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A294860 Solution of the equation a(n) = a(n-2) + b(n-2), where a( ) and b( ) are increasing sequences of positive integers such that every positive integer is in one of them and only one term is in both.

%I A294860 #8 Dec 02 2017 21:00:42
%S A294860 1,2,4,6,9,13,17,23,28,35,42,50,58,68,77,88,98,110,122,135,148,162,
%T A294860 177,192,208,224,241,258,277,295,315,334,355,375,398,419,443,465,490,
%U A294860 513,539,564,591,617,645,672,701,729,760,789,821,851,884,915,949,981
%N A294860 Solution of the equation a(n) = a(n-2) + b(n-2), where a( ) and b( ) are increasing sequences of positive integers such that every positive integer is in one of them and only one term is in both.
%C A294860 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values sequences in the following guide are a(0) = 1, a(1) = 2, b(0) = 3.
%C A294860 A294860: a(n) = a(n-2) + b(n-2); not quite complementary
%C A294860 A022939: a(n) = a(n-2) + b(n-2); offset 1, complementary
%C A294860 A294861: a(n) = a(n-2) + b(n-2) + 1
%C A294860 A294862: a(n) = a(n-2) + b(n-2) + 2
%C A294860 A294863: a(n) = a(n-2) + b(n-2) + 3
%C A294860 A294864: a(n) = a(n-2) + b(n-2) + n
%C A294860 A294865: a(n) = a(n-2) + 2*b(n-2)
%C A294860 A294866: a(n) = 2*a(n-1) - a(n-2) + b(n-1)
%C A294860 A294867: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 1
%C A294860 A294868: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 2
%C A294860 A294869: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 1
%C A294860 A294870: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2
%C A294860 A294871: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3
%C A294860 A294872: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + n
%C A294860 A022942: a(n) = a(n-2) + b(n-1); offset 1
%C A294860 A295998: a(n) = 2*a(n-2) + b(n-2)
%H A294860 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A294860 a(0) = 1, a(1) = 2, b(0) = 3, so that a(2) = 4
%e A294860 (b(n)) = (3,4,5,7,8,10,11,12,14,15,...)
%t A294860 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294860 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294860 a[n_] := a[n] = a[n - 2] + b[n - 2];
%t A294860 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294860 Table[a[n], {n, 0, 18}]  (* A294860 *)
%t A294860 Table[b[n], {n, 0, 10}]
%Y A294860 Cf. A294861, A294864, A294865.
%K A294860 nonn,easy
%O A294860 0,2
%A A294860 _Clark Kimberling_, Nov 16 2017
%E A294860 Edited by _Clark Kimberling_, Dec 02 2017