A205593 -id:A205593 - cp's OEIS frontend

A205592 a(2) = 1, a(3k) = a(3k+1) = a(2k), a(3k+2) = 2a(2k+1) for k >= 1.

1, 1, 1, 2, 1, 1, 4, 1, 1, 2, 4, 4, 2, 1, 1, 4, 4, 4, 8, 2, 2, 2, 1, 1, 8, 4, 4, 8, 8, 8, 4, 2, 2, 4, 1, 1, 2, 8, 8, 8, 4, 4, 16, 8, 8, 16, 4, 4, 4, 2, 2, 8, 1, 1, 2, 2, 2, 16, 8, 8, 16, 4, 4, 8, 16, 16, 16, 8, 8, 32, 4, 4, 8, 4, 4, 4, 2, 2, 16, 1, 1, 2, 2, 2, 4

Offset: 2

Joseph Myers, Jan 29 2012

Joseph Myers, Table of n, a(n) for n = 2..1000
2011/12 British Mathematical Olympiad Round 2, Problem 2.
Index to sequences related to Olympiads and other Mathematical competitions

Cf. A205591, A205593, A205594, A205595, A205596.

Cf. A335933.

a(n) = A205591(n) - A205591(n-1).

A205591 a(1) = 1, a(n) = a(floor((2n-1)/3)) + a(floor(2n/3)) for n > 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 13, 14, 16, 20, 24, 26, 27, 28, 32, 36, 40, 48, 50, 52, 54, 55, 56, 64, 68, 72, 80, 88, 96, 100, 102, 104, 108, 109, 110, 112, 120, 128, 136, 140, 144, 160, 168, 176, 192, 196, 200, 204, 206, 208, 216, 217, 218, 220, 222, 224, 240, 248, 256

Offset: 1

Joseph Myers, Jan 29 2012

In other words, a(1)=1 and then any term is a sum of two earliest possible previous terms (not necessarily distinct), given that each term must be used in summation no more than three times. So a(2)=1+1 (thus 1 gets used twice), a(3)=1+2 (thus 1 gets used for the third and final time, then 2 steps in), and so on. - Ivan Neretin, Jul 09 2015

Joseph Myers, Table of n, a(n) for n = 1..1000
2011/12 British Mathematical Olympiad Round 2, Problem 2.

Cf. A205592, A205593, A205594, A205595, A205596.

A205594 n such that A205592(n) > n.

Original entry on oeis.org

242, 818, 1841, 2762, 6215, 9323, 13985, 14003, 14012, 20978, 21005, 26243, 29888, 31467, 31468, 31481, 31508, 31529, 31589, 39365, 47201, 47202, 47203, 47222, 47262, 47263, 47294, 47384, 59048, 59156, 59183, 61235, 70802, 70805, 70895, 88572, 88573, 88775, 91853

Offset: 1

Joseph Myers, Jan 29 2012

nonn

Joseph Myers, Table of n, a(n) for n = 1..1000
2011/12 British Mathematical Olympiad Round 2, Problem 2.

Cf. A205591, A205592, A205593, A205595, A205596.

A205595 A205592(A205594(n)).

Original entry on oeis.org

256, 1024, 2048, 4096, 8192, 16384, 32768, 16384, 16384, 65536, 32768, 32768, 32768, 65536, 65536, 32768, 65536, 32768, 32768, 65536, 131072, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 131072, 65536, 65536, 65536, 262144, 131072, 131072, 131072, 131072, 131072

Offset: 1

Joseph Myers, Jan 29 2012

nonn

Joseph Myers, Table of n, a(n) for n = 1..1000
2011/12 British Mathematical Olympiad Round 2, Problem 2.

Cf. A205591, A205592, A205593, A205594, A205596.

A205596 Least k such that A205592(k) = 2^n.

Original entry on oeis.org

2, 5, 8, 20, 44, 71, 107, 161, 242, 545, 818, 1841, 2762, 6215, 9323, 13985, 20978, 47201, 70802, 159305, 238958, 358910, 807548, 1814615, 2721923, 4082885, 6124328, 13779557, 20669336, 40222412, 87267041

Offset: 0

Joseph Myers, Jan 29 2012

nonn

2011/12 British Mathematical Olympiad Round 2, Problem 2.

Cf. A205591, A205592, A205593, A205594, A205595.

A335933 A fractal function, related to ruler functions. a(1) = 0; otherwise for m >= 0, a(3m) = 1, a(3m-1) = a(2m-1) + sign(a(2m-1)), a(3m+1) = a(2m+1) + sign(a(2m+1)).

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 6, 6, 1, 4, 4, 1, 2, 2, 1, 3, 3, 1, 7, 7, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 4, 4, 1, 8, 8, 1, 2, 2, 1, 3, 3, 1, 6, 6, 1, 2, 2, 1, 4, 4, 1, 3, 3, 1, 2, 2, 1, 5, 5, 1

Offset: 0

Peter Munn, Jun 30 2020

We choose a form for the definition that shows clearly its relationship to A307744.
The odd bisection is essentially A087088.
If we add a(-1) = 0 to the definition and allow negative m (and therefore n), we get a symmetric function, that is a(n) = a(-n).
For k >= 1 numbers 1..k occur with the same periodic and mirror symmetries as in A307744 and in ruler function A051064. In A051064, k occurs 3 times more frequently than k+1. Here, and in A307744, k occurs 3/2 times more frequently than k+1, precisely 2^(k-1) times in every 3^k terms.

Sequences with similar definitions: A205593, A307744.
A051064 has matching symmetries.
Odd bisection: A087088.

a(n) = if (n==1, 0, if ((n%3) == 0, 1, if ((n%3)==1, my(k=(n-1)/3, aa = a(2*k+1)); aa+sign(aa),  my(k=(n+1)/3, aa = a(2*k-1)); aa+sign(aa)))); \\ Michel Marcus, Jul 03 2020

cp's OEIS Frontend

A205592 a(2) = 1, a(3k) = a(3k+1) = a(2k), a(3k+2) = 2a(2k+1) for k >= 1.

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A205591 a(1) = 1, a(n) = a(floor((2n-1)/3)) + a(floor(2n/3)) for n > 1.

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A205594 n such that A205592(n) > n.

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A205595 A205592(A205594(n)).

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A205596 Least k such that A205592(k) = 2^n.

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A335933 A fractal function, related to ruler functions. a(1) = 0; otherwise for m >= 0, a(3m) = 1, a(3m-1) = a(2m-1) + sign(a(2m-1)), a(3m+1) = a(2m+1) + sign(a(2m+1)).

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