|
%I #61 Oct 14 2023 11:10:13
%S 0,0,6,42,156,420,930,1806,3192,5256,8190,12210,17556,24492,33306,
%T 44310,57840,74256,93942,117306,144780,176820,213906,256542,305256,
%U 360600,423150,493506,572292,660156,757770,865830,985056,1116192,1260006,1417290,1588860
%N a(n) = n*(n+1)*(n*(n+1)+1).
%C Partial sums of A061804. - _Bruno Berselli_, Feb 10 2017
%H Vincenzo Librandi, <a href="/A169938/b169938.txt">Table of n, a(n) for n = -1..1000</a>
%H Nathan Kaplan, Susie Kimport, Rachel Lawrence, Luke Peilen, and Max Weinreich, <a href="https://doi.org/10.1007/s00022-017-0391-1">Counting arcs in projective planes via Glynn’s algorithm</a>, J. Geom. 108, No. 3 (2017), 1013-1029, Th. 1.4, C_2.
%H Daniel Poveda Parrilla, <a href="/A169938/a169938.png">Illustration of initial terms seen as cuboids</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n+1) = a(n) + 2*(n+1)*(2*(n+1)^2+1). - _Robert Munafo_, Jul 27 2010
%F G.f.: 6*x*(1 + 2*x + x^2)/(1-x)^5. - _Vincenzo Librandi_, Dec 18 2012, corrected Aug 29 2022
%F From _Daniel Poveda Parrilla_, Jun 08 2017 and Jun 11 2017: (Start)
%F a(n) = A002378(n)*A002061(n+1) for n > -1.
%F a(n) = A002061(A002061(n+1)) - 1. (End)
%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Wesley Ivan Hurt_, Jan 25 2022
%F a(n) = 6*A006325(n+1). - _R. J. Mathar_, Aug 29 2022
%F Sum_{n>=1} 1/a(n) = 2 - tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - _Amiram Eldar_, Sep 22 2022
%p A169938 := proc(n)
%p n*(n+1)*(n*(n+1)+1);
%p end proc:
%p seq(A169938(n),n=-1..40) ; # _R. J. Mathar_, Aug 29 2022
%t CoefficientList[Series[6*x^2(1 + 2*x + x^2)/(1-x)^5,{x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 18 2012 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,0,6,42,156},40] (* _Harvey P. Dale_, Oct 14 2023 *)
%o (Magma) [n*(n+1)*(n*(n+1)+1): n in [-1..40]]; // _Vincenzo Librandi_, Dec 18 2012
%o (PARI) a(n) = n + 2*n^2 + 2*n^3 + n^4; \\ _Altug Alkan_, Feb 10 2017
%o (PARI) a(n) = n*=n+1;n*=n+1 \\ _David A. Corneth_, Jun 11 2017
%o (Python)
%o def A169938(n): return n*(n*(n*(n + 2) + 2) + 1) # _Chai Wah Wu_, Aug 29 2022
%Y A variant of A176780.
%Y Cf. A002061, A002378, A061804.
%Y A lower bound on A082986.
%K nonn,easy
%O -1,3
%A _N. J. A. Sloane_, Jul 25 2010, based on an email from Terry Stickel
%E The terms were also computed by _Robert Munafo_, Jul 25 2010
|
