A296003 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 2, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 10, 32, 94, 278, 824, 2440, 7228, 21408, 63406, 187800, 556234, 1647478, 4879574, 14452538, 42806168, 126785206, 375518042, 1112225982, 3294240212, 9757026674, 28898794076, 85593729210, 253515301048, 750872855508, 2223968505284, 6587048494582

Offset

0,1

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 2.961844324... (as in A296004). See A296000 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..999

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

Example

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

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 4; b[0] = 1; a[n_] := a[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}]]];
Table[a[n], {n, 0, 100}];  (* A296003 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296004 *)

Crossrefs

Cf. A296000, A296004.

Sequence in context: A243931 A005269 A070900 * A263662 A151400 A363138

Adjacent sequences: A296000 A296001 A296002 * A296004 A296005 A296006

Keyword

nonn,easy

Author

Clark Kimberling, Dec 07 2017

Status

approved