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Numbers congruent to 13 or 17 mod 30.
1

%I #26 Jul 30 2024 04:13:26

%S 13,17,43,47,73,77,103,107,133,137,163,167,193,197,223,227,253,257,

%T 283,287,313,317,343,347,373,377,403,407,433,437,463,467,493,497,523,

%U 527,553,557,583,587,613,617,643,647,673,677,703,707,733,737,763,767,793,797

%N Numbers congruent to 13 or 17 mod 30.

%C 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).

%H Karl V. Keller, Jr., <a href="/A248474/b248474.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F From _Colin Barker_, Oct 07 2014: (Start)

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

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

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

%F E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - _David Lovler_, Sep 10 2022

%F 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

%t Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* _Alonso del Arte_, Oct 06 2014 *)

%o (Python)

%o for n in range(1,101):

%o ..print (n*30-17),

%o ..print (n*30-13),

%o (PARI)

%o Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ _Colin Barker_, Oct 07 2014

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

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

%K nonn,easy

%O 1,1

%A _Karl V. Keller, Jr._, Oct 06 2014