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

Author

Ilya Gutkovskiy, Mar 22 2016

Keywords

• sign
• easy

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)

Examples

			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.
		

Links

• 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).

Crossrefs

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

Cf. A000290.

Programs

GAP

List([0..50], n -> 3*n^2+2*n-1); # Bruno Berselli, Feb 16 2018

Magma

[3*n^2+2*n-1: n in [0..50]]; // Bruno Berselli, Mar 25 2016

Mathematica

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

Maxima

makelist(3*n^2+2*n-1, n, 0, 50); /* Bruno Berselli, Mar 25 2016 */

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

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