A047222 Numbers that are congruent to {0, 2, 3} mod 5.

0, 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 107

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Row sum of a triangle where the top value is 2 and every elementary triangle or triple is required to have the values 1,2,2 (see link below). Compare with A008854 where the triple contains 1,2,2 with 1 at the top. - Craig Knecht, Oct 18 2015

Also, numbers k such that k*(k^2+1)/5 is a nonnegative integer. - Bruno Berselli, Jan 16 2016

Conjecture: Apart from 0, the sequence gives the values for c/6, such that an infinite number of primes, p, result in both p^2-c and p^2+c being positive primes, except when c is a square. When c is square solutions exist for c (both within and outside of the a(n) set), but occur at only a single prime p. See A274609. Other c values with only one prime providing a solution occur when p^2-c=3. See A274610. The only remaining c values with single p solutions are: c=2 (with p=3) and c=6 (with p=5). - Richard R. Forberg, Jun 26 2016

See A047363 for case of p^3 +- c. See A005097 and A177735 for observations on the general case p^q +- c. - Richard R. Forberg, Aug 11 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Craig Knecht, Row sum for the 1,2,2 triangle with 2 at the top.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A008854, A028738, A042965, A047363, A005097.

Programs

Magma

[n : n in [0..150] | n mod 5 in [0, 2, 3]]; // Wesley Ivan Hurt, Jun 14 2016

Maple

A047222 := n -> 2*n - 2 - iquo(n,3): seq(A047222(n), n=1..100); # Wesley Ivan Hurt, Nov 07 2013

Mathematica

Floor[(Range[5, 305, 5] - 4)/3] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
Flatten[Table[5n + {0, 2, 3}, {n, 0, 19}]] (* Alonso del Arte, Nov 07 2013 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 3, 5}, 100] (* Vincenzo Librandi, Jun 15 2016 *)

PARI

a(n)=(5*n-4)\3 \\ Charles R Greathouse IV, Oct 28 2011

PARI

concat(0, Vec(x^2*(2+x+2*x^2)/((1-x)^2*(1+x+x^2)) + O(x^100))) \\ Altug Alkan, Oct 26 2015

Formula

From R. J. Mathar, Oct 18 2008: (Start)

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

a(n) = A028738(n-2), 1 < n < 16. (End)

a(n) = floor((5*n-4)/3). - Gary Detlefs, Oct 28 2011

a(n) = 2*n - 2 - floor(n/3). - Wesley Ivan Hurt, Nov 07 2013

From Wesley Ivan Hurt, Jun 14 2016: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = (15*n-15-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 5k-2, a(3k-1) = 5k-3, a(3k-2) = 5k-5. (End)

a(n) = n - 1 + floor((2n-1)/3). - Wesley Ivan Hurt, Dec 27 2016

Sum_{n>=2} (-1)^n/a(n) = arccoth(3/sqrt(5))/sqrt(5) - log(2)/5. - Amiram Eldar, Dec 10 2021

From Peter Bala, Aug 04 2022: (Start)

a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 4 with a(1) = 0, a(2) = 2 and a(3) = 3.

a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A008854 and A042965. (End)