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A055112 a(n) = n*(n+1)*(2*n+1). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 4x^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(A028896(i), i = 0..n-1) 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)(2n+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

REFERENCES

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.5.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Mathematics Magazine, 63 (5) 1990, 291-306.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = n*(n+1)*(2*n+1).

G.f.: 6*x*(1+x)/(1-x)^4. - Bruno Berselli, Mar 31 2012

a(n) = 6*A000330(n) = A007531(2*n)/4 = 3*A000292(2*n-1)/2 = A005408(n)*A046092(n)/2 = A005408(n)*(A001844(n)-1)/2 sum(n > 0, 1/a(n)) = 3 - 4*log(2). - Benoit Cloitre, Apr 30 2002

a(n) = Sum_{i = 1..n} 6*i^2 = Sum_{i = 1..n} A033581(i). - Jonathan Vos Post, Mar 15 2006

a(n) = A000217(2*n)*A000217(2*n+1)/(2*n+1). - Charlie Marion, Feb 17 2012

a(n) = Sum_{i = 1..2*n + 1} (n^2 + (i-1)). - Charlie Marion, Sep 14 2012

a(n) = n(n + 1)(2n + 1) = 2n^3 + 3n^2 + n. With a different offset, a(n) = n(n - 1)(2n - 1) = 2n^3 - 3n^2 + n. - Alonso del Arte, Oct 13 2013

Sum_{n >= 1} (-1)^(n+1)/a(n) = Pi - 3, due to Nilakantha, circa 1500. See Roy p. 304. - Peter Bala, Feb 19 2015

a(n) = A002378(n) * (2n+1). - Bruce J. Nicholson, Aug 31 2017

EXAMPLE

.  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 .

- Reinhard Zumkeller, Jan 24 2013

In this triangle, the center column is A002378 (oblong numbers) or 2 * A000217 (triangular numbers). - Bruce J. Nicholson, Aug 31 2017

MATHEMATICA

Table[n(n + 1)(2n + 1), {n, 0, 39}] (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)

PROG

(PARI) a(n)=n*(n+1)*(2*n+1);

CROSSREFS

Cf. A005408 (X values), A046092 (Y values), A001844 (Z values), A002939 (perimeter), A033581.

Sequence in context: A259918 A002444 A152788 * A094143 A217260 A009775

Adjacent sequences:  A055109 A055110 A055111 * A055113 A055114 A055115

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jun 15 2000

STATUS

approved

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Last modified February 25 22:27 EST 2018. Contains 299662 sequences. (Running on oeis4.)