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A135705
a(n) = 10*binomial(n,2) + 9*n.
14
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
0,2
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
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
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
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 *)
PROG
(PARI) a(n) = 10*binomial(n, 2) + 9*n \\ Charles R Greathouse IV, Jun 11 2015
(Magma) [n*(5*n+4): n in [0..50]]; // G. C. Greubel, Jul 04 2019
(Sage) [n*(5*n+4) for n in (0..50)] # G. C. Greubel, Jul 04 2019
(GAP) List([0..50], n-> n*(5*n+4)) # G. C. Greubel, Jul 04 2019
CROSSREFS
Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.
Cf. A195162.
Sequence in context: A031308 A063155 A366863 * A321559 A041359 A034126
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 04 2008
STATUS
approved