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 A270710 a(n) = 3*n^2 + 2*n - 1. 3

%I

%S -1,4,15,32,55,84,119,160,207,260,319,384,455,532,615,704,799,900,

%T 1007,1120,1239,1364,1495,1632,1775,1924,2079,2240,2407,2580,2759,

%U 2944,3135,3332,3535,3744,3959,4180,4407,4640,4879,5124,5375,5632,5895,6164,6439,6720,7007,7300,7599

%N a(n) = 3*n^2 + 2*n - 1.

%C In general, the ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k, is (k + (p + q - 2*k)*x + (p - q + k)*x^2)/(1 - x)^3.

%C From _Bruno Berselli_, Mar 25 2016: (Start)

%C This sequence and A140676 provide all integer m such that 3*m + 4 is a square.

%C Numbers related to A135713 by A135713(n) = n*a(n) - Sum_{k=0..n-1} a(k).

%C After -1, second bisection of A184005. (End)

%H Ilya Gutkovskiy, <a href="/A270710/a270710.pdf">Examples of the ordinary generating function for the values of quadratic polynomial</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (-1 + 7*x)/(1 - x)^3.

%F E.g.f.: exp(x)*(-1 + 5*x + 3*x^2).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = A033428(n) + A060747(n).

%F a(n) = A045944(n) - 1 = A056109(n) - 2.

%F a(-n) = A140676(n-1), with A140676(-1) = -1.

%F Sum_{n>=0} 1/a(n) = 3*(log(3) - 2)/8 - Pi/(8*sqrt(3)) = -0.564745312278736...

%F a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - _Bruno Berselli_, Feb 16 2018

%e a(0) = 3*0^2 + 2*0 - 1 = -1;

%e a(1) = 3*1^2 + 2*1 - 1 = 4;

%e a(2) = 3*2^2 + 2*2 - 1 = 15;

%e a(3) = 3*3^2 + 2*3 - 1 = 32, etc.

%t Table[3 n^2 + 2 n - 1, {n, 0, 50}]

%t LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]

%o (PARI) Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ _Michel Marcus_, Mar 22 2016

%o (PARI) lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ _Altug Alkan_, Mar 25 2016

%o (PARI) vector(50, n, n--; 3*n^2+2*n-1) \\ _Bruno Berselli_, Mar 25 2016

%o (Sage) [3*n^2+2*n-1 for n in (0..50)] # _Bruno Berselli_, Mar 25 2016

%o (Maxima) makelist(3*n^2+2*n-1, n, 0, 50); /* _Bruno Berselli_, Mar 25 2016 */

%o (MAGMA) [3*n^2+2*n-1: n in [0..50]]; // _Bruno Berselli_, Mar 25 2016

%o (GAP) List([0..50], n -> 3*n^2+2*n-1); # _Bruno Berselli_, Feb 16 2018

%Y Cf. A033428, A045944, A056105, A056109, A060747, A135713, A140676, A184005.

%K sign,easy

%O 0,2

%A _Ilya Gutkovskiy_, Mar 22 2016

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Last modified October 17 19:36 EDT 2018. Contains 316293 sequences. (Running on oeis4.)