login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A270710 a(n) = 3*n^2 + 2*n - 1. 6
-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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

Sequence in context: A322740 A121914 A321490 * A322571 A110341 A317614

Adjacent sequences:  A270707 A270708 A270709 * A270711 A270712 A270713

KEYWORD

sign,easy

AUTHOR

Ilya Gutkovskiy, Mar 22 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 5 16:07 EDT 2020. Contains 335473 sequences. (Running on oeis4.)