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A063488 a(n) = (2*n-1)*(n^2 -n +2)/2.

%I A063488 #43 Jul 18 2025 04:07:18
%S A063488 1,6,20,49,99,176,286,435,629,874,1176,1541,1975,2484,3074,3751,4521,
%T A063488 5390,6364,7449,8651,9976,11430,13019,14749,16626,18656,20845,23199,
%U A063488 25724,28426,31311,34385,37654,41124,44801,48691,52800,57134,61699,66501,71546,76840
%N A063488 a(n) = (2*n-1)*(n^2 -n +2)/2.
%C A063488 Sum of two consecutive terms of A006003(n) = n*(n^2+1)/2. a(n) = A006003(n-1) + A006003(n). - _Alexander Adamchuk_, Jun 03 2006
%C A063488 If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 5-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%H A063488 Harry J. Smith, <a href="/A063488/b063488.txt">Table of n, a(n) for n = 1..1000</a>
%H A063488 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A063488 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A063488 T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).
%H A063488 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A063488 G.f.: (1 + x)*(1 + x + x^2)/(1 - x)^4. - _Jaume Oliver Lafont_, Aug 30 2009
%F A063488 a(n) = A000217(A000217(n)) - A000217(A000217(n-2)). - _Bruno Berselli_, Oct 14 2016
%F A063488 E.g.f.: (-2 + 4*x + 3*x^2 + 2*x^3)*exp(x)/2 + 1. - _G. C. Greubel_, Dec 01 2017
%t A063488 Table[(2 n - 1) (n^2 - n + 2)/2, {n, 1, 40}] (* _Bruno Berselli_, Oct 14 2016 *)
%t A063488 LinearRecurrence[{4,-6,4,-1}, {1,6,20,49}, 50] (* _G. C. Greubel_, Dec 01 2017 *)
%o A063488 (PARI) a(n) = { (2*n - 1)*(n^2 - n + 2)/2 } \\ _Harry J. Smith_, Aug 23 2009
%o A063488 (PARI) my(x='x+O('x^30)); Vec(serlaplace((-2 + 4*x + 3*x^2 + 2*x^3)*exp(x)/2 + 1)) \\ _G. C. Greubel_, Dec 01 2017
%o A063488 (Magma) [(2*n-1)*(n^2 -n +2)/2: n in [1..30]]; // _G. C. Greubel_, Dec 01 2017
%Y A063488 1/12*t*n*(2*n^2 - 3*n + 1) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
%Y A063488 Cf. A000217, A006003.
%Y A063488 Partial sums of A005918.
%K A063488 nonn,easy
%O A063488 1,2
%A A063488 _N. J. A. Sloane_, Aug 01 2001