A379211 List of positive integers that are congruent to {2, 7, 8, 13} mod 15.

2, 7, 8, 13, 17, 22, 23, 28, 32, 37, 38, 43, 47, 52, 53, 58, 62, 67, 68, 73, 77, 82, 83, 88, 92, 97, 98, 103, 107, 112, 113, 118, 122, 127, 128, 133, 137, 142, 143, 148, 152, 157, 158, 163, 167, 172, 173, 178, 182, 187, 188, 193, 197, 202, 203, 208, 212, 217, 218, 223, 227, 232, 233, 238, 242, 247, 248, 253, 257, 262

Offset: 1

Author

Peter Bala, Dec 18 2024

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A001622, A072703, A151972, A204542, A315211, A379210.

Programs

Maple

a := proc(n) option remember;
      `if`(n < 5, [0, 2, 7, 8, 13][n+1], 15 + a(n-4))
     end:
seq(a(n), n = 1..70);

Mathematica

LinearRecurrence[{1, 0, 0, 1, -1}, {2, 7, 8, 13, 17}, 70] (* Amiram Eldar, Dec 24 2024 *)

Formula

a(n) = 15 + a(n-4); a(n) = - a(1-n).

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 6.

G.f.: x*(x^2 + 3*x + 1)*(2*x^2 - x + 2)/((1 + x)*(1 - x)^2*(1 + x^2)).

a(n)^2 = 15 * A379210(n) + 4.

For n >= 2, a(n-1) + a(n+1) = A072703(n).

It appears that a(n) + a(n+1) = (3/2) * A315211(n).

E.g.f.: (8 - 3*cos(x) + 5*(3*x - 1)*cosh(x) + 3*sin(x) + 5*(3*x - 2)*sinh(x))/4. - Stefano Spezia, Dec 23 2024

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(5*sqrt(3)*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 24 2024