cp's OEIS Frontend

A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.

%I A297469 #17 Jan 22 2022 23:43:27
%S A297469 2,7,11,17,22,27,31,37,41,47,51,57,62,67,71,77,82,87,91,97,102,107,
%T A297469 111,117,121,127,131,137,142,147,151,157,161,167,171,177,182,187,191,
%U A297469 197,201,207,211,217,222,227,231,237,242,247,251,257,262,267,271,277
%N A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.
%C A297469 Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
%C A297469 aa(n) = least new;
%C A297469 bb(n) = aa(n) + cc(n-1);
%C A297469 cc(n) = least new;
%C A297469 where "least new k" means the least positive integer not yet placed.
%C A297469 ***
%C A297469 The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >= 0,
%C A297469 (1) 5*n - 1 - 2*aa(n) is in {0,1,2},
%C A297469 (2) (aa(n) mod 5) is in {1,2,4},
%C A297469 (3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
%C A297469 (4) (bb(n) mod 5) is in {1,2}.
%C A297469 From _N. J. A. Sloane_, Nov 05 2019: (Start)
%C A297469 Conjecture: For t >= 0, bb(2t) = 10t + 1 (+1 if binary expansion of t ends in an odd number of 0's), bb(2t+1) = 10t + 7.
%C A297469 The first part may also be written as bb(2t) = 10t + 1 + A328789(t-1).
%C A297469 (End)
%H A297469 Clark Kimberling, <a href="/A297469/b297469.txt">Table of n, a(n) for n = 0..10000</a> [This is the sequence bb]
%e A297469 n:  0 1 2 3 4 5 6 7 8 9 10
%e A297469 aa: 1 4 6 9 12 14 16 19 21 24 26
%e A297469 bb: 2 7 11 17 22 27 31 37 41 47 51
%e A297469 cc: 3 5 8 10 13 15 18 20 23 25 28
%t A297469 z = 500;
%t A297469 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297469 a = {1}; b = {2}; c = {3};
%t A297469 Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
%t A297469   AppendTo[b, Last[a] + Last[c]];
%t A297469   AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
%t A297469 Take[a, 100] (* A298468 *)
%t A297469 Take[b, 100] (* A297469 *)
%t A297469 Take[c, 100] (* A047218 *)
%t A297469 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A297469 Cf. A299634, A298468 (aa), A047218 (cc), A328789.
%K A297469 nonn,easy
%O A297469 0,1
%A A297469 _Clark Kimberling_, May 04 2018
%E A297469 Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - _N. J. A. Sloane_, Nov 03 2019