signature (23,21,-4,-1).

FORMULA

a(n) = 23a(n-1)+21a(n-2)-4a(n-3)-a(n-4) for n>=4; a(0)=1, a(1)=30, a(2)=707, a(3)=16886 (follows from the minimal polynomial x^4-23x^3-21x^2+4x+1 of the matrix M).

G.f.: -(x^3+4*x^2-7*x-1) / ((x+1)*(x^3+3*x^2-24*x+1)). [Colin Barker, Dec 07 2012]

MAPLE

with(linalg): M[1]:=matrix(4, 4, [10, 9, 7, 4, 9, 8, 6, 3, 7, 6, 4, 2, 4, 3, 2, 1]): for n from 2 to 15 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(M[n][1, 1]+M[n][1, 2]+M[n][1, 3]+M[n][1, 4], n=1..15);

a[0]:=1: a[1]:=30: a[2]:=707: a[3]:=16886: for n from 4 to 15 do a[n]:=23*a[n-1]+21*a[n-2]-4*a[n-3]-a[n-4] od: seq(a[n], n=0..15);

MATHEMATICA

M = {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}; v[1] = {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

CROSSREFS

Cf. A015448.

Sequence in context: A027475 A180801 A035520 * A166840 A166828 A053509

Adjacent sequences:  A122183 A122184 A122185 * A122187 A122188 A122189

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

EXTENSIONS

Edited by N. J. A. Sloane, Nov 07 2006

STATUS

approved

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Last modified April 5 11:49 EDT 2020. Contains 333241 sequences. (Running on oeis4.)