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A037270
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n^2*(n^2+1)/2.
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20
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0, 1, 10, 45, 136, 325, 666, 1225, 2080, 3321, 5050, 7381, 10440, 14365, 19306, 25425, 32896, 41905, 52650, 65341, 80200, 97461, 117370, 140185, 166176, 195625, 228826, 266085, 307720, 354061, 405450
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sum of first n^2 integers.
Start from xanthene and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - rgwv, Aug 02 2002 - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2002
Sum of the next n multiples of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2002
The sum of the terms in an n X n spiral. These are also triangular numbers. - William A. Tedeschi (fynmun(AT)hotmail.com), Feb 27 2008
Hypotenuse of Pythagorean triangles with smallest side a cube: A000578(n)^2 + A083374(n)^2 = a(n)^2. -- [Martin Renner, Nov 12 2011]
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REFERENCES
| C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See p. 5.
Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
R. A. Wilson, Cosmic Trigger, epilogue of S.-P. Sirag.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795
Th. Gruner, A. Kerber, R. Laue, M. Meringer, Mathematics for Combinatorial Chemistry
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FORMULA
| a(n) = a(n-1) + n^3 + (n-1)^3.
a(n) = A000537(n)+A000537(n-1), i.e. square of sum of first n integers plus square of sum of first n-1 integers. - Henry Bottomley (se16(AT)btinternet.com), Oct 15 2001
a(n) = Sum{k=0..n^2, k} - William A. Tedeschi (fynmun(AT)hotmail.com), Feb 27 2008
a(n)=(1/8)Sinh[2*ArcSinh[n]] [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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MAPLE
| a:=n->add(n+add(binomial(n, 2), j=0..n), j=1..n):seq(a(n), n=0..35); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
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MATHEMATICA
| Table[ n^2*((n^2 + 1)/2), {n, 0, 30} ]
Table[(1/8) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}] (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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CROSSREFS
| Sequence in context: A105938 A022605 A179095 * A027800 A005714 A175705
Adjacent sequences: A037267 A037268 A037269 * A037271 A037272 A037273
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KEYWORD
| easy,nonn,nice
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AUTHOR
| Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)
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EXTENSIONS
| Reference from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
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