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%I #65 Nov 20 2023 12:17:15
%S 1,10,25,46,73,106,145,190,241,298,361,430,505,586,673,766,865,970,
%T 1081,1198,1321,1450,1585,1726,1873,2026,2185,2350,2521,2698,2881,
%U 3070,3265,3466,3673,3886,4105,4330,4561,4798,5041,5290,5545,5806,6073,6346,6625
%N a(n) = 3*n^2 - 2.
%C Integers k such that 3*k + 6 is a perfect square. - _Gary Detlefs_, Feb 22 2010
%C Binomial transform of (1, 9, 6, 0, 0, 0, 0, 0, 0, 0, ...). - _Philippe Deléham_, Mar 16 2014
%H G. C. Greubel, <a href="/A100536/b100536.txt">Table of n, a(n) for n = 1..5000</a>
%H A. J. C. Cunningham, <a href="/A056107/a056107.pdf">Factorisation of N and N' = (x^n -+ y^n) / (x -+ y [when x-y=n]</a>, Messenger Math., 54 (1924), 17-21 [Incomplete annotated scanned copy]
%H John Elias, <a href="/A100536/a100536.png">Illustration of Initial Terms: triple diamond configuration</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1)
%F a(n) = a(n-1) + 6*n - 3 for n>1. - _Vincenzo Librandi_, Nov 17 2010
%F G.f.: x*(1+7*x-2*x^2) / (1-x)^3. - _R. J. Mathar_, Oct 03 2011
%F -a(n) = (k-1)^2 + k^2 + (k+1)^2, where k = n*sqrt(-1). - _Bruno Berselli_, Jan 24 2014
%F a(T(n)+1) = T(n+1)^2 + T(n)^2 + T(n-1)^2, where T = A000217. - _Bruno Berselli_, May 14 2014
%F a(n+1) = binomial(n,0) + 9*binomial(n,1) + 6*binomial(n,2). - _Philippe Deléham_, Mar 16 2014
%F a(n) = floor(1/(n*tan(1/n) - 1)). - _Clark Kimberling_, Dec 02 2014
%F E.g.f.: 2 - (2 - 3*x - 3*x^2)*exp(x). - _G. C. Greubel_, Mar 27 2023
%e From _Philippe Deléham_, Mar 16 2014: (Start)
%e a(2)=10 after the evaluation of a(2) = 3*(2^2) - 2 = 3*(4) - 2 = 12 - 2 = 10.
%e a(1) = 1*1 = 1;
%e a(2) = 1*1 + 9*1 = 10;
%e a(3) = 1*1 + 9*2 + 6*1 = 25;
%e a(4) = 1*1 + 9*3 + 6*3 = 46;
%e a(5) = 1*1 + 9*4 + 6*6 = 73; etc. (End)
%t 3*Range[50]^2-2 (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2011 *)
%t CoefficientList[Series[x (1+7x-2x^2)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1},{1,10,25},50] (* _Harvey P. Dale_, Nov 20 2023 *)
%o (PARI) a(n)=3*n^2-2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [3*n^2-2: n in [1..50]]; // _G. C. Greubel_, Mar 27 2023
%o (SageMath) [3*n^2 -2 for n in range(1,51)] # _G. C. Greubel_, Mar 27 2023
%Y Cf. A000217, A000124, A054000.
%K nonn,easy
%O 1,2
%A Tyler J Newman (Tylerjnewman(AT)adelphia.net), Nov 27 2004
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