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A098931 a(0) = 1, a(n) = 1 + 2*3 + 4*5 + 6*7 + ... + (2n)*(2n+1) for n > 0.

%I A098931 #20 Sep 08 2022 08:45:15
%S A098931 1,7,27,69,141,251,407,617,889,1231,1651,2157,2757,3459,4271,5201,
%T A098931 6257,7447,8779,10261,11901,13707,15687,17849,20201,22751,25507,28477,
%U A098931 31669,35091,38751,42657,46817,51239,55931,60901,66157,71707,77559,83721
%N A098931 a(0) = 1, a(n) = 1 + 2*3 + 4*5 + 6*7 + ... + (2n)*(2n+1) for n > 0.
%C A098931 If a(n) = a0, a1, a2, a3, ... then Sum(a(n))= a0, a0+a1, a0+a1+a2, a0+a1+a2+a3, ...
%H A098931 Robert Israel, <a href="/A098931/b098931.txt">Table of n, a(n) for n = 0..10000</a>
%H A098931 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A098931 a(n) = 1 + 3*n^2 + n*(5 + 4*n^2)/3.
%F A098931 G.f.: (1 + 3*x + 5*x^2 - x^3)/(1-x)^4.
%F A098931 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Vincenzo Librandi_, Jul 28 2015
%F A098931 From _Robert Israel_, Jul 28 2015: (Start)
%F A098931 E.g.f.: (1+6*x+7*x^2+(4/3)*x^3)*exp(x).
%F A098931 a(n) = 1 + Sum(A068377(i),i=1..n+1). (End)
%e A098931 a(0) = 1;
%e A098931 a(1) = 1 + 2*3 = 7;
%e A098931 a(2) = 1 + 2*3 + 4*5 = 27, etc.
%p A098931 seq((4/3)*n^3+3*n^2+(5/3)*n+1, n=0..100); # _Robert Israel_, Jul 28 2015
%t A098931 Table[1 + 3 n^2 + n (5 + 4 n^2)/3, {n, 0, 40}] (* _Robert G. Wilson v_, Oct 23 2004 *)
%t A098931 LinearRecurrence[{4, -6, 4, -1}, {1, 7, 27, 69}, 40] (* _Vincenzo Librandi_, Jul 28 2015 *)
%o A098931 (Magma) [1+3*n^2+n*(5+4*n^2)/3: n in [0..40]]; // _Vincenzo Librandi_, Jul 28 2015
%o A098931 (PARI) a(n)=n*(4*n^2+9*n+5)/3+1 \\ _Charles R Greathouse IV_, Jul 28 2015
%Y A098931 Cf. A068377.
%K A098931 nonn,easy
%O A098931 0,2
%A A098931 _Miklos Kristof_, Oct 20 2004
%E A098931 Edited and extended by _Robert G. Wilson v_, Oct 23 2004