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%I #36 Jul 02 2022 11:36:59
%S 0,11,38,81,140,215,306,413,536,675,830,1001,1188,1391,1610,1845,2096,
%T 2363,2646,2945,3260,3591,3938,4301,4680,5075,5486,5913,6356,6815,
%U 7290,7781,8288,8811,9350,9905,10476,11063,11666,12285,12920
%N a(n) = n*(8*n+3).
%C Sequence found by reading the line from 0, in the direction 0, 11,..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139272 in the same spiral.
%H G. C. Greubel, <a href="/A139276/b139276.txt">Table of n, a(n) for n = 0..5000</a>
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 8*n^2 + 3*n.
%F Sequences of the form a(n)=8*n^2+c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n)= 3a(n-1)-3a(n-2)+a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - _R. J. Mathar_, May 12 2008
%F a(n) = 16*n+a(n-1)-5 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F From _G. C. Greubel_, Jul 18 2017: (Start)
%F G.f.: x*(5*x + 11)/(1-x)^3.
%F E.g.f.: (8*x^2 + 11*x)*exp(x). (End)
%F Sum_{n>=1} 1/a(n) = 8/9 - (sqrt(2)-1)*Pi/6 - 4*log(2)/3 + sqrt(2)*log(sqrt(2)+1)/3. - _Amiram Eldar_, Mar 17 2022
%e a(1)=16*1+0-5=11; a(2)=16*2+11-5=38; a(3)=16*3+38-5=81. - _Vincenzo Librandi_, Aug 03 2010
%t a[n_]:=n*(8*n+3); a[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)
%t LinearRecurrence[{3,-3,1},{0,11,38},50] (* _Harvey P. Dale_, Jul 02 2022 *)
%o (PARI) a(n)=n*(8*n+3) \\ _Charles R Greathouse IV_, Jun 16 2017
%Y Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139274, A139275, A139277, A139278, A139279, A139280, A139281, A139282.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, Apr 26 2008
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