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A296843 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n+1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 3
1, 2, 9, 18, 35, 63, 109, 184, 306, 504, 825, 1345, 2187, 3551, 5758, 9330, 15110, 24463, 39597, 64085, 103708, 167820, 271556, 439405, 710991, 1150427, 1861450, 3011910, 4873394, 7885340, 12758771, 20644149, 33402959, 54047148, 87450148, 141497338 (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. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

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

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6

a(2) = a(0) + a(1) + b(3) = 9

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, ...)

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];

j = 1; While[j < 16, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

u = Table[a[n], {n, 0, k}];  (* A296843 *)

Table[b[n], {n, 0, 20}] (* complement *)

CROSSREFS

Cf. A001622, A296245, A296844, A296845.

Sequence in context: A294535 A294543 A295956 * A200085 A083708 A280588

Adjacent sequences:  A296840 A296841 A296842 * A296844 A296845 A296846

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 12 2018

STATUS

approved

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Last modified May 10 01:36 EDT 2021. Contains 343747 sequences. (Running on oeis4.)