A178976 -id:A178976 - cp's OEIS frontend

A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2m), ..., a(t-(k-1)m), for 0=1, are equal.

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

Offset: 1

Scott R. Shannon, Oct 13 2023

nonn

The terms form quickly form a repetitive pattern of arithmetic progressions of increasing length, see the graph. This leads to any given value t eventually being in a progression of length t+1 which then never increases.

See A366724 for the index where a number first appears.

			a(3) = 2 as a(2) = 1 and a(2) = a(1) = 1.
a(11) = 3 as a(10) = 2 and a(7) = a(6) = a(5) = 2.
a(18) = 4 as a(17) = 3 and a(17) = a(15) = a(13) = a(11) = 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A366724, A178976, A308638, A229037.

A179510 Number of equally-spaced triples (x,y,z) of preceding terms with x+y=z.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 49, 50, 50, 51, 52, 53, 53, 54, 55, 56, 56, 57, 57, 58, 59, 60, 60, 61, 62, 62, 63, 64, 64, 65

Offset: 0

Alex Abercrombie, Jan 08 2011

a(n) is the number of 3-element subsets (i,j,k) of {0,...,n-1} such that k-j=j-i>0 and a(i)+a(j)=a(k). An induction argument can show simultaneously that a(n)-a(n-1) is always 0 or 1 and that the only case where three values of a(n) are equal is a(0)=a(1)=a(2)=0. Numerical evidence suggests that a(n) is asymptotic to 2n/3.

			For n=9, just the triples (0,1,2),(2,3,4),(3,4,5),(4,5,6),(4,6,8) satisfy the stated conditions, so a(9) = 5

A178976 has a somewhat similar definition

cp's OEIS Frontend

A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2m), ..., a(t-(k-1)m), for 0=1, are equal.

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A179510 Number of equally-spaced triples (x,y,z) of preceding terms with x+y=z.

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cp's OEIS Frontend

A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2*m), ..., a(t-(k-1)*m), for 0=1, are equal.

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A179510 Number of equally-spaced triples (x,y,z) of preceding terms with x+y=z.

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Author

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A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2m), ..., a(t-(k-1)m), for 0=1, are equal.