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A296005 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) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences. 3
2, 3, 11, 33, 104, 323, 1007, 3136, 9769, 30431, 94791, 295274, 919773, 2865082, 8924690, 27800290, 86597525, 269750118, 840267961, 2617423311, 8153238141, 25397226311, 79112015761, 246432856920, 767635009499, 2391172651130, 7448470401642, 23201884354901 (list; graph; refs; listen; history; text; internal format)
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) -> 3.114986447390302... (as in A296006). See A296000 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..27.

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

EXAMPLE

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

a(2) = a(0)*b(1) + a(1)*b(0) = 11

Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)

MATHEMATICA

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

a[0] = 2; a[1] = 3; 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}];  (* A296005 *)

t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]

Take[RealDigits[Last[t], 10][[1]], 100]  (* A296006 *)

CROSSREFS

Cf. A296000, A296006.

Sequence in context: A268687 A062630 A300771 * A159458 A305846 A057838

Adjacent sequences:  A296002 A296003 A296004 * A296006 A296007 A296008

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 07 2017

EXTENSIONS

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

STATUS

approved

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Last modified November 14 23:27 EST 2018. Contains 317221 sequences. (Running on oeis4.)