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%I #44 Sep 08 2022 08:45:33
%S 0,5,26,63,116,185,270,371,488,621,770,935,1116,1313,1526,1755,2000,
%T 2261,2538,2831,3140,3465,3806,4163,4536,4925,5330,5751,6188,6641,
%U 7110,7595,8096,8613,9146,9695,10260,10841,11438,12051,12680
%N a(n) = n*(8*n - 3).
%C Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139277 in the same spiral.
%C Also, sequence of numbers of the form d*A000217(n-1) + 5*n with generating functions x*(5+(d-5)*x)/(1-x)^3; the inverse binomial transform is 0,5,d,0,0,.. (0 continued). See Crossrefs. - _Bruno Berselli_, Feb 11 2011
%C Even decagonal numbers divided by 2. - _Omar E. Pol_, Aug 19 2011
%H G. C. Greubel, <a href="/A139273/b139273.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 <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) = 3*a(n-1) - 3*a(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) - 11 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F From _Bruno Berselli_, Feb 11 2011: (Start)
%F G.f.: x*(5 + 11*x)/(1 - x)^3.
%F a(n) = 4*A000217(n) + A051866(n). (End)
%F a(n) = A028994(n)/2. - _Omar E. Pol_, Aug 19 2011
%F a(0)=0, a(1)=5, a(2)=26; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Feb 02 2012
%F E.g.f.: (8*x^2 + 5*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F Sum_{n>=1} 1/a(n) = 4*log(2)/3 - (sqrt(2)-1)*Pi/6 - sqrt(2)*arccoth(sqrt(2))/3. - _Amiram Eldar_, Jul 03 2020
%t Table[n (8 n - 3), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 26}, 40] (* _Harvey P. Dale_, Feb 02 2012 *)
%o (Magma) [ n*(8*n-3) : n in [0..40] ]; // _Bruno Berselli_, Feb 11 2011
%o (PARI) a(n)=n*(8*n-3) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%Y Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Apr 26 2008
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