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A122074
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a(0)=1, a(1)=6, a(n)=7*a(n-1)-2*a(n-2).
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3
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1, 6, 40, 268, 1796, 12036, 80660, 540548, 3622516, 24276516, 162690580, 1090281028, 7306586036, 48965540196, 328145609300, 2199088184708, 14737326074356, 98763106151076, 661867090908820, 4435543424059588
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| First row sum of the matrix M^n, where M is the 3 X 3 matrix {{2,2,2},{2,3,2},{2,2,3}}.
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REFERENCES
| Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 0..200
P. Steinbach, Golden Fields: A Case for the Heptagon (see References).
Index to sequences with linear recurrences with constant coefficients, signature (7,-2).
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FORMULA
| a(n)=8*a(n-1)-9*a(n-2)+2*a(n-3); a(0)=1, a(1)=6, a(2)=40 (follows from the minimal polynomial x^3-8x^2+9x-2 of M).
a(n) = (1/2+5*sqrt(41)/82)*(7/2+sqrt(41)/2)^n+(1/2-5*sqrt(41)/82)*(7/2-sqrt(41)/2)^n. - Antonio Alberto Olivares, Jun 06 2011
G.f.: (1-x)/(1-7*x+2*x^2). [Colin Barker, Feb 08 2012]
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EXAMPLE
| a(2)=40 because M^2={{12,14,14},{14,17,16},{14,16,17}} and 12+14+14=40.
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MAPLE
| with(linalg): M[1]:=matrix(3, 3, [2, 2, 2, 2, 3, 2, 2, 2, 3]): for n from 2 to 20 do M[n]:=multiply(M[n-1], M[1]) od: 1, seq(M[n][1, 1]+M[n][1, 2]+M[n][1, 3], n=1..20);
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MATHEMATICA
| M = {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
Transpose[NestList[{Last[#], 7*Last[#]-2*First[#]}&, {1, 6}, 25]] [[1]] (* From Harvey P. Dale, Mar 11 2011 *)
f[s_] := Append[s, 7*s[[-1]] - 2*s[[-2]]]; Nest[f, {1, 6}, 18] (* Rober G. Wilson v, Mar 12 2011 *)
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CROSSREFS
| Sequence in context: A065113 A052518 A135032 * A123357 A081016 A083426
Adjacent sequences: A122071 A122072 A122073 * A122075 A122076 A122077
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Gary Adamson (qntmpkt(AT)yahoo.com), Oct 16 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 29 2006 and Dec 04 2006
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