cp's OEIS Frontend

A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.

%I A268685 #27 Apr 30 2023 02:07:49
%S A268685 6,126,630,1950,4680,9576,17556,29700,47250,71610,104346,147186,
%T A268685 202020,270900,356040,459816,584766,733590,909150,1114470,1352736,
%U A268685 1627296,1941660,2299500,2704650,3161106,3673026,4244730,4880700,5585580,6364176,7221456,8162550
%N A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.
%C A268685 a(n) is the total volume of the family of (n+1) rectangular prisms, where the k-th prism has dimensions (3k) X (3k-1) X (3k-2). - _Wesley Ivan Hurt_, Oct 02 2018
%H A268685 Felix Fröhlich, <a href="/A268685/b268685.txt">Table of n, a(n) for n = 0..10000</a>
%H A268685 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A268685 G.f.: -6*(10*x^2 + 16*x + 1)/(x - 1)^5.
%F A268685 a(n) = Sum_{k = 0..n} (3*k + 1)(3*k + 2)(3*k + 3).
%F A268685 Sum {n>=0} 1/a(n) = 2*(sqrt(3)*Pi + 9*log(3) - 14)/15 = 0.1771878254287521...
%F A268685 a(n) mod 6 = 0.
%F A268685 a(n) = 6*A116689(n+1). - _R. J. Mathar_, Jun 07 2016
%F A268685 E.g.f.: 3*exp(x)*(8 + 160*x +256*x^2 + 96*x^3 + 9*x^4)/4. - _Stefano Spezia_, Apr 18 2023
%F A268685 Sum_{n>=0} (-1)^n/a(n) = 28/15 - 8*Pi/(15*sqrt(3)) - 16*log(2)/15. - _Amiram Eldar_, Apr 30 2023
%e A268685 a(0) = 1*2*3 = 6;
%e A268685 a(1) = 1*2*3 + 4*5*6 = 126;
%e A268685 a(2) = 1*2*3 + 4*5*6 + 7*8*9 = 630;
%e A268685 a(3) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
%e A268685 a(4) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
%e A268685 a(5) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576, etc.
%t A268685 Table[3 (n + 1) (n + 2) (3 n + 1) ((3 n + 4)/4), {n, 0, 32}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {6, 126, 630, 1950, 4680}, 32]
%t A268685 CoefficientList[Series[6 (10 x^2 + 16 x + 1) / (1 - x)^5, {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 11 2016 *)
%o A268685 (Magma) [3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // _Vincenzo Librandi_, Feb 11 2016
%o A268685 (PARI) a(n) = 3*(n+1)*(n+2)*(3*n+1)*(3*n+4)/4 \\ _Felix Fröhlich_, Jun 07 2016
%Y A268685 Cf. A000027, A054776, A116689, A135036.
%Y A268685 Trisection of A319014 and A319867.
%K A268685 nonn,easy
%O A268685 0,1
%A A268685 _Ilya Gutkovskiy_, Feb 11 2016