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A177071 a(n) = (7*n + 3)*(7*n + 4).

%I A177071 #30 Oct 28 2024 17:39:10
%S A177071 12,110,306,600,992,1482,2070,2756,3540,4422,5402,6480,7656,8930,
%T A177071 10302,11772,13340,15006,16770,18632,20592,22650,24806,27060,29412,
%U A177071 31862,34410,37056,39800,42642,45582,48620,51756,54990,58322,61752,65280,68906,72630,76452
%N A177071 a(n) = (7*n + 3)*(7*n + 4).
%C A177071 Cf. Zumkeller's contribution in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2, therefore a(n) = 49*A002061(n+1) - 37. - _Bruno Berselli_, Aug 24 2010
%H A177071 Harvey P. Dale, <a href="/A177071/b177071.txt">Table of n, a(n) for n = 0..1000</a>
%H A177071 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A177071 a(n) = 98*n + a(n-1) with n > 0, a(0)=12.
%F A177071 From _Harvey P. Dale_, Oct 09 2011: (Start)
%F A177071 a(0)=12, a(1)=110, a(2)=306, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A177071 G.f.: -((2*(x+6)*(6*x+1))/(x-1)^3). (End)
%F A177071 From _Amiram Eldar_, Feb 19 2023: (Start)
%F A177071 a(n) = A017017(n)*A017029(n).
%F A177071 Sum_{n>=0} 1/a(n) = tan(Pi/14)*Pi/7.
%F A177071 Product_{n>=0} (1 - 1/a(n)) = sec(Pi/14)*cos(sqrt(5)*Pi/14).
%F A177071 Product_{n>=0} (1 + 1/a(n)) = sec(Pi/14)*cosh(sqrt(3)*Pi/14). (End)
%F A177071 From _Elmo R. Oliveira_, Oct 27 2024: (Start)
%F A177071 E.g.f.: exp(x)*(12 + 49*x*(2 + x)).
%F A177071 a(n) = 2*A061792(n). (End)
%t A177071 Table[(7n+3)(7n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{12,110,306},40] (* _Harvey P. Dale_, Oct 09 2011 *)
%o A177071 (PARI) a(n)=2*binomial(7*n+4,2) \\ _Charles R Greathouse IV_, Jan 11 2012
%Y A177071 Cf. A002061, A017017, A017029, A061792, A177059.
%K A177071 nonn,easy
%O A177071 0,1
%A A177071 _Vincenzo Librandi_, May 31 2010
%E A177071 Edited by _N. J. A. Sloane_, Jun 22 2010