A294400 - OEIS

%I #10 Feb 06 2018 13:29:59

%S 1,3,7,14,23,34,48,64,82,102,124,148,175,204,235,268,303,340,379,420,

%T 464,510,558,608,660,714,770,828,888,950,1015,1082,1151,1222,1295,

%U 1370,1447,1526,1607,1690,1775,1862,1951,2043,2137,2233,2331,2431,2533,2637

%N Solution of the complementary equation a(n) = a(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%C The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

%H 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 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

%e a(2)  = a(1) + b(0) + 2 = 7

%e Complement: (b(n)) = (2, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

%t a[n_] := a[n] = a[n - 1] + b[n - 2] + n;

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 40}]  (* A294400 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A293076, A293765, A022940.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 31 2017