

A296215


Solution of the complementary equation a(n) = a(1)*b(n2) + a(2)*b(n3) + ... + a(n1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.


4



1, 3, 6, 24, 87, 321, 1176, 4314, 15822, 58032, 212847, 780672, 2863317, 10501959, 38518662, 141277197, 518170812, 1900526031, 6970672818, 25566752964, 93772706622, 343935755925, 1261473710904, 4626782461218, 16969926331719, 62241612204120, 228287277978756
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OFFSET

0,2


COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.


LINKS



EXAMPLE

a(0) =1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
a(2) = a(1)*b(0) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...)


MATHEMATICA

a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n  k  1], {k, 1, n  1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
u = Table[a[n], {n, 0, 200}]; (* A296215 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n  1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296216 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



