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a(n) = 10*binomial(n,2) + 9*n.
14

%I #54 Jun 14 2023 18:20:56

%S 0,9,28,57,96,145,204,273,352,441,540,649,768,897,1036,1185,1344,1513,

%T 1692,1881,2080,2289,2508,2737,2976,3225,3484,3753,4032,4321,4620,

%U 4929,5248,5577,5916,6265,6624,6993,7372,7761,8160,8569,8988,9417,9856,10305,10764

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

%C 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

%C 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

%C Bisection of A195162. - _Omar E. Pol_, Aug 04 2012

%H Ivan Panchenko, <a href="/A135705/b135705.txt">Table of n, a(n) for n = 0..1000</a>

%H L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.

%H Leo Tavares, <a href="/A135705/a135705.jpg">Illustration: Diamond Clipped Stars</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _R. J. Mathar_, Mar 06 2008: (Start)

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

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

%F a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - _Vincenzo Librandi_, Nov 24 2009

%F a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - _Bruno Berselli_, Jan 15 2011

%F a(n) = A131242(10n+8). - _Philippe Deléham_, Mar 27 2013

%F 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

%F From _G. C. Greubel_, Oct 29 2016: (Start)

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

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

%F a(n) = A003154(n+1) - A000290(n+1). - _Leo Tavares_, Mar 29 2022

%t LinearRecurrence[{3,-3,1}, {0,9,28}, 50] (* or *) Table[5*n^2 + 4*n, {n,0,50}] (* _G. C. Greubel_, Oct 29 2016 *)

%t Table[10 Binomial[n,2]+9n,{n,0,60}] (* _Harvey P. Dale_, Jun 14 2023 *)

%o (PARI) a(n) = 10*binomial(n,2) + 9*n \\ _Charles R Greathouse IV_, Jun 11 2015

%o (Magma) [n*(5*n+4): n in [0..50]]; // _G. C. Greubel_, Jul 04 2019

%o (Sage) [n*(5*n+4) for n in (0..50)] # _G. C. Greubel_, Jul 04 2019

%o (GAP) List([0..50], n-> n*(5*n+4)) # _G. C. Greubel_, Jul 04 2019

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

%Y Cf. A003154, A000290.

%Y Cf. A195162.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 04 2008