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 A124080 #63 Feb 21 2023 02:12:56
%S A124080 0,10,30,60,100,150,210,280,360,450,550,660,780,910,1050,1200,1360,
%T A124080 1530,1710,1900,2100,2310,2530,2760,3000,3250,3510,3780,4060,4350,
%U A124080 4650,4960,5280,5610,5950,6300,6660,7030,7410,7800,8200,8610,9030,9460,9900,10350
%N A124080 10 times triangular numbers: a(n) = 5*n*(n + 1).
%C A124080 If Y is a 5-subset of an n-set X then, for n >= 5, a(n-4) is equal to the number of 5-subsets of X having exactly three elements in common with Y. Y is a 5-subset of an n-set X then, for n >= 6, a(n-6) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - _Milan Janjic_, Dec 28 2007
%C A124080 Also sequence found by reading the line from 0, in the direction 0, 10, ... and the same line from 0, in the direction 0, 30, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Axis perpendicular to A195148 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%H A124080 Ivan Panchenko, <a href="/A124080/b124080.txt">Table of n, a(n) for n = 0..1000</a>
%H A124080 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A124080 a(n) = 10*C(n,2), n >= 1.
%F A124080 a(n) = A049598(n) - A002378(n). - _Zerinvary Lajos_, Mar 06 2007
%F A124080 a(n) = 5*n*(n + 1), n >= 0. - _Zerinvary Lajos_, Mar 06 2007
%F A124080 a(n) = 5*n^2 + 5*n = 10*A000217(n) = 5*A002378(n) = 2*A028895(n). - _Omar E. Pol_, Dec 12 2008
%F A124080 a(n) = 10*n + a(n-1) (with a(0) = 0). - _Vincenzo Librandi_, Nov 12 2009
%F A124080 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 10, a(2) = 30. - _Harvey P. Dale_, Jul 21 2011
%F A124080 a(n) = A062786(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F A124080 a(n) = A131242(10*n+9). - _Philippe Deléham_, Mar 27 2013
%F A124080 From _G. C. Greubel_, Aug 22 2017: (Start)
%F A124080 G.f.: 10*x/(1 - x)^3.
%F A124080 E.g.f.: 5*x*(x + 2)*exp(x). (End)
%F A124080 From _Amiram Eldar_, Sep 04 2022: (Start)
%F A124080 Sum_{n>=1} 1/a(n) = 1/5.
%F A124080 Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2)-1)/5. (End)
%F A124080 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A124080 Product_{n>=1} (1 - 1/a(n)) = -(5/Pi)*cos(3*Pi/(2*sqrt(5))).
%F A124080 Product_{n>=1} (1 + 1/a(n)) = (5/Pi)*cos(Pi/(2*sqrt(5))). (End)
%p A124080 [seq(10*binomial(n,2),n=1..51)];
%p A124080 seq(n*(n+1)*5, n=0..39); # _Zerinvary Lajos_, Mar 06 2007
%t A124080 10*Accumulate[Range[0,50]] (* or *) LinearRecurrence[{3,-3,1},{0,10,30},50] (* _Harvey P. Dale_, Jul 21 2011 *)
%o A124080 (Magma) [ 5*n*(n+1) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o A124080 (PARI) a(n)=5*n*(n+1) \\ _Charles R Greathouse IV_, Sep 28 2015
%Y A124080 Cf. A028895, A046092, A045943, A002378, A028896, A024966, A033996, A027468, A049598, A000217.
%K A124080 easy,nonn
%O A124080 0,2
%A A124080 _Zerinvary Lajos_, Nov 24 2006