A028375 Squares of (odd numbers not divisible by 5).

1, 9, 49, 81, 121, 169, 289, 361, 441, 529, 729, 841, 961, 1089, 1369, 1521, 1681, 1849, 2209, 2401, 2601, 2809, 3249, 3481, 3721, 3969, 4489, 4761, 5041, 5329, 5929, 6241, 6561, 6889, 7569, 7921, 8281, 8649, 9409, 9801, 10201, 10609, 11449, 11881, 12321

Offset: 1

Author

ems (nibor(AT)ix.netcom.com)

Keywords

• nonn
• easy

Comments

Catalan stated empirically that the triple of any odd square not divisible by 5 is a sum of squares of three primes other than 2 and 3. - Jonathan Vos Post, Mar 03 2010 [Reference?]

Links

• Rodolfo Ruiz-Huidobro, Table of n, a(n) for n = 1..250
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Programs

Mathematica

Select[Range[1, 191, 2], Mod[#, 5] != 0 &]^2 (* or *) LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {1, 9, 49, 81, 121, 169, 289, 361, 441}, 50] (* Harvey P. Dale, Feb 26 2017 *)
Complement[2Range[100] - 1, 5Range[20]]^2 (* Alonso del Arte, Dec 23 2019 *)

Scala

((1 to 99 by 2).diff(5 to 100 by 5)).map(n => (n * n)) // Alonso del Arte, Dec 23 2019

Formula

a(n) = (A045572(n))^2.

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9). - R. J. Mathar, Sep 22 2009

G.f.: x*(1 + 8*x + 40*x^2 + 32*x^3 + 38*x^4 + 32*x^5 + 40*x^6 + 8*x^7 + x^8)/((1 + x)^2 * (x^2 + 1)^2 * (1 - x)^3). - R. J. Mathar, Sep 22 2009

Sum_{n>=1} 1/a(n) = 3*Pi^2/25. - Amiram Eldar, Dec 19 2020

Extensions

Definition corrected by R. J. Mathar, Sep 22 2009