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A037270
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n^2*(n^2+1)/2.
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23
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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
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OFFSET
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0,3
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COMMENTS
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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. - Robert G. Wilson v, 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, 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]
For n>1, triangular numbers that can be represented as a sum of a square and a triangular number. For example, a(2)=10=4+6=9+1 [Ivan N. Ianakiev, Apr 24 2012]
A037270 can be constructed in the following manner: Take A000217 and for every n not in A000290 delete the corresponding A000217(n). [Ivan N. Ianakiev, Apr 26 2012]
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REFERENCES
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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
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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
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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, Oct 15 2001
a(n) = Sum{k=0..n^2, k}. - William A. Tedeschi, Feb 27 2008
a(n) = (1/8)*Sinh[2*ArcSinh[n]]. [From Artur Jasinski, Feb 10 2010]
G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^5. [Colin Barker, Mar 22 2012]
a(n) = a(n-1) + A005898(n-1). [Ivan N. Ianakiev, May 13 2012]
a(n) = 2 * A000217(n-1) * A000217(n) + A000290(n). [Ivan N. Ianakiev, May 26 2012] \
a(n) = A000217(n^2). - J. M. Bergot, Jun 07 2012
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MAPLE
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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
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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, Feb 10 2010]
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PROG
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(PARI) a(n)=binomial(n^2+1, 2) \\ Charles R Greathouse IV, Apr 25 2012
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CROSSREFS
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Sequence in context: A211032 A179095 A213188 * A027800 A005714 A175705
Adjacent sequences: A037267 A037268 A037269 * A037271 A037272 A037273
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)
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EXTENSIONS
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Reference from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
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STATUS
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approved
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