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%I #67 Apr 01 2024 11:42:05
%S 0,7,30,69,124,195,282,385,504,639,790,957,1140,1339,1554,1785,2032,
%T 2295,2574,2869,3180,3507,3850,4209,4584,4975,5382,5805,6244,6699,
%U 7170,7657,8160,8679,9214,9765,10332,10915,11514,12129,12760,13407,14070,14749
%N a(n) = n*(8*n-1).
%C Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the triangular numbers A000217.
%C Polygonal number connection: 2*P_n + 5*S_n where P_n is the n-th pentagonal number and S_n is the n-th square. - _William A. Tedeschi_, Sep 12 2010
%H G. C. Greubel, <a href="/A139274/b139274.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 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) - 9 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F a(n) = (1/3) * Sum_{i=n..(7*n-1)} i. - _Wesley Ivan Hurt_, Dec 04 2016
%F From _G. C. Greubel_, Jul 18 2017: (Start)
%F G.f.: x*(9*x+7)/(1-x)^3.
%F E.g.f.: (8*x^2 + 7*x)*exp(x). (End)
%F Sum_{n>=1} 1/a(n) = 4*log(2) + sqrt(2)*log(sqrt(2)+1) - (sqrt(2)+1)*Pi/2. - _Amiram Eldar_, Mar 18 2022
%e a(1) = 16*1 + 0 - 9 = 7; a(2) = 16*2 + 7 - 9 = 30; a(3) = 16*3 + 30 - 9 = 69. - _Vincenzo Librandi_, Aug 03 2010
%p A139274:=n->n*(8*n-1): seq(A139274(n), n=0..100); # _Wesley Ivan Hurt_, Dec 04 2016
%t CoefficientList[Series[x (9 x + 7)/(1 - x)^3, {x, 0, 43}], x] (* _Michael De Vlieger_, Jan 11 2020 *)
%t Table[n(8n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,30},50] (* _Harvey P. Dale_, Apr 01 2024 *)
%o (Magma) [n*(8*n-1) : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 04 2016
%o (PARI) a(n)=n*(8*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, Apr 26 2008
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