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