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0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217 ) - the n-th partial sum of triangular numbers(A000217 ). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n) - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 05 2002
Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x)=prod(i=0,n-1,i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 15 2003
The sequence allows us to find Y values of the equation: (X-Y)^3-XY=0. Sequence gives Y values. To find X values: a(n)=n*(n+1)^2. (see A045991) - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 09 2006
a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos Sep 06 2006
Number of units of a(n) belongs to a periodic sequence: 0, 2, 2, 6, 0.We conclude that a(n) and a(n+5) have the same number of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]
Contribution from Bruno Berselli, Nov 18 2010: (Start)
This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(k), k=0..n-1); this is the case i=3 in the identity n^2*(i*n+i-2)/2-sum(k*(i*k+i-2)/2, k=0..n-1) = n*(n+1)*(2*i*n+i-3)/6. (End)
Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). [J.M. Bergot, Sep 22 2011]
a(n+1) = sum of n-th row of the triangle in A195437. [Reinhard Zumkeller, Nov 23 2011]
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REFERENCES
| L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
B. Berselli, a description of the recurrence in Comments: website Matem@ticamente
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FORMULA
| a(n) = 2*A002411(n).
a(n) = Sum[Sum[(i+j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 12 2004
sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340.. [Jolley eq 272] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 22 2006
a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). [Arkadiusz Wesolowski, Feb 10 2012]
G.f.: 2*(x + 2*x^2)/(1 - x)^4. [Arkadiusz Wesolowski, Feb 11 2012]
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EXAMPLE
| a(3)=3^2+3^3=36.
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MATHEMATICA
| lst={}; Do[AppendTo[lst, n^3+n^2], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 03 2009]
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PROG
| (MAGMA) [n^2+n^3: n in [0..40]]; // Vincenzo Librandi, May 02 2011
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CROSSREFS
| Cf. A022549.
Cf. A045991.
Sequence in context: A062094 A200543 A176583 * A073404 A141208 A181825
Adjacent sequences: A011376 A011377 A011378 * A011380 A011381 A011382
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Glen Burch; Felice Russo (frusso(AT)micron.com)
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