A033537 a(n) = n*(2*n+5).

0, 7, 18, 33, 52, 75, 102, 133, 168, 207, 250, 297, 348, 403, 462, 525, 592, 663, 738, 817, 900, 987, 1078, 1173, 1272, 1375, 1482, 1593, 1708, 1827, 1950, 2077, 2208, 2343, 2482, 2625, 2772, 2923, 3078, 3237, 3400, 3567, 3738, 3913, 4092, 4275, 4462, 4653, 4848, 5047, 5250, 5457, 5668

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Permutations avoiding 12-3 that contain the pattern 32-1 exactly once.

a(n) = A014107(n) + 8*n^2; A100035(a(n)) = 3 for n>1. - Reinhard Zumkeller, Oct 31 2004

If Y is a 3-subset of an (2n+1)-set X then, for n>=1, a(n-1) is the number of (2n-1)-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A100036, A100037, A100038, A100039.

Programs

GAP

List([0..60], n-> n*(2*n+5) ); # G. C. Greubel, Oct 14 2019

Magma

[n*(2*n+5): n in [0..60]]; // G. C. Greubel, Oct 14 2019

Maple

seq(n*(2*n+5), n=0..60); # G. C. Greubel, Oct 14 2019

Mathematica

Table[n*(2*n+5), {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3,-3,1},{0,7,18},60] (* Harvey P. Dale, Nov 19 2021 *)

PARI

a(n)=n*(2*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n*(2*n+5) for n in (0..60)] # G. C. Greubel, Oct 14 2019

Formula

a(n) = a(n-1) + 4*n + 3 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010

From L. Edson Jeffery, Oct 14 2012: (Start)

G.f.: x*(7-3*x)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=7, a(2)=18. (End)

E.g.f.: x*(7 + 2*x)*exp(x). - G. C. Greubel, Jul 15 2017

From Amiram Eldar, Feb 06 2022: (Start)

Sum_{n>=1} 1/a(n) = 46/75 - 2*log(2)/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/10 + log(2)/5 - 26/75. (End)