cp's OEIS Frontend

A294871 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294871 #4 Nov 18 2017 20:53:47
%S A294871 1,2,10,26,51,86,132,190,262,349,452,572,710,867,1044,1242,1462,1705,
%T A294871 1972,2264,2582,2927,3300,3703,4137,4603,5102,5635,6203,6807,7448,
%U A294871 8127,8845,9603,10402,11243,12127,13055,14028,15047,16113,17227,18390,19603,20867
%N A294871 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294871 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 A294871 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 A294871 a(0) = 1, a(1) = 2, b(0) = 3
%e A294871 b(1) = 4 (least "new number")
%e A294871 a(2) = 2*a(1) - a(0) + b(1) + 3 = 10
%e A294871 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
%t A294871 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294871 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294871 a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 3;
%t A294871 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294871 Table[a[n], {n, 0, 18}]  (* A294871 *)
%t A294871 Table[b[n], {n, 0, 10}]
%Y A294871 Cf. A294860.
%K A294871 nonn,easy
%O A294871 0,2
%A A294871 _Clark Kimberling_, Nov 18 2017