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

-1, 4, 15, 32, 55, 84, 119, 160, 207, 260, 319, 384, 455, 532, 615, 704, 799, 900, 1007, 1120, 1239, 1364, 1495, 1632, 1775, 1924, 2079, 2240, 2407, 2580, 2759, 2944, 3135, 3332, 3535, 3744, 3959, 4180, 4407, 4640, 4879, 5124, 5375, 5632, 5895, 6164, 6439, 6720, 7007, 7300, 7599

Offset

0,2

Comments

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.

From Bruno Berselli, Mar 25 2016: (Start)

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

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

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

Links

Table of n, a(n) for n=0..50.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quadratic polynomial.

Leo Tavares, Triple Diamond Illustration.

Eric Weisstein's World of Mathematics, Quadratic Polynomial.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

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

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

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

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

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

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

a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - Bruno Berselli, Feb 16 2018

a(n) = A000290(n+1) + 2*A000290(n) - 2. - Leo Tavares, May 28 2023

Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + 3/4. - Amiram Eldar, Jul 20 2023

Example

a(0) = 3*0^2 + 2*0 - 1 = -1;
a(1) = 3*1^2 + 2*1 - 1 =  4;
a(2) = 3*2^2 + 2*2 - 1 = 15;
a(3) = 3*3^2 + 2*3 - 1 = 32, etc.

Mathematica

Table[3 n^2 + 2 n - 1, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]

Prog

(PARI) Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ Michel Marcus, Mar 22 2016
(PARI) lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ Altug Alkan, Mar 25 2016
(PARI) vector(50, n, n--; 3*n^2+2*n-1) \\ Bruno Berselli, Mar 25 2016
(Sage) [3*n^2+2*n-1 for n in (0..50)] # Bruno Berselli, Mar 25 2016
(Maxima) makelist(3*n^2+2*n-1, n, 0, 50); /* Bruno Berselli, Mar 25 2016 */
(Magma) [3*n^2+2*n-1: n in [0..50]]; // Bruno Berselli, Mar 25 2016
(GAP) List([0..50], n -> 3*n^2+2*n-1); # Bruno Berselli, Feb 16 2018

Crossrefs

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

Cf. A000290.

Sequence in context: A121914 A336607 A321490 * A322571 A110341 A317614

Adjacent sequences: A270707 A270708 A270709 * A270711 A270712 A270713

Keyword

sign,easy

Author

Ilya Gutkovskiy, Mar 22 2016

Status

approved