OFFSET
0,2
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
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).
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
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)
EXAMPLE
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
MAPLE
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 *)
PROG
(Python) for n in range (0, 100): print (9*n**2+1)
(PARI) a(n)=9*n^2+1 \\ Charles R Greathouse IV, Sep 26 2014
(Magma) [9*n^2+1: n in [0..60]]; // Vincenzo Librandi, Sep 27 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karl V. Keller, Jr., Sep 23 2014
STATUS
approved