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A177230
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a(n) = determinant of n X n circulant matrix whose first row consists of the first n squares (beginning with 1).
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1
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1, -15, 686, -62400, 9406375, -2117816064, 665460727820, -278158506983424, 149228699913026685, -99940926131200000000, 81720620766038589115418, -80119979953874981093376000, 92770427931597143858070722691, -125252587064115948721297529241600
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OFFSET
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1,2
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COMMENTS
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This sequence is the solution to Problem #11467, proposed by Xiang Qian Chang, in the December 2009 issue of the American Mathematical Monthly.
Please notice that "The Wohascum County Problem Book" predates the Problem #11467 by 16 years. - Robert G. Wilson v, Aug 31 2014
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REFERENCES
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George T. Gilbert, Mark I. Krusemeyer and Loren C. Larson, The Wohascum County Problem Book, The Mathematical Association of America, Dolciani Mathematical Expositions No. 14, 1993, problem 130 "Prove that det(...) = (-1)^(n-1)n^(n-2)(n+1)(2n+1)((n+2)^n-n^n)/12", page 31 and solution on page 216.
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LINKS
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Xiang Qian Chang, Problem 11467, The American Mathematical Monthly, Vol. 116, No. 10 (Dec., 2009), p. 940.
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FORMULA
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a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n - n^n)/12.
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EXAMPLE
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a(4) = -62400 = determinant
| 1, 4, 9, 16|
|16, 1, 4, 9|
| 9, 16, 1, 4|
| 4, 9, 16, 1|
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MATHEMATICA
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a[n_] := (-1)^(n - 1) n^(n - 2) (n + 1) (2 n + 1) ((n + 2)^n - n^n)/12; Array[a, 14] (* Robert G. Wilson v, Aug 31 2014 *)
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PROG
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(Magma)
[(-1)^n*n^(n-2)*(n^n-(n+2)^n)*Binomial(2*n+2, 2)/12: n in [1..30]]; // G. C. Greubel, Apr 12 2024
(SageMath)
[(-1)^n*n^(n-2)*(n^n-(n+2)^n)*binomial(2*n+2, 2)/12 for n in range(1, 31)] # G. C. Greubel, Apr 12 2024
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CROSSREFS
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KEYWORD
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easy,sign,changed
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AUTHOR
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Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010
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STATUS
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approved
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