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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, 29725069786599476, 199204401658077156
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OFFSET
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0,2
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COMMENTS
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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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LINKS
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FORMULA
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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
E.g.f.: (1/41)*exp(7*x/2)*(41*cosh(sqrt(41)*x/2) + 5*sqrt(41)*sinh(sqrt(41)*x/2)). - Stefano Spezia, Oct 03 2019
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EXAMPLE
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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
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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);
# alternative:
f:= gfun:-rectoproc({a(n+2)-7*a(n+1)+2*a(n), a(0)=1, a(1)=6}, a(n), remember):
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MATHEMATICA
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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]] (* Harvey P. Dale, Mar 11 2011 *)
f[s_] := Append[s, 7*s[[-1]] - 2*s[[-2]]]; Nest[f, {1, 6}, 18] (* Robert G. Wilson v, Mar 12 2011 *)
LinearRecurrence[{7, -2}, {1, 6}, 25] (* Harvey P. Dale, Jan 04 2014 *)
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PROG
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(PARI) Vec((1-x)/(1-7*x+2*x^2) + O(x^30)) \\ Michel Marcus, Oct 03 2015
(Magma) I:=[1, 6]; [n le 2 select I[n] else 7*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 03 2015
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-x)/(1-7*x+2*x^2)).list()
(GAP) a:=[1, 6];; for n in [3..30] do a[n]:=7*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 02 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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