A177230 a(n) = determinant of n X n circulant matrix whose first row consists of the first n squares (beginning with 1).
1, -15, 686, -62400, 9406375, -2117816064, 665460727820, -278158506983424, 149228699913026685, -99940926131200000000, 81720620766038589115418, -80119979953874981093376000, 92770427931597143858070722691, -125252587064115948721297529241600
Offset: 1
Examples
a(4) = -62400 = determinant | 1, 4, 9, 16| |16, 1, 4, 9| | 9, 16, 1, 4| | 4, 9, 16, 1|
References
- 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.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..200
- Xiang Qian Chang, Problem 11467, The American Mathematical Monthly, Vol. 116, No. 10 (Dec., 2009), p. 940.
Crossrefs
Cf. A118707.
Programs
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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
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Mathematica
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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PARI
a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n-n^n)/12 \\ Charles R Greathouse IV, Aug 31 2014
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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
Formula
a(n) = (-1)^(n-1)*(n+1)*(2*n+1)*n^(n-2)*((n+2)^n - n^n)/12.
Comments