This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330261 #13 Feb 27 2020 23:16:34 %S A330261 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 A330261 -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 A330261 -5,1,5,1,-4,1,0,1,3,1,-3,1,1,1,0,1,-2,1,-2 %N 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. %C A330261 This sequence is a variant of A308551. %C A330261 After processing n, S has A268289(n) elements. %C A330261 Every integer appears infinitely many times in the sequence: %C A330261 - the effect of the binary string b(0) = "110" is to leave 0 on top of S, %C A330261 - the effect of the binary string b(1) = "1" is to leave 1 on top of S, %C A330261 - the effect of the binary string b(-1) = "11100" is to leave -1 on top of S, %C A330261 - let "|" denote the binary concatenation, %C A330261 - for any k > 0: %C A330261 - the effect of b(k+1) = b(-1)|b(k)|"0" is to leave k+1 on top of S, %C A330261 - the effect of b(-k-1) = b(1)|b(-k)|"0" is to leave -k-1 on top of S, %C A330261 - for any k, for any n > 0, if the binary representation of n ends with b(k), then a(n) = k, QED, %C A330261 - see A330264 for the values in order of appearance. %H A330261 Rémy Sigrist, <a href="/A330261/b330261.txt">Table of n, a(n) for n = 1..8192</a> %H A330261 Rémy Sigrist, <a href="/A330261/a330261.png">Scatterplot of the first 2^20 terms</a> %H A330261 Rémy Sigrist, <a href="/A330261/a330261.gp.txt">PARI program for A330261</a> %F A330261 a(2*k-1) = 1 for any k > 0. %e A330261 The first terms, alongside the binary representation of n and the evolution of stack S, are: %e A330261 n a(n) bin(n) S %e A330261 -- ---- ------ ------------------------------------------------------------ %e A330261 1 1 1 () -> (1) %e A330261 2 0 10 (1) -> (1,1) -> (0) %e A330261 3 1 11 (0) -> (0,1) -> (0,1,1) %e A330261 4 -1 100 (0,1,1) -> (0,1,1,1) -> (0,1,0) -> (0,-1) %e A330261 5 1 101 (0,-1) -> (0,-1,1) -> (0,2) -> (0,2,1) %e A330261 6 0 110 (0,2,1) -> (0,2,1,1) -> (0,2,1,1,1) -> (0,2,1,0) %e A330261 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 A330261 (PARI) See Links section. %Y A330261 Cf. A268289, A308551, A330261, A330264. %K A330261 sign,base %O A330261 1,8 %A A330261 _Rémy Sigrist_, Dec 07 2019