A178977 - OEIS

%I #43 Oct 31 2024 06:49:14

%S 5,20,44,77,119,170,230,299,377,464,560,665,779,902,1034,1175,1325,

%T 1484,1652,1829,2015,2210,2414,2627,2849,3080,3320,3569,3827,4094,

%U 4370,4655,4949,5252,5564,5885,6215,6554,6902,7259,7625,8000,8384,8777,9179,9590,10010

%N a(n) = (3*n+2)*(3*n+5)/2.

%C Companion to A145910.

%H Vincenzo Librandi, <a href="/A178977/b178977.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = a(n-1) + 6 + 9*n.

%F a(n) = A178971(3*n+2).

%F a(n) = A145910(n) + 3 + 3*n = A145910(n) + A008585(n+1).

%F a(n) = A168233(n+1)*A168300(n+1).

%F G.f.: (-5-5*x+x^2)/(x-1)^3. [Adapted to the offset by _Bruno Berselli_, Apr 14 2011]

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Apr 19 2013

%F From _Amiram Eldar_, Mar 10 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 1/3.

%F Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) - 1/3 - 4*log(2)/9. (End)

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

%F E.g.f.: exp(x)*exp(x)*(5 + 15*x + 9*x^2/2).

%F a(n) = A016789(n)*A016789(n+1)/2. (End)

%p A178977:=n->(3*n+2)*(3*n+5)/2: seq(A178977(n), n=0..50); # _Wesley Ivan Hurt_, Oct 23 2014

%t f[n_] := (3 n + 2) (3 n + 5)/2; Array[f, 45, 0]

%t LinearRecurrence[{3,-3,1},{5,20,44},50] (* _Harvey P. Dale_, Apr 19 2013 *)

%o (Magma) [n*(n+3)/2: n in [2..135 by 3]]; // _Bruno Berselli_, Apr 14 2011

%o (PARI) a(n)=(3*n+2)*(3*n+5)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A008585, A145910, A168233, A168300, A016789, A178971, A235332.

%K nonn,easy,changed

%O 0,1

%A _Paul Curtz_, Jan 02 2011