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A330261 Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S, say u on top of v, with u-v; a(n) gives the number on top of S after processing n. 3

%I #13 Feb 27 2020 23:16:34

%S 1,0,1,-1,1,0,1,-2,1,1,1,-1,1,0,1,-3,1,4,1,0,1,0,1,-2,1,1,1,-1,1,0,1,

%T -4,1,5,1,7,1,0,1,-1,1,0,1,0,1,0,1,-3,1,2,1,0,1,0,1,-2,1,1,1,-1,1,0,1,

%U -5,1,5,1,-4,1,0,1,3,1,-3,1,1,1,0,1,-2,1,-2

%N Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S, say u on top of v, with u-v; a(n) gives the number on top of S after processing n.

%C This sequence is a variant of A308551.

%C After processing n, S has A268289(n) elements.

%C Every integer appears infinitely many times in the sequence:

%C - the effect of the binary string b(0) = "110" is to leave 0 on top of S,

%C - the effect of the binary string b(1) = "1" is to leave 1 on top of S,

%C - the effect of the binary string b(-1) = "11100" is to leave -1 on top of S,

%C - let "|" denote the binary concatenation,

%C - for any k > 0:

%C - the effect of b(k+1) = b(-1)|b(k)|"0" is to leave k+1 on top of S,

%C - the effect of b(-k-1) = b(1)|b(-k)|"0" is to leave -k-1 on top of S,

%C - for any k, for any n > 0, if the binary representation of n ends with b(k), then a(n) = k, QED,

%C - see A330264 for the values in order of appearance.

%H Rémy Sigrist, <a href="/A330261/b330261.txt">Table of n, a(n) for n = 1..8192</a>

%H Rémy Sigrist, <a href="/A330261/a330261.png">Scatterplot of the first 2^20 terms</a>

%H Rémy Sigrist, <a href="/A330261/a330261.gp.txt">PARI program for A330261</a>

%F a(2*k-1) = 1 for any k > 0.

%e The first terms, alongside the binary representation of n and the evolution of stack S, are:

%e n a(n) bin(n) S

%e -- ---- ------ ------------------------------------------------------------

%e 1 1 1 () -> (1)

%e 2 0 10 (1) -> (1,1) -> (0)

%e 3 1 11 (0) -> (0,1) -> (0,1,1)

%e 4 -1 100 (0,1,1) -> (0,1,1,1) -> (0,1,0) -> (0,-1)

%e 5 1 101 (0,-1) -> (0,-1,1) -> (0,2) -> (0,2,1)

%e 6 0 110 (0,2,1) -> (0,2,1,1) -> (0,2,1,1,1) -> (0,2,1,0)

%e 7 1 111 (0,2,1,0) -> (0,2,1,0,1) -> (0,2,1,0,1,1) -> (0,2,1,0,1,1,1)

%o (PARI) See Links section.

%Y Cf. A268289, A308551, A330261, A330264.

%K sign,base

%O 1,8

%A _Rémy Sigrist_, Dec 07 2019

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)