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A294865 Solution of the complementary equation a(n) = a(n-2) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294865 #4 Nov 18 2017 09:06:13
%S A294865 1,2,7,10,17,22,33,40,55,64,81,92,111,124,147,162,187,204,233,252,283,
%T A294865 304,337,360,395,420,457,484,525,554,597,628,673,706,755,790,841,878,
%U A294865 931,970,1025,1066,1123,1166,1225,1270,1331,1378,1443,1492,1559,1610
%N A294865 Solution of the complementary equation a(n) = a(n-2) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294865 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.
%H A294865 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 A294865 a(0) = 1, a(1) = 2, b(0) = 3
%e A294865 b(1) = 4 (least "new number")
%e A294865 a(2) = a(0) + 2*b(0) = 7
%e A294865 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, ...)
%t A294865 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294865 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294865 a[n_] := a[n] = a[n - 2] + 2*b[n - 2];
%t A294865 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294865 Table[a[n], {n, 0, 18}]  (* A294865 *)
%t A294865 Table[b[n], {n, 0, 10}]
%Y A294865 Cf. A294860.
%K A294865 nonn,easy
%O A294865 0,2
%A A294865 _Clark Kimberling_, Nov 16 2017