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%I #55 Sep 08 2022 08:45:01
%S 0,6,13,21,30,40,51,63,76,90,105,121,138,156,175,195,216,238,261,285,
%T 310,336,363,391,420,450,481,513,546,580,615,651,688,726,765,805,846,
%U 888,931,975,1020,1066,1113,1161,1210,1260,1311,1363,1416,1470,1525
%N a(n) = n*(n+11)/2.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H G. C. Greubel, <a href="/A056115/b056115.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 G.f.: x*(6-5*x)/(1-x)^3.
%F a(n) = A000096(n) + 4*A001477(n) = A056000(n) + A001477(n) = A056119(n) - A001477(n). - _Zerinvary Lajos_, Oct 01 2006
%F a(n) = A126890(n,5) for n>4. - _Reinhard Zumkeller_, Dec 30 2006
%F Equals A119412/2. - _Zerinvary Lajos_, Feb 12 2007
%F If we define f(n,i,a) = Sum_{k=0..n-i} ( binomial(n,k)*stirling1(n-k,i) *Product_{j=0..k-1} (-a-j) ), then a(n) = -f(n,n-1,6), for n>=1. - _Milan Janjic_, Dec 20 2008
%F a(n) = a(n-1) + n + 5 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F Sum_{n>=1} 1/a(n) = 83711/152460. - _R. J. Mathar_, Jul 14 2012
%F a(n) = 6*n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F E.g.f.: x*(12 + x)*exp(x)/2. - _G. C. Greubel_, Jan 18 2020
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/11 - 20417/152460. - _Amiram Eldar_, Jan 10 2021
%t ((2*Range[0,50]+11)^2 -11^2)/8 (* _G. C. Greubel_, Jan 18 2020 *)
%o (PARI) a(n)=n*(n+11)/2; \\ _Joerg Arndt_, Oct 25 2014
%o (Magma) [n*(n+11)/2: n in [0..50]]; // _G. C. Greubel_, Jan 18 2020
%o (Sage) [n*(n+11)/2 for n in (0..50)] # _G. C. Greubel_, Jan 18 2020
%o (GAP) List([0..50], n-> n*(n+11)/2 ); # _G. C. Greubel_, Jan 18 2020
%Y Cf. A000096, A001477, A055999, A056000, A056119.
%Y Third column of Pascal (1, 6) triangle A096956.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jul 04 2000
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