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A055112
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a(n) = n*(n+1)*(2*n+1).
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14
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0, 6, 30, 84, 180, 330, 546, 840, 1224, 1710, 2310, 3036, 3900, 4914, 6090, 7440, 8976, 10710, 12654, 14820, 17220, 19866, 22770, 25944, 29400, 33150, 37206, 41580, 46284, 51330, 56730, 62496, 68640, 75174, 82110, 89460, 97236, 105450
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OFFSET
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0,2
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COMMENTS
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Original name: Areas of Pythagorean triangles (X, Y, Z = Y + 1) with X^2 + Y^2 = Z^2.
a(n) is the set of possible y values for 4*x^3 + x^2 = y^2 with the x values being A002378(n). - Gary Detlefs, Feb 22 2010
This sequence is related to A028896 by a(n) = n*A028896(n) - Sum_{i = 0..n-1} A028896(i) and this is the case d = 3 in the identity n*(d*(d+1)*n*(n+1)/4) - Sum_{i = 0..n-1} d*(d+1)*i*(i+1)/4 = d*(d+1)*n*(n+1)*(2*n+1)/12. - Bruno Berselli, Mar 31 2012
Also sums of rows of natural numbers (cf. A001477) seen as triangle with an odd numbers of terms per row, see example. - Reinhard Zumkeller, Jan 24 2013
Without mentioning the connection to Pythagorean triangles, Bolker (1967) gives it as an exercise to prove that these numbers are always divisible by 6. This is easy to prove from the formula that he gives, n(n - 1)(2n - 1): obviously either n or (n - 1) must be even; then, if n is congruent to 2 mod 3 it means that (2n - 1) is a multiple of 3, otherwise either n or (n - 1) is a multiple of 3; thus both prime divisors of 6 are accounted for in a(n). - Alonso del Arte, Oct 13 2013
a(n) = n*(n+1)*(n+(n+1)) is the product of two consecutive integers multiplied by the sum of those two consecutive integers. - Charles Kusniec, Sep 04 2022
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REFERENCES
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Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.5.
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LINKS
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FORMULA
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a(n) = n*(n+1)*(2*n+1).
Sum_{n > 0} 1/a(n) = 3 - 4*log(2). (End)
a(n) = Sum_{i = 1..2*n + 1} (n^2 + (i-1)). - Charlie Marion, Sep 14 2012
Sum_{n >= 1} (-1)^(n+1)/a(n) = Pi - 3, due to Nilakantha, circa 1500. See Roy p. 304. - Peter Bala, Feb 19 2015
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EXAMPLE
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. n A001477(n) as triangle with row lengths = 2*n+1 Row sums = a(n)
. 0 0 0
. 1 1 2 3 6
. 2 4 5 6 7 8 30
. 3 9 10 11 12 13 14 15 84
. 4 16 17 18 19 20 21 22 23 24 180
. 5 25 26 27 28 29 30 31 32 33 34 35 330
. 6 36 37 38 39 40 41 42 43 44 45 46 47 48 546
. 7 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 840 .
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MATHEMATICA
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PROG
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(PARI) a(n)=n*(n+1)*(2*n+1);
(Python)
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CROSSREFS
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Similar sequences are listed in A316224.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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