A247792 a(n) = 9*n^2 + 1.

1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, 2026, 2305, 2602, 2917, 3250, 3601, 3970, 4357, 4762, 5185, 5626, 6085, 6562, 7057, 7570, 8101, 8650, 9217, 9802, 10405, 11026, 11665, 12322, 12997, 13690, 14401, 15130, 15877, 16642, 17425, 18226, 19045, 19882

Offset: 0

Author

Karl V. Keller, Jr., Sep 23 2014

Keywords

• nonn
• easy

Comments

The odd numbers of the form 9n^2 + 1 are listed in A158591 (36n^2 + 1).

The even numbers of the form 9n^2 + 1 are given by 36x^2 - 36x + 10, x > 0.

Every integer n>0 give three perfect squares and consecutives from 2^2. The formulas for each value of n are: a(n)-6n, a(n)-1 and a(n)+6n. - Miquel Cerda, Sep 19 2016

These squares are, for n>0, A000290(3*n-1), 3*n and (3n+1) and the sum of them is 3*a(n) - 1. - Miquel Cerda, Sep 26 2016

Examples

			a(1) = (2^2 + 4^2)/2 = 3^2 + 1 = 10, a(2) = (5^2 + 7^2)/2 = 6^2 + 1 = 37, a(3) = (8^2 + 10^2)/2 = 9^2 + 1 = 82. - _Miquel Cerda_, Jun 25 2016
		

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A016766, A158591 (36n^2 + 1), A156226 (primes of the form 9n^2 + 1).

Cf. also A000290.

Programs

Magma

[9*n^2+1: n in [0..60]]; // Vincenzo Librandi, Sep 27 2014

Maple

A247792:=n->9*n^2 + 1: seq(A247792(n), n=0..80); # Wesley Ivan Hurt, Jun 25 2016

Mathematica

(3Range[0, 49])^2 + 1 (* Alonso del Arte, Sep 24 2014 *)
CoefficientList[Series[(1 + 7 x + 10 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 27 2014 *)

PARI

a(n)=9*n^2+1 \\ Charles R Greathouse IV, Sep 26 2014

Python

for n in range (0,100): print (9*n**2+1)

Formula

a(n) = (3n)^2 + 1 = 9n^2 + 1 = A016766(n) + 1.

G.f.: (1+7*x+10*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 27 2014

a(n) = ((3n-1)^2 + (3n+1)^2)/2 = (A016790(n-1) + A016778(n))/2. - Miquel Cerda, Jun 25 2016

From Ilya Gutkovskiy, Jun 25 2016: (Start)

E.g.f.: (1 + 9*x + 9*x^2)*exp(x).

Dirichlet g.f.: 9*zeta(s-2) + zeta(s).

Sum_{n>=0} 1/a(n) = (3 + Pi*coth(Pi/3))/6. (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Jun 25 2016

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/3)*csch(Pi/3))/2. - Amiram Eldar, Jul 15 2020

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/3)*sinh(sqrt(2)*Pi/3).

Product_{n>=1} (1 - 1/a(n)) = (Pi/3)*csch(Pi/3). (End)