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A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3.

%I A269342 #15 Sep 08 2022 08:46:15
%S A269342 3,26,85,196,375,638,1001,1480,2091,2850,3773,4876,6175,7686,9425,
%T A269342 11408,13651,16170,18981,22100,25543,29326,33465,37976,42875,48178,
%U A269342 53901,60060,66671,73750,81313,89376,97955,107066,116725,126948,137751,149150,161161
%N A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3.
%H A269342 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A269342 G.f.: (3 + 14*x - x^2)/(1 - x)^4.
%F A269342 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A269342 a(n) = Sum_{k=0..n} (8*k^2 + 12*k + 3).
%F A269342 Sum_{n>=0} 1/a(n) = 3*(80*log(2) + 5*Pi - 48)/175 = 0.397024075075621559...
%e A269342 a(0) = 0*2 + 1*3 = 3;
%e A269342 a(1) = 0*2 + 1*3 + 2*4 + 3*5 = 26;
%e A269342 a(2) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 = 85;
%e A269342 a(3) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 = 196;
%e A269342 a(4) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 + 8*10 + 9*11 = 375, etc.
%t A269342 Table[(n + 1) (2 n + 1) (4 n + 9)/3, {n, 0, 38}]
%t A269342 LinearRecurrence[{4, -6, 4, -1}, {3, 26, 85, 196}, 39]
%t A269342 Table[Sum[8 k^2 + 12 k + 3, {k, 0, n}], {n, 0, 38}]
%o A269342 (PARI) Vec((3 + 14*x - x^2)/(1 - x)^4 + O(x^50)) \\ _Michel Marcus_, Feb 25 2016
%o A269342 (Magma) [(n+1)*(2*n+1)*(4*n+9)/3: n in [0..50]]; // _Vincenzo Librandi_, Feb 25 2016
%Y A269342 Cf. A000466, A005563, A033996, A195241.
%K A269342 nonn,easy
%O A269342 0,1
%A A269342 _Ilya Gutkovskiy_, Feb 24 2016