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A125641
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Square of the modulus of the (3,1)-entry of the 3 X 3 matrix M^n, where M=[1,0,0; 1,1,0, 1,i,1].
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0
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1, 5, 18, 52, 125, 261, 490, 848, 1377, 2125, 3146, 4500, 6253, 8477, 11250, 14656, 18785, 23733, 29602, 36500, 44541, 53845, 64538, 76752, 90625, 106301, 123930, 143668, 165677, 190125, 217186, 247040, 279873, 315877, 355250, 398196, 444925
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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FORMULA
| a(n) = |(b(n)|^2, where b(n)=3b(n-1)-3b(n-2)+b(n-3) for n>=4; b(1)=1, b(2)=2+i, b(3)=3+3i (the recurrence relation follows from the minimal polynomial t^3-3t^2+3t-1 of the matrix M).
a(n) = n^2(n^2-2n+5)/4. - T. D. Noe, Feb 09 2007
O.g.f.: -x*(1+3*x^2+2*x^3)/(-1+x)^5. - R. J. Mathar, Dec 05 2007
a(n) = binomial(n,2)^2 + n^2, n>1. - Gary Detlefs, Nov 23 2011
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EXAMPLE
| a(5)=25 because M^5=[1,0,0; 5,1,0; 5+10i, 5i, 1] and |5+10i|^2=125.
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MAPLE
| b[1]:=1: b[2]:=2+I: b[3]:=3+3*I: for n from 4 to 45 do b[n]:=3*b[n-1]-3*b[n-2]+b[n-3] od: seq(abs(b[j])^2, j=1..45);
with(linalg): M[1]:=matrix(3, 3, [1, 0, 0, 1, 1, 0, 1, I, 1]): for n from 2 to 45 do M[n]:=multiply(M[1], M[n-1]) od: seq(abs(M[j][3, 1])^2, j=1..45);
seq(sum((binomial(n, m))^2, m=1..2), n=1..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 19 2008
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MATHEMATICA
| Table[n^2 (n^2-2n+5)/4, {n, 30}] (* Vincenzo Librandi, Feb 14 2012 *)
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CROSSREFS
| Sequence in context: A146213 A176145 A036893 * A006479 A127983 A056782
Adjacent sequences: A125638 A125639 A125640 * A125642 A125643 A125644
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2006
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EXTENSIONS
| Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2006
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