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A177072 a(n) = (9*n+2)*(9*n+7).

%I A177072 #33 Oct 19 2024 09:33:41
%S A177072 14,176,500,986,1634,2444,3416,4550,5846,7304,8924,10706,12650,14756,
%T A177072 17024,19454,22046,24800,27716,30794,34034,37436,41000,44726,48614,
%U A177072 52664,56876,61250,65786,70484,75344,80366,85550,90896,96404,102074,107906,113900,120056
%N A177072 a(n) = (9*n+2)*(9*n+7).
%C A177072 Cf. comment of _Reinhard Zumkeller_ in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 67. - _Bruno Berselli_, Aug 24 2010
%H A177072 Vincenzo Librandi, <a href="/A177072/b177072.txt">Table of n, a(n) for n = 0..1000</a>
%H A177072 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A177072 a(n) = 162*n + a(n-1) with n > 0, a(0)=14.
%F A177072 From _Vincenzo Librandi_, Apr 08 2013: (Start)
%F A177072 G.f.: 2*(7+67*x+7*x^2)/(1-x)^3.
%F A177072 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A177072 From _Amiram Eldar_, Feb 19 2023: (Start)
%F A177072 a(n) = A017185(n)*A017245(n).
%F A177072 Sum_{n>=0} 1/a(n) = cot(2*Pi/9)*Pi/45.
%F A177072 Product_{n>=0} (1 - 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(29)*Pi/18).
%F A177072 Product_{n>=0} (1 + 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(21)*Pi/18). (End)
%F A177072 E.g.f.: exp(x)*(14 + 81*x*(2 + x)). - _Elmo R. Oliveira_, Oct 18 2024
%t A177072 CoefficientList[Series[2(7 + 67 x + 7 x^2)/(1-x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 08 2013 *)
%t A177072 Table[(9*n + 2)*(9*n + 7), {n, 0, 40}] (* _Amiram Eldar_, Feb 19 2023 *)
%t A177072 LinearRecurrence[{3,-3,1},{14,176,500},50] (* _Harvey P. Dale_, Jun 10 2023 *)
%o A177072 (Magma) I:=[14, 176, 500]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Apr 08 2013
%o A177072 (PARI) a(n)=(9*n+2)*(9*n+7) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A177072 Cf. A002061, A017185, A017245, A177059.
%K A177072 nonn,easy
%O A177072 0,1
%A A177072 _Vincenzo Librandi_, May 31 2010
%E A177072 Edited by _N. J. A. Sloane_, Jun 22 2010