A047247 Numbers that are congruent to {2, 3, 4, 5} (mod 6).

2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 64, 65, 68, 69, 70, 71, 74, 75, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 95, 98, 99

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence is the interleaving of A047235 with A047270. - Guenther Schrack, Feb 10 2019

Numbers k for which A276076(k) and A276086(k) are multiples of three. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10015
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
• Index entries for two-way infinite sequences.

Crossrefs

Cf. A047227, A047235, A047246, A047270, A047257.

Cf. A007623, A049345, A276076, A276086.

Complement: A047225.

Programs

Magma

[n : n in [0..100] | n mod 6 in [2, 3, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

Maple

A047247:=n->(6*n-1-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047247(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Mathematica

Table[(6n-1-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},{2,3,4,5,8},70] (* Harvey P. Dale, May 25 2024 *)

PARI

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

Sage

a=(x*(2+x+x^2+x^3+x^4)/((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+x+x^2+x^3+x^4) / ( (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 - 1 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i = sqrt(-1).

a(2*n) = A047270(n), a(2*n-1) = A047235(n).

a(n) = A047227(n) + 1, a(1-n) = - A047227(n). (End)

From Guenther Schrack, Feb 10 2019: (Start)

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

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

a(n) = A047227(n) + 1. a(n) = A047246(n) + 2. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 - 2*log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016