A296227 - OEIS

A296227

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

1, 2, 8, 34, 146, 628, 2703, 11632, 50057, 215415, 927016, 3989317, 17167612, 73879038, 317930779, 1368182139, 5887829959, 25337665679, 109038016813, 469233798454, 2019298993572, 8689843823858, 37395841786394, 160929127296116, 692541811472532

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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

Table of n, a(n) for n=0..24.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4

a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8

Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

MATHEMATICA

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];

a[0] = 1; a[1] = 2; b[0] = 3;

a[n_] := a[n] = - n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];

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