This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135705 #57 May 09 2025 19:43:47
%S A135705 0,9,28,57,96,145,204,273,352,441,540,649,768,897,1036,1185,1344,1513,
%T A135705 1692,1881,2080,2289,2508,2737,2976,3225,3484,3753,4032,4321,4620,
%U A135705 4929,5248,5577,5916,6265,6624,6993,7372,7761,8160,8569,8988,9417,9856,10305,10764
%N A135705 a(n) = 10*binomial(n,2) + 9*n.
%C A135705 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 A135705 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 A135705 Bisection of A195162. - _Omar E. Pol_, Aug 04 2012
%H A135705 Ivan Panchenko, <a href="/A135705/b135705.txt">Table of n, a(n) for n = 0..1000</a>
%H A135705 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 A135705 Leo Tavares, <a href="/A135705/a135705.jpg">Illustration: Diamond Clipped Stars</a>
%H A135705 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A135705 From _R. J. Mathar_, Mar 06 2008: (Start)
%F A135705 O.g.f.: x*(9+x)/(1-x)^3.
%F A135705 a(n) = n*(5*n+4). (End)
%F A135705 a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - _Vincenzo Librandi_, Nov 24 2009
%F A135705 a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - _Bruno Berselli_, Jan 15 2011
%F A135705 a(n) = A131242(10n+8). - _Philippe Deléham_, Mar 27 2013
%F A135705 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 A135705 From _G. C. Greubel_, Oct 29 2016: (Start)
%F A135705 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A135705 E.g.f.: x*(9 + 5*x)*exp(x). (End)
%F A135705 a(n) = A003154(n+1) - A000290(n+1). - _Leo Tavares_, Mar 29 2022
%t A135705 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 A135705 Table[10 Binomial[n,2]+9n,{n,0,60}] (* _Harvey P. Dale_, Jun 14 2023 *)
%o A135705 (PARI) a(n) = 10*binomial(n,2) + 9*n \\ _Charles R Greathouse IV_, Jun 11 2015
%o A135705 (Magma) [n*(5*n+4): n in [0..50]]; // _G. C. Greubel_, Jul 04 2019
%o A135705 (Sage) [n*(5*n+4) for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o A135705 (GAP) List([0..50], n-> n*(5*n+4)); # _G. C. Greubel_, Jul 04 2019
%Y A135705 Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.
%Y A135705 Cf. A003154, A000290.
%Y A135705 Cf. A195162.
%K A135705 nonn,easy
%O A135705 0,2
%A A135705 _N. J. A. Sloane_, Mar 04 2008