A239325 - OEIS

%I #23 May 11 2019 09:58:24

%S 1,15,41,79,129,191,265,351,449,559,681,815,961,1119,1289,1471,1665,

%T 1871,2089,2319,2561,2815,3081,3359,3649,3951,4265,4591,4929,5279,

%U 5641,6015,6401,6799,7209,7631,8065,8511,8969,9439,9921,10415,10921,11439,11969

%N a(n) = 6*n^2 + 8*n + 1.

%C Binomial transform of 1, 14, 12, 0, 0, 0 (0 continued).

%C Sum_{n>=0} 1/a(n) = (Psi(0,(4+sqrt(10))/6) - Psi(0,(4-sqrt(10))/6))/(2*sqrt(10)) = 1.14373625509612753878..., where Psi(n,k) is the n^th derivative of the digamma function. - _Bruno Berselli_, Mar 16 2014

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

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

%F a(0) = 1, a(1) = 15, a(2) = 41; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = C(n,0) + 14*C(n,1) + 12*C(n,2).

%F a(n) = (A069133(n+1) + A100536(n+1) - A000290(n))/2.

%F a(n) = A139267(n+1) - 1. - _Yuriy Sibirmovsky_, Oct 04 2016

%e a(0) = 1*1 = 1;

%e a(1) = 1*1 + 14*1 = 15;

%e a(2) = 1*1 + 14*2 + 12*1 = 41;

%e a(3) = 1*1 + 14*3 + 12*3 = 79;

%e a(4) = 1*1 + 14*4 + 12*6 = 129; etc.

%t Table[6 n^2 + 8 n + 1, {n, 0, 44}] (* or *)

%t CoefficientList[Series[(1 + 12 x - x^2)/(1 - x)^3, {x, 0, 44}], x] (* _Michael De Vlieger_, Oct 04 2016 *)

%t LinearRecurrence[{3,-3,1},{1,15,41},50] (* _Harvey P. Dale_, May 11 2019 *)

%o (PARI) a(n)=6*n^2+8*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A069133, A100536, A139267.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Mar 16 2014