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A364639
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Irregular triangle read by rows: T(n,k) = A237591(n,k) - A237591(n-1,k).
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5
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1, 1, 0, 1, 1, 0, 0, 1, 1, -1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, -1, 1, 0, 1, 0, 0, 1, -1, 1, 0, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, 1, -1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 1, 0, -1, 1, 1, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0
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OFFSET
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1
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COMMENTS
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It appears that every column (except for its first term) is a periodic sequence.
Conjecture: column k gives 1 together with period A002378(k) repeat S_k, where S_k is the k-th row of the triangle A364746.
Some connections with other sequences are as follows:
n is a power of 2 (A000079) if and only if row n is [1] or if row n starts with 1 and the remaining terms are zeros.
n is an odd prime (A065091) if and only if row n is [0, 1] or if row n starts with [0, 1] and the remaining terms are zeros.
n is a nonzero multiple of 6 (A008588 > 0) if and only if row n starts with [1, -1].
n is an even perfect number (A000396 assuming there are no odd perfect numbers) if and only if row n is [1, -1, 1] or if row n starts with [1] and ends in [-1, 1] and the remaining terms are zeros and row n has an odd number of terms and row n-1 has an even number of terms.
n is a term >= 10 of A091999 (and thus n is also a term >= 10 of A063221) if and only if row n starts with [1, 0, -1].
n is a term of A362866 (or, equivalently, a prime doubled (A100484) >= 10, also an even semiprime >= 10) if and only if row n is [1, 0, -1, 1] or if row n starts with [1, 0, -1, 1] and the remaining terms are zeros.
n is a term of A364414 if and only if row n starts with [0, 0, 1, 0, -1].
n is a term of A365081 if and only if row n is [0, 0, 1, 0, -1, 1] or if row n starts with [0, 0, 1, 0, -1, 1] and the remaining terms are zeros.
T(n,k) is also the difference between the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(n) and the length of the k-th line segment of its smallest Dyck paths, from left to the axis of symmetry. For further information about these Dyck paths see A237593.
The partial sums of column k give the k-th column of A237591.
Since A237591 can be defined using partitions into consecutive parts (see A299765 and A286000), this sequence can be associated to partitions of that kind.
All connections mentioned above are due to the connections between geometry and number theory given by the stepped pyramid described in A245092. The diagram called "symmetric representation of sigma(n)" is essentially the shape of the terraces in the n-th level starting from the top of the mentioned pyramid.
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LINKS
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EXAMPLE
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Triangle begins:
1;
1;
0, 1;
1, 0;
0, 1;
1, -1, 1;
0, 1, 0;
1, 0, 0;
0, 0, 1;
1, 0, -1, 1;
0, 1, 0, 0;
1, -1, 1, 0;
0, 1, 0, 0;
1, 0, -1, 1;
0, 0, 1, -1, 1;
1, 0, 0, 0, 0;
0, 1, 0, 0, 0;
1, -1, 0, 1, 0;
0, 1, 0, 0, 0;
1, 0, 0, -1, 1;
0, 0, 1, 0, -1, 1;
1, 0, -1, 1, 0, 0;
0, 1, 0, 0, 0, 0;
1, -1, 1, 0, 0, 0;
0, 1, 0, -1, 1, 0;
1, 0, -1, 1, 0, 0;
0, 0, 1, 0, -1, 1;
1, 0, 0, 0, 0, -1, 1;
...
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PROG
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(PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i); \\ A235791
row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); } \\ A237591
row(n) = my(vb=row237591(n), va=row237591(n-1)); vector(#vb, k, if (k<=#va, vb[k]-va[k], vb[k]));
lista(nn) = for (n=1, nn, my(v=row(n)); for (k=1, #v, print1(v[k], ", "))); \\ Michel Marcus, Aug 28 2023
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CROSSREFS
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For the position of powers of 2, odd primes, and even perfect numbers see also A249351, A347529, A360022.
Cf. A000079, A000396, A002378, A008588, A063221, A065091, A091999, A100484, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A286000, A286001, A299765, A362866, A364414, A364746, A365081.
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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