login
a(n) = (7*n+1)*(7*n+6).
1

%I #32 Oct 25 2024 07:44:48

%S 6,104,300,594,986,1476,2064,2750,3534,4416,5396,6474,7650,8924,10296,

%T 11766,13334,15000,16764,18626,20586,22644,24800,27054,29406,31856,

%U 34404,37050,39794,42636,45576,48614,51750,54984,58316,61746,65274,68900,72624,76446

%N a(n) = (7*n+1)*(7*n+6).

%H T. D. Noe, <a href="/A001526/b001526.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) = 98*n + a(n-1) with a(0)=6. - _Vincenzo Librandi_, Nov 12 2010

%F From _Amiram Eldar_, Feb 19 2023: (Start)

%F a(n) = A016993(n)*A017053(n).

%F Sum_{n>=0} 1/a(n) = cot(Pi/7)*Pi/35 = 0.186388....

%F Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/7)*cos(sqrt(29)*Pi/14).

%F Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/7)*cos(sqrt(3/7)*Pi/2). (End)

%F G.f.: -2*(3+43*x+3*x^2)/(x-1)^3. - _R. J. Mathar_, Apr 23 2024

%F From _Elmo R. Oliveira_, Oct 25 2024: (Start)

%F E.g.f.: exp(x)*(6 + 49*x*(2 + x)).

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

%t a[n_] := (7*n + 1)*(7*n + 6); Array[a, 40, 0] (* _Amiram Eldar_, Feb 19 2023 *)

%o (PARI) a(n)=(7*n+1)*(7*n+6) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A016993, A017053.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_