A144459 a(n) = (3*n+1)*(5*n+1).

1, 24, 77, 160, 273, 416, 589, 792, 1025, 1288, 1581, 1904, 2257, 2640, 3053, 3496, 3969, 4472, 5005, 5568, 6161, 6784, 7437, 8120, 8833, 9576, 10349, 11152, 11985, 12848, 13741, 14664, 15617, 16600, 17613, 18656, 19729, 20832, 21965, 23128, 24321, 25544, 26797, 28080

Offset: 0

Author

Paul Curtz, Oct 08 2008

Keywords

• nonn
• easy

Comments

This appears in a "diagonal" scan through the numerators of the fractions of the hydrogen spectrum: A005563(4), A061037(9), A061039(13), etc.

a(n) mod 9 is a sequence of period length 9: repeat 1, 6, 5, 7, 3, 2, 4, 0, 8 (a permutation of A142069).

Links

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

Crossrefs

Cf. A001622, A005563, A016777, A016861, A061037, A061039, A131579, A142069.

Programs

Magma

[(3*n+1)*(5*n+1): n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Mathematica

Table[(3n+1)(5n+1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,24 ,77}, 50] (* Harvey P. Dale, Jul 16 2014 *)

PARI

a(n)=(3*n+1)*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = A016777(n)*A016861(n).

a(n) mod 10 = A131579(n+7).

G.f.: (1+21*x+8*x^2) / (1-x)^3 . - R. J. Mathar, Jul 01 2011

a(0)=1, a(1)=24, a(2)=77, a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Harvey P. Dale, May 02 2015

E.g.f.: (1 + 23*x + 15*x^2)*exp(x). - G. C. Greubel, Sep 20 2018

Sum_{n>=0} 1/a(n) = sqrt(2 + 3/sqrt(5) - sqrt(3 + 6/sqrt(5)))*Pi/(2*sqrt(6)) + sqrt(5)*log(phi)/4 + 5*log(5)/8 - 3*log(3)/4, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023