A248474 Numbers congruent to 13 or 17 mod 30.

13, 17, 43, 47, 73, 77, 103, 107, 133, 137, 163, 167, 193, 197, 223, 227, 253, 257, 283, 287, 313, 317, 343, 347, 373, 377, 403, 407, 433, 437, 463, 467, 493, 497, 523, 527, 553, 557, 583, 587, 613, 617, 643, 647, 673, 677, 703, 707, 733, 737, 763, 767, 793, 797

Offset: 1

Author

Karl V. Keller, Jr., Oct 06 2014

Keywords

• nonn
• easy

Comments

The combination of A082369(30*n+13) and A128468(30*n+17) is the base sequence for A140533(Primes congruent to 13 or 17 mod 30).

Links

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

Crossrefs

Cf. A082369 (30*n+13), A128468 (30*n+17).

Cf. A039949 (Primes of the form 30n-13), A132233 (Primes congruent to 13 mod 30), A140533 (Primes congruent to 13 or 17 mod 30).

Programs

Mathematica

Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* Alonso del Arte, Oct 06 2014 *)

PARI

Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2014

Python

for n in range(1,101):
  print (n*30-17),
  print (n*30-13),

Formula

From Colin Barker, Oct 07 2014: (Start)

a(n) = (-15-11*(-1)^n+30*n)/2.

a(n) = a(n-1)+a(n-2)-a(n-3).

G.f.: x*(13*x^2+4*x+13) / ((x-1)^2*(x+1)). (End)

E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - David Lovler, Sep 10 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2*(5+sqrt(5)))+sqrt(3)-sqrt(15))*Pi / (30*(sqrt(6*(5+sqrt(5)))+sqrt(5)-1)). - Amiram Eldar, Jul 30 2024