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

1,3

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).

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

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 *)

Prog

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

Crossrefs

Cf. A007310, A047233, A047269.

Complement: A047243.

Sequence in context: A010419 A332486 A255137 * A284790 A172999 A354489

Adjacent sequences: A047257 A047258 A047259 * A047261 A047262 A047263

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

Status

approved