A273513 -id:A273513 - cp's OEIS frontend

A262097 a(n) is the number of arithmetic triples k

0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 5, 2, 2, 4, 2, 2, 5, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0

Offset: 0

Max Barrentine, Sep 11 2015

This is a recursive sequence that gives the number of times n is rejected from A005836, if n is the largest member of an arithmetic triple whose initial two terms are contained in A005836.
This is similar to both A002487, which has a similar recurrence relation and counts hyperbinary representations of n, and A000119, which counts representations of n as a sum of distinct Fibonacci numbers.
a(n) is the number of times n occurs in A262096.
Indices of maxima between a(n)=0 and a(k)=0 (choose the smallest k) appear to converge to (1/12)*(k-n) and (1/4)*(k-n). - Max Barrentine, May 24 2016

Max Barrentine, Table of n, a(n) for n = 0..19683

Cf. A000119, A002487, A005836, A262096, A262256, A273513, A273514.

a(0)=0, a(n) = a(3n) = a(3n+1); if a(n)=0, a(3n+2) = a(n+1) + 1, otherwise a(3n+2) = a(n+1) + a(n). - Max Barrentine, May 24 2016

Name improved by Max Barrentine, Jun 23 2016

A273514 a(n) is the number of arithmetic progressions m < n < p (three numbers in arithmetic progression) such that m and p contain no 2's in their ternary representation.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 8, 8, 8, 8, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2

Offset: 0

Max Barrentine, May 23 2016

This is a recursive sequence that gives the number of times n is rejected from A005836, if n is the middle member of an arithmetic triple whose first and last terms are contained in A005836.
Also, a(n) is the number of unordered pairs of members of A005836 whose average (arithmetic mean) is n.
It appears that when A273513(n) and A262097(n) are coprime, a(n) = 2.
Local maxima occur at a(A125857(n)).

			a(2) = 2 because there are two arithmetic triples a < 2 < b such that a and b are members of A005836: 0, 2, 4 and 1, 2, 3.

Max Barrentine, Table of n, a(n) for n = 0..19683

Cf. A005836, A125857, A262097, A273513.

precCantor(n)=my(v=digits(n,3)); for(i=1,#v, if(v[i]==2, for(j=i,#v,v[j]=1); break)); fromdigits(v,2)
a(n)=if(n==0, return(0)); sum(i=0,precCantor(n-1), my(m=fromdigits(digits(i,2),3)); vecmax(digits(2*n-m,3))<2) \\ Charles R Greathouse IV, Jun 17 2016

a(0) = 0, a(n) = a(3n) = a(3n+1); if a(n) = 0, a(9n + 2) = 2, otherwise a(9n + 2) = 4a(n); a(9n + 5) = a(9n + 6) = a(9n + 7) = a(9n + 8) = a(3n + 2).

A273515 The number of times n is rejected from the Stanley sequence S(0,1), A005836.

Original entry on oeis.org

0, 0, 4, 0, 0, 5, 4, 4, 5, 0, 0, 4, 0, 0, 6, 5, 5, 7, 4, 4, 12, 4, 4, 7, 5, 5, 6, 0, 0, 4, 0, 0, 5, 4, 4, 5, 0, 0, 4, 0, 0, 7, 6, 6, 9, 5, 5, 14, 5, 5, 10, 7, 7, 9, 4, 4, 12, 4, 4, 14, 12, 12, 14, 4, 4, 12, 4, 4, 9, 7, 7, 10, 5, 5, 14, 5, 5, 9, 6, 6, 7, 0, 0

Offset: 0

Max Barrentine, May 23 2016

This sequence gives the number of times n is a member of an arithmetic triple aA005836. A005836 is the minimal sequence that disallows any arithmetic triples.

Max Barrentine, Table of n, a(n) for n = 0..19683

Cf. A005836, A262097, A273513, A273514, A273516.

a(n) = A273513(n) + A273514(n) + A262097(n).

cp's OEIS Frontend

A262097 a(n) is the number of arithmetic triples k

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A273514 a(n) is the number of arithmetic progressions m < n < p (three numbers in arithmetic progression) such that m and p contain no 2's in their ternary representation.

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A273515 The number of times n is rejected from the Stanley sequence S(0,1), A005836.

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