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A118383 Unrefined Orloj clock sequences; row n sums to n. 4
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 4, 2, 1, 2, 3, 3, 1, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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 unrefined sequence whose values sum to each n; all other Orloj clock sequences summing to n can be obtained by refining this one. Refining means splitting one or more terms into values summing to that term. (The unrefined sequence for n = 2^k*(2m-1) is the sequence for 2m-1 repeated 2^k times, but any single refinement - possible unless m = 1 - will produce an aperiodic sequence summing to n.) The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

LINKS

Table of n, a(n) for n=1..105.

Wikipedia, Prague astronomical clock

FORMULA

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. The sequence for 2n is the sequence for n repeated.

EXAMPLE

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, 1;

1, 2;

1, 1, 1, 1;

1, 2, 2;

1, 2, 1, 2;

1, 2, 3, 1.

PROG

(PARI) {Orloj(n) = my(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}

for (n=1, 10, print(Orloj(n)))

CROSSREFS

Cf. A028355, A118382.

Length of row n is A117484(n).

Sequence in context: A329037 A279794 A025900 * A115766 A108339 A138559

Adjacent sequences:  A118380 A118381 A118382 * A118384 A118385 A118386

KEYWORD

nonn,tabf

AUTHOR

Franklin T. Adams-Watters, Apr 26 2006

STATUS

approved

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Last modified June 5 09:58 EDT 2020. Contains 334840 sequences. (Running on oeis4.)