A118382 Primitive Orloj clock sequences; row n sums to 2n-1.

1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 3, 1, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 1, 3, 1, 2, 1, 5, 2, 2, 1, 2, 3, 1, 3, 3, 2, 6, 1, 2, 2, 1, 3, 1, 3, 2, 5, 1, 1, 1, 1, 2, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 1, 2, 1, 2, 1, 1, 2, 5, 1, 2, 2

Offset: 1

Author

Franklin T. Adams-Watters, Apr 26 2006

Keywords

• nonn
• tabf

Comments

An Orloj clock sequence is a finite sequence of positive integers that, when iterated, can be grouped so that the groups sum to successive natural numbers. There is one primitive sequence whose values sum to each odd m; all other sequences can be obtained by repeating and refining these. Refining means splitting one or more terms into values summing to that term. The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

These are known in some papers as Sindel sequences. It appears that this sequence was submitted prior to the first such publication.

Examples

			For a sum of 5, we have 1,2,2, which groups as 1, 2, 2+1, 2+2, 1+2+2, 1+2+2+1, .... This could be refined by splitting the second 2, to give the sequence 1,2,1,1; note that when this is grouped, the two 1's from the refinement always wind up in the same sum.
The array starts:
  1;
  1,2;
  1,2,2;
  1,2,3,1;
  1,2,3,3;
  1,2,1,2,4,1;
  ...
		

Links

• Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.

Crossrefs

Cf. A028355, A118383. Length of row n is A117484(2n-1) = A000224(2n-1).

Programs

PARI

{Orloj(n) = local(found,tri,i,last,r); found = vector(n,i,0); found[n] = 1; tri = 0; for(i = 1, if(n%2==0,n-1,n\2), tri += i; if(tri >= n, tri -= n); found[tri] = 1); last = 0; r = []; for(i = 1, n, if(found[i], r = concat(r, [i-last]); last = i)); r}

Formula

Let b(i),0<=i

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