|
%I #4 Nov 18 2017 20:54:21
%S 1,2,8,14,26,39,60,83,115,150,195,245,306,373,452,538,637,744,865,995,
%T 1140,1295,1467,1650,1851,2064,2296,2541,2806,3085,3385,3700,4037,
%U 4390,4767,5161,5580,6017,6480,6962,7471,8000,8557,9135,9742,10371,11030,11712
%N Solution of the complementary equation a(n) = a(n-2) + b(0) + b(1) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 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) = 2, b(0) = 3
%e b(1) = 4 (least "new number")
%e a(2) = a(0) + b(0) + b(1) = 8
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 2; b[0] = 3;
%t a[n_] := a[n] = a[n - 2] + Sum[b[k], {k, 0, n - 1}];
%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, 18}] (* A295055 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A295053.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 18 2017
|
