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A123224
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a(n) = the first row sum of M^(n-1), where M = matrix(4,4, [1,1,1,1;0,1,2,3;0,1,3,6;0,1,4,10]).
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0
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1, 4, 35, 428, 5499, 71022, 917743, 11859586, 153256817, 1980478970, 25592969721, 330728127324, 4273872685405, 55229616782848, 713711145491519, 9223015274606792, 119185487424396663, 1540188321242549706, 19903262688728749387, 257202226632231192190
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1) = 1; a(2) = 4; a(3) = 35; a(4) = 428; a(n) = 15a(n-1)-28a(n-2)+15a(n-3)-a(n-4) for n>= 5. The minimal polynomial of M is x^4-15x^3+28x^2-15x+1, the coefficients of which yield the coefficients of the recurrence relation.
O.g.f.: x*(1-11*x+3*x^2)/((x^2-13*x+1)*(x-1)^2). - R. J. Mathar, Dec 05 2007
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MAPLE
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a[1]:=1:a[2]:=4:a[3]:=35:a[4]:=428: for n from 5 to 18 do a[n]:=15*a[n-1]-28*a[n-2]+15*a[n-3]-a[n-4] od: seq(a[n], n=1..18);
with(linalg): M[1]:=matrix(4, 4, [1, 1, 1, 1, 0, 1, 2, 3, 0, 1, 3, 6, 0, 1, 4, 10]): for n from 2 to 17 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(sum(M[n][1, i], i=1..4), n=1..17);
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MATHEMATICA
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M = {{ 1, 1, 1, 1 }, { 0, 1, 2, 3 }, {0, 1, 3, 6 }, { 0, 1, 4, 10 }}; v[1] = {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 25}]
LinearRecurrence[{15, -28, 15, -1}, {1, 4, 35, 428}, 50] (* Vincenzo Librandi, Jun 15 2015 *)
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PROG
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(Magma) I:=[1, 4, 35, 428]; [n le 4 select I[n] else 15*Self(n-1)-28*Self(n-2)+15*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 15 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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