A157870 - OEIS

%I #34 Oct 31 2024 06:49:18

%S 2,30,90,182,306,462,650,870,1122,1406,1722,2070,2450,2862,3306,3782,

%T 4290,4830,5402,6006,6642,7310,8010,8742,9506,10302,11130,11990,12882,

%U 13806,14762,15750,16770,17822,18906,20022,21170,22350,23562,24806,26082,27390,28730,30102,31506,32942

%N a(n) = (4*n+1)*(4*n+2) = (4*n+2)!/(4*n)!.

%H Vincenzo Librandi, <a href="/A157870/b157870.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A002378(4*n+1) = 2*A014634(n). - _R. J. Mathar_, Mar 11 2009

%F From _Vincenzo Librandi_, Jul 10 2012: (Start)

%F G.f.: 2*(1+12*x+3*x^2)/(1-x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

%F From _Amiram Eldar_, Mar 01 2022: (Start)

%F Sum_{n>=0} 1/a(n) = Pi/8 + log(2)/4.

%F Sum_{n>=0} (-1)^n/a(n) = ((sqrt(2)-1)*Pi + sqrt(2)*log((2+sqrt(2))/(2-sqrt(2))))/8. (End)

%F From _Elmo R. Oliveira_, Oct 30 2024: (Start)

%F E.g.f.: 2*exp(x)*(1 + 14*x + 8*x^2).

%F a(n) = A016813(n)*A016825(n). (End)

%t Table[(4n+1)*(4n+2),{n,0,50}] (* _Vincenzo Librandi_, Jul 10 2012 *)

%o (Magma) (4*n+1)*(4*n+2); // _Vincenzo Librandi_ Jul 10 2012

%o (PARI) a(n)=(4*n+1)*(4*n+2) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A002378, A014634, A016813, A016825, A157870.

%K nonn,easy

%O 0,1

%A SUNKU Sai Swaroop (sai2020(AT)gmail.com), Mar 08 2009

%E Definition corrected and sequence extended by _R. J. Mathar_, Mar 11 2009