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A033994 a(n) = n*(n+1)*(5*n+1)/6.

%I A033994 #66 Jul 02 2025 16:01:56
%S A033994 2,11,32,70,130,217,336,492,690,935,1232,1586,2002,2485,3040,3672,
%T A033994 4386,5187,6080,7070,8162,9361,10672,12100,13650,15327,17136,19082,
%U A033994 21170,23405,25792,28336,31042,33915,36960,40182,43586,47177,50960,54940
%N A033994 a(n) = n*(n+1)*(5*n+1)/6.
%C A033994 Partial sums of A005476.
%C A033994 a(n) is the dot product of the vectors of the first n positive integers and the next n integers. - _Michel Marcus_, Sep 02 2020
%D A033994 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A033994 Harvey P. Dale, <a href="/A033994/b033994.txt">Table of n, a(n) for n = 1..1000</a>
%H A033994 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A033994 G.f.: x*(2+3*x)/(1-x)^4.
%F A033994 a(n) = A132121(n,1). - _Reinhard Zumkeller_, Aug 12 2007
%F A033994 a(n) = A000292(n) + A002412(n) = A000330(n) + A002411(n). - _Omar E. Pol_, Jan 11 2013
%F A033994 a(n) = Sum_{i=1..n} Sum_{j=1..n} i+min(i,j). - _Enrique Pérez Herrero_, Jan 15 2013
%F A033994 a(n) = Sum_{i=1..n} i*(n+i). - _Charlie Marion_, Apr 10 2013
%F A033994 Sum_{n>=1} 1/a(n) = 36 - 3*Pi*5^(3/4)*phi^(3/2)/4 - 15*sqrt(5)*log(phi)/4 - 75*log(5)/8 = 0.66131826232008423794478..., where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 01 2018
%F A033994 E.g.f.: exp(x)*x*(12 + 21*x + 5*x^2)/6. - _Stefano Spezia_, Feb 21 2024
%p A033994 [n*(n+1)*(5*n+1)/6$n=1..40]; # _Muniru A Asiru_, Jan 01 2019
%t A033994 Table[Range[x].Range[x+1,2x],{x,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,11,32,70},40] (* _Harvey P. Dale_, Jun 01 2018 *)
%o A033994 (PARI) a(n) = n*(n+1)*(5*n+1)/6;
%o A033994 (Magma) [n*(n+1)*(5*n+1)/6 : n in [1..40]]; // _Vincenzo Librandi_, Jan 01 2019
%o A033994 (GAP) a:=List([1..40],n->n*(n+1)*(5*n+1)/6);; Print(a); # _Muniru A Asiru_, Jan 01 2019
%Y A033994 Cf. A005476, A016873, A000330, A132124, A132112, A050409.
%Y A033994 Cf. A000292, A001622, A002411, A002412.
%K A033994 easy,nonn
%O A033994 1,1
%A A033994 _Barry E. Williams_, Dec 16 1999
%E A033994 More terms from _James Sellers_, Jan 19 2000