A179436 - OEIS

%I #46 Oct 31 2024 06:49:22

%S 7,25,52,88,133,187,250,322,403,493,592,700,817,943,1078,1222,1375,

%T 1537,1708,1888,2077,2275,2482,2698,2923,3157,3400,3652,3913,4183,

%U 4462,4750,5047,5353,5668,5992,6325,6667,7018,7378,7747,8125,8512,8908,9313,9727,10150

%N a(n) = (3*n+7)*(3*n+2)/2.

%C Trisection of A055998.

%H Vincenzo Librandi, <a href="/A179436/b179436.txt">Table of n, a(n) for n = 0..10000</a>

%H Leo Tavares, <a href="/A179436/a179436.jpg">Illustration: Triangulated Triangles</a>.

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

%F G.f.: (-7-4*x+2*x^2)/(x-1)^3.

%F a(n) = a(n-1) + 9*(n+1) = (14 + 27*n + 9*n^2)/2.

%F a(n) = 2*a(n-1) - a(n-2) + 9.

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

%F a(n) mod 9 = A153466(n) mod 9 = 7.

%F Sum_{n>=0} 1/a(n) = 1/2-2*Pi*sqrt(3)/45 = 0.2581600... - _R. J. Mathar_, Apr 07 2011

%F a(n) = A133694(n+1) + 6*A000217(n+1). - _Leo Tavares_, Mar 24 2022

%F Sum_{n>=0} (-1)^n/a(n) = 3/10 - 4*log(2)/15. - _Amiram Eldar_, Mar 27 2022

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

%F E.g.f.: exp(x)*(7 + 18*x + 9*x^2/2).

%F a(n) = A016777(n+2)*A016789(n)/2. (End)

%p A179436:=n->(3*n+7)*(3*n+2)/2: seq(A179436(n), n=0..100); # _Wesley Ivan Hurt_, Apr 24 2017

%t a[n_] := (3*n + 7)*(3*n + 2)/2; Array[a, 50, 0] (* _Amiram Eldar_, Mar 27 2022 *)

%o (Magma) [(3*n+7)*(3*n+2)/2: n in [0..50]]; // _Vincenzo Librandi_, Aug 04 2011

%o (PARI) a(n)=(3*n+7)*(3*n+2)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000217, A016777, A016789, A055998, A133694, A153466.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Jan 12 2011