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

1,1

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.

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

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 *)

Prog

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

Crossrefs

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

Cf. A007623, A049345, A276076, A276086.

Complement: A047225.

Sequence in context: A046892 A068406 A276878 * A169606 A140769 A364380

Adjacent sequences: A047244 A047245 A047246 * A047248 A047249 A047250

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

Status

approved