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A071910 a(n) = t(n)*t(n+1)*t(n+2), where t() are the triangular numbers.

%I A071910 #32 Aug 08 2025 18:25:08
%S A071910 0,18,180,900,3150,8820,21168,45360,89100,163350,283140,468468,745290,
%T A071910 1146600,1713600,2496960,3558168,4970970,6822900,9216900,12273030,
%U A071910 16130268,20948400,26910000,34222500,43120350,53867268,66758580,82123650,100328400,121777920
%N A071910 a(n) = t(n)*t(n+1)*t(n+2), where t() are the triangular numbers.
%C A071910 a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - _Alejandro Rodriguez_, Oct 20 2020
%H A071910 Harvey P. Dale, <a href="/A071910/b071910.txt">Table of n, a(n) for n = 0..1000</a>
%H A071910 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A071910 a(n) = 18*A006542(n+3). - _Vladeta Jovovic_, Jun 14 2002
%F A071910 G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - _Vladeta Jovovic_, Jun 14 2002
%F A071910 a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - _Jon Perry_, Feb 13 2004
%F A071910 a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - _Zerinvary Lajos_, Jul 29 2005
%F A071910 a(n) = A059827(n+1) - A000537(n+1). - _Michel Marcus_, Oct 21 2015
%t A071910 Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* _Harvey P. Dale_, Aug 08 2025 *)
%o A071910 (PARI) t(n) = n*(n+1)/2;
%o A071910 a(n) = t(n)*t(n+1)*t(n+2); \\ _Michel Marcus_, Oct 21 2015
%Y A071910 Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.
%K A071910 nonn
%O A071910 0,2
%A A071910 _N. J. A. Sloane_, Jun 13 2002