A135713 a(n) = n*(n+1)*(4*n+1)/2.

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

Offset: 0

Author

N. J. A. Sloane, Mar 05 2008

Keywords

• nonn
• easy

Comments

This sequence is related to A045944 by a(n) = n*A045944(n)-Sum_{i=0..n-1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d-2)/2 - sum(k*(d*k+d-2)/2, k=0..n-1) = n*(n+1)*(2*d*n+d-3)/6 . - Bruno Berselli, Nov 19 2010

Bisection (even part) of A002717. See the Conway and Guy reference. - Wolfdieter Lang, Apr 16 2020

References

• J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.

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), 370-392.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Bisection of A002717 (even part).

Cf. A045944, A160378; A132124, A006331.

Programs

Magma

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

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 *)
Table[PolygonalNumber[n](4n+1),{n,0,40}] (* Harvey P. Dale, Apr 26 2025 *)

Formula

O.g.f.: x*(7*x+5)/(x-1)^4. - R. J. Mathar, Apr 22 2008.

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n>3. - Bruno Berselli, Nov 19 2010

a(-n-1) = -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