1,3

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.

Let b(i),0<=i<k be all the residues of triangular numbers modulo n in order, with b(k)=n. The differences b(i+1)-b(i) are the sequence for n.

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

(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}

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

Sequence in context: A053260 A140223 A014643 this_sequence A007723 A067437 A029315

Adjacent sequences: A118379 A118380 A118381 this_sequence A118383 A118384 A118385

nonn,tabf

Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 26 2006