

A135713


a(n) = n*(n+1)*(4*n+1)/2.


5



0, 5, 27, 78, 170, 315, 525, 812, 1188, 1665, 2255, 2970, 3822, 4823, 5985, 7320, 8840, 10557, 12483, 14630, 17010, 19635, 22517, 25668, 29100, 32825, 36855, 41202, 45878, 50895, 56265, 62000, 68112, 74613, 81515, 88830, 96570, 104747, 113373, 122460, 132020
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OFFSET

0,2


COMMENTS

This sequence is related to A045944 by a(n) = n*A045944(n)Sum_{i=0..n1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d2)/2  sum(k*(d*k+d2)/2, k=0..n1) = n*(n+1)*(2*d*n+d3)/6 .  Bruno Berselli, Nov 19 2010


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
M. E. Larsen, The eternal triangle  a history of a counting problem, College Math. J., 20 (1989), 370392.
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

O.g.f.: x*(7*x+5)/(x1)^4.  R. J. Mathar, Apr 22 2008.
a(n) = 4*a(n1) 6*a(n2) +4*a(n3) a(n4) for n>3.  Bruno Berselli, Nov 19 2010
a(n1) = A051895(n).  Bruno Berselli, Aug 23 2011
E.g.f.: (1/2)*x*(10 + 17*x + 4*x^2)*exp(x).  G. C. Greubel, Oct 29 2016
Sum_{n>=1} 1/a(n) = 2*(5  2*Pi/3  4*log(2)) = 0.26603235073404654...  Ilya Gutkovskiy, Oct 29 2016


MATHEMATICA

LinearRecurrence[{4, 6, 4, 1}, {0, 5, 27, 78}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Table[n*(n+1)*(4*n+1)/2, {n, 0, 25}] (* G. C. Greubel, Oct 29 2016 *)


PROG

(MAGMA) [n*(n+1)*(4*n+1)/2: n in [0..40]]; // Bruno Berselli, Aug 23 2011


CROSSREFS

Bisection of A002717.
Cf. A045944, A160378; A132124, A006331.
Sequence in context: A137116 A137117 A064675 * A085740 A212783 A226315
Adjacent sequences: A135710 A135711 A135712 * A135714 A135715 A135716


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Mar 05 2008


STATUS

approved



