A047225 Numbers that are congruent to {0, 1} mod 6.

0, 1, 6, 7, 12, 13, 18, 19, 24, 25, 30, 31, 36, 37, 42, 43, 48, 49, 54, 55, 60, 61, 66, 67, 72, 73, 78, 79, 84, 85, 90, 91, 96, 97, 102, 103, 108, 109, 114, 115, 120, 121, 126, 127, 132, 133, 138, 139, 144, 145, 150

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also: 0 followed by partial sums of A010686. - R. J. Mathar, Feb 23 2008

Expansion of 1/(1 + x + x^2 + x^3 + x^4 + x^5) = 1 - x + x^6 - x^7 + x^12 - x^13 + ... and the exponents are the terms of this sequence. - Gary W. Adamson, Apr 04 2011

Numbers k such that floor(k/2) = 3*floor(k/6). - Bruno Berselli, Oct 05 2017

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A010686, A030308, A082505.

Programs

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+6 od: seq(a[n], n=0..50); # Zerinvary Lajos, Mar 16 2008

Mathematica

{#,#+1}&/@(6Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,1,6},60] (* Harvey P. Dale, Aug 24 2019 *)

PARI

forstep(n=0,200,[1,5],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

From R. J. Mathar, Feb 23 2008: (Start)

O.g.f.: 1/(1+x) + 3/(-1+x)^2 + 4/(-1+x).

a(n) = a(n-2) + 6, n >= 2. (End)

a(n) = 6*n - a(n-1) - 11 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010

a(n+1) = Sum_{k>=0} A030308(n,k)*A082505(k+1). - Philippe Deléham, Oct 17 2011

Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/12 + log(2)/3 + log(3)/4. - Amiram Eldar, Dec 13 2021

E.g.f.: 5 + (3*x - 4)*exp(x) - exp(-x). - David Lovler, Aug 25 2022

Extensions

Formula corrected by Paolo P. Lava, Oct 12 2010