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 A139274 a(n) = n*(8*n-1). 10
 0, 7, 30, 69, 124, 195, 282, 385, 504, 639, 790, 957, 1140, 1339, 1554, 1785, 2032, 2295, 2574, 2869, 3180, 3507, 3850, 4209, 4584, 4975, 5382, 5805, 6244, 6699, 7170, 7657, 8160, 8679, 9214, 9765, 10332, 10915, 11514, 12129, 12760, 13407, 14070, 14749 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the triangular numbers A000217. 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..5000 Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019). Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA 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 a(n) = 16*n + a(n-1) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010 a(n) = (1/3) * Sum_{i=n..(7*n-1)} i. - Wesley Ivan Hurt, Dec 04 2016 From G. C. Greubel, Jul 18 2017: (Start) G.f.: x*(9*x+7)/(1-x)^3. E.g.f.: (8*x^2 + 7*x)*exp(x). (End) EXAMPLE 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 MAPLE A139274:=n->n*(8*n-1): seq(A139274(n), n=0..100); # Wesley Ivan Hurt, Dec 04 2016 MATHEMATICA s=0; lst={s}; Do[s+=n++ +7; AppendTo[lst, s], {n, 0, 8!, 16}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *) CoefficientList[Series[x (9 x + 7)/(1 - x)^3, {x, 0, 43}], x] (* Michael De Vlieger, Jan 11 2020 *) PROG (MAGMA) [n*(8*n-1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016 (PARI) a(n)=n*(8*n-1) \\ Charles R Greathouse IV, Jun 17 2017 CROSSREFS Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139275, A139276, A139278, A139279, A139280, A139281, A139282. Sequence in context: A063128 A063148 A116283 * A086640 A083474 A030440 Adjacent sequences:  A139271 A139272 A139273 * A139275 A139276 A139277 KEYWORD easy,nonn AUTHOR Omar E. Pol, Apr 26 2008 STATUS approved

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Last modified October 28 04:06 EDT 2020. Contains 338048 sequences. (Running on oeis4.)