A135705 a(n) = 10*binomial(n,2) + 9*n.

0, 9, 28, 57, 96, 145, 204, 273, 352, 441, 540, 649, 768, 897, 1036, 1185, 1344, 1513, 1692, 1881, 2080, 2289, 2508, 2737, 2976, 3225, 3484, 3753, 4032, 4321, 4620, 4929, 5248, 5577, 5916, 6265, 6624, 6993, 7372, 7761, 8160, 8569, 8988, 9417, 9856, 10305, 10764

Offset: 0

Author

N. J. A. Sloane, Mar 04 2008

Keywords

• nonn
• easy

Comments

Also, second 12-gonal (or dodecagonal) numbers. Identity for the numbers b(n)=n*(h*n+h-2)/2 (see Crossrefs): Sum_{i=0..n} (b(n)+i)^2 = (Sum_{i=n+1..2*n} (b(n)+i)^2) + h*(h-4)*A000217(n)^2 for n>0. - Bruno Berselli, Jan 15 2011

Sequence found by reading the line from 0, in the direction 0, 28, ..., and the line from 9, in the direction 9, 57, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 24 2012

Bisection of A195162. - Omar E. Pol, Aug 04 2012

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
• Leo Tavares, Illustration: Diamond Clipped Stars
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.

Cf. A003154, A000290.

Cf. A195162.

Programs

GAP

List([0..50], n-> n*(5*n+4)); # G. C. Greubel, Jul 04 2019

Magma

[n*(5*n+4): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Mathematica

LinearRecurrence[{3,-3,1}, {0,9,28}, 50] (* or *) Table[5*n^2 + 4*n, {n,0,50}] (* G. C. Greubel, Oct 29 2016 *)
Table[10 Binomial[n,2]+9n,{n,0,60}] (* Harvey P. Dale, Jun 14 2023 *)

PARI

a(n) = 10*binomial(n,2) + 9*n \\ Charles R Greathouse IV, Jun 11 2015

Sage

[n*(5*n+4) for n in (0..50)] # G. C. Greubel, Jul 04 2019

Formula

From R. J. Mathar, Mar 06 2008: (Start)

O.g.f.: x*(9+x)/(1-x)^3.

a(n) = n*(5*n+4). (End)

a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 24 2009

a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - Bruno Berselli, Jan 15 2011

a(n) = A131242(10n+8). - Philippe Deléham, Mar 27 2013

Sum_{n>=1} 1/a(n) = 5/16 + sqrt(1 + 2/sqrt(5))*Pi/8 - 5*log(5)/16 - sqrt(5)*log((1 + sqrt(5))/2)/8 = 0.2155517745488486003038... . - Vaclav Kotesovec, Apr 27 2016

From G. C. Greubel, Oct 29 2016: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

E.g.f.: x*(9 + 5*x)*exp(x). (End)

a(n) = A003154(n+1) - A000290(n+1). - Leo Tavares, Mar 29 2022