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A296215 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*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 (list; graph; refs; listen; history; text; internal format)
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..26.

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

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

Cf. A296000, A296217.

Sequence in context: A054718 A132390 A327643 * A152761 A295761 A262349

Adjacent sequences:  A296212 A296213 A296214 * A296216 A296217 A296218

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified September 20 20:12 EDT 2019. Contains 327247 sequences. (Running on oeis4.)