A056115 a(n) = n*(n+11)/2.

0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, 121, 138, 156, 175, 195, 216, 238, 261, 285, 310, 336, 363, 391, 420, 450, 481, 513, 546, 580, 615, 651, 688, 726, 765, 805, 846, 888, 931, 975, 1020, 1066, 1113, 1161, 1210, 1260, 1311, 1363, 1416, 1470, 1525

Offset: 0

Author

Barry E. Williams, Jul 04 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A001477, A055999, A056000, A056119.

Third column of Pascal (1, 6) triangle A096956.

Programs

GAP

List([0..50], n-> n*(n+11)/2 ); # G. C. Greubel, Jan 18 2020

Magma

[n*(n+11)/2: n in [0..50]]; // G. C. Greubel, Jan 18 2020

Mathematica

((2*Range[0,50]+11)^2 -11^2)/8 (* G. C. Greubel, Jan 18 2020 *)

PARI

a(n)=n*(n+11)/2; \\ Joerg Arndt, Oct 25 2014

Sage

[n*(n+11)/2 for n in (0..50)] # G. C. Greubel, Jan 18 2020

Formula

G.f.: x*(6-5*x)/(1-x)^3.

a(n) = A000096(n) + 4*A001477(n) = A056000(n) + A001477(n) = A056119(n) - A001477(n). - Zerinvary Lajos, Oct 01 2006

a(n) = A126890(n,5) for n>4. - Reinhard Zumkeller, Dec 30 2006

Equals A119412/2. - Zerinvary Lajos, Feb 12 2007

If we define f(n,i,a) = Sum_{k=0..n-i} ( binomial(n,k)*stirling1(n-k,i) *Product_{j=0..k-1} (-a-j) ), then a(n) = -f(n,n-1,6), for n>=1. - Milan Janjic, Dec 20 2008

a(n) = a(n-1) + n + 5 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010

Sum_{n>=1} 1/a(n) = 83711/152460. - R. J. Mathar, Jul 14 2012

a(n) = 6*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

E.g.f.: x*(12 + x)*exp(x)/2. - G. C. Greubel, Jan 18 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/11 - 20417/152460. - Amiram Eldar, Jan 10 2021