A047260 Numbers that are congruent to {0, 1, 4, 5} mod 6.

0, 1, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 28, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 43, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 78, 79, 82, 83, 84, 85, 88, 89, 90, 91, 94, 95, 96, 97

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers x which are not a solution to 3^x - 2^x == 5 mod 7. - Cino Hilliard, May 14 2003

The sequence is the interleaving of A047233 with A007310. - Guenther Schrack, Feb 13 2019

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10006
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A007310, A047233, A047269.

Complement: A047243.

Programs

GAP

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 4 or n mod 6 = 5); # Muniru A Asiru, Feb 19 2019

Magma

[n : n in [0..100] | n mod 6 in [0, 1, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

Maple

A047260:=n->(6*n-5-I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/4: seq(A047260(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Mathematica

Table[(6n-5-I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,5,6},70] (* Harvey P. Dale, Sep 20 2023 *)

PARI

my(x='x+O('x^70)); concat([0], Vec(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019

Sage

a=(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Formula

G.f.: x^2*(1+3*x+x^2+x^3) / ((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, May 21 2016: (Start)

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

a(n) = (6*n - 5 - i^(2*n) + (1-i)*i^(-n) + (1+i)*i^n)/4 where i=sqrt(-1).

a(2*n) = A007310(n), a(2*n-1) = A047233(n). (End)

From Guenther Schrack, Feb 13 2019: (Start)

a(n) = (6*n - 5 - (-1)^n + 2*(-1)^(n*(n + 1)/2))/4.

a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=4, a(4)=5, for n > 4.

a(-n) = -A047269(n+2). (End)

Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4 + 2*log(2)/3. - Amiram Eldar, Dec 16 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016