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a(n) = n*(9*n-2).
4

%I #41 Apr 22 2024 06:30:36

%S 0,7,32,75,136,215,312,427,560,711,880,1067,1272,1495,1736,1995,2272,

%T 2567,2880,3211,3560,3927,4312,4715,5136,5575,6032,6507,7000,7511,

%U 8040,8587,9152,9735,10336,10955,11592,12247,12920,13611,14320,15047,15792,16555

%N a(n) = n*(9*n-2).

%C For n>0, numbers such that sqrt(a(n)) has the continued fraction {k;[1,1,1,2k]}, where the part in [] is repeated and k is of the form 3m+2 (A016789). - _Bruno Berselli_, May 30 2013

%C For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n-1; {3, 3n-1, 3, 12n-2}]. - _Magus K. Chu_, Sep 18 2022

%H G. C. Greubel, <a href="/A013656/b013656.txt">Table of n, a(n) for n = 0..5000</a>

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

%F a(n+1) = A144454(9*n+7) = A061039(27*n+21). - _Paul Curtz_, Nov 05 2008

%F a(n) = a(n-1) + 18*n - 11 with n>0, a(0)=0. - _Vincenzo Librandi_, Nov 22 2010

%F a(0)=0, a(1)=7, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jul 07 2012

%F From _G. C. Greubel_, Mar 11 2022: (Start)

%F G.f.: x*(7 - 11*x)/(1-x)^3.

%F E.g.f.: x*(7 + 9*x)*exp(x). (End)

%F Sum_{n>=1} 1/a(n) = -(psi(7/9)+gamma)/2 = (A354640-A001620)/2 = 0.22000753... - _R. J. Mathar_, Apr 22 2024

%t Table[n(9n-2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,32},50] (* _Harvey P. Dale_, Jul 07 2012 *)

%o (PARI) a(n)=n*(9*n-2) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [n*(9*n-2): n in [0..60]]; // _G. C. Greubel_, Mar 11 2022

%Y Cf. A010701, A017257, A185019.

%Y Cf. A016789, A061039, A144454.

%K nonn,easy

%O 0,2

%A _David W. Wilson_