A047239 Numbers that are congruent to {1, 2} (mod 6).

1, 2, 7, 8, 13, 14, 19, 20, 25, 26, 31, 32, 37, 38, 43, 44, 49, 50, 55, 56, 61, 62, 67, 68, 73, 74, 79, 80, 85, 86, 91, 92, 97, 98, 103, 104, 109, 110, 115, 116, 121, 122, 127, 128, 133, 134, 139, 140, 145, 146, 151

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

If a(n) is the n-th Towers of Hanoi move, the smallest disc (#1) is on peg C. If n == (3,4) (mod 6), disc #1 is on peg B; and if n == (0,5) (mod 6) disc #1 is on peg A. Disc #1 moves every 1,3,5,7,...-th move in a rotational cycle CBACBACBA such that it's on peg C the first TOH move and stays there for the 2nd move (in which case disc #2 moves). Therefore disc #1 is on peg C in moves (1, 2, 7, 8, 13, ...). - Gary W. Adamson, Jun 22 2012

Conjecture: a(n) is the least positive integer > a(n-1) that is not equal to a(i) + a(j) + a(k) for any i <= j <= k <= n. - Clark Kimberling, Oct 09 2019

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A047264.

Programs

Mathematica

Select[Range[200],MemberQ[{1,2},Mod[#,6]]&] (* or *) LinearRecurrence[ {1,1,-1},{1,2,7},80] (* Harvey P. Dale, Nov 23 2011 *)

Formula

a(n) = 3*(n-1) - (-1)^n. - Rolf Pleisch, Aug 04 2009

a(n) = 6*n - a(n-1) - 9 (with a(1)=1). - Vincenzo Librandi, Aug 05 2010

G.f. x*(1+x+4*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = a(n-1) + a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=7. - Harvey P. Dale, Nov 23 2011

E.g.f.: 4 + 3*exp(x)*(x - 1) - exp(-x). - Stefano Spezia, Oct 09 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/18 + log(2)/3. - Amiram Eldar, Dec 13 2021