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A295953 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 5
1, 3, 10, 20, 38, 67, 115, 194, 322, 530, 867, 1413, 2297, 3728, 6044, 9792, 15858, 25673, 41555, 67253, 108834, 176114, 284976, 461119, 746125, 1207275, 1953432, 3160740, 5114206, 8274981, 13389223, 21664241, 35053502, 56717783, 91771326, 148489151 (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 A295862 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..2000

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

b(3) = 6 (least "new number")

a(2) = a(1) + a(0) + b(2) + 1 = 10

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

MATHEMATICA

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

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

j = 1; While[j < 6, k = a[j] - j - 1;

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

Table[a[n], {n, 0, k}];  (* A295953 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A338631 A272764 A293765 * A005997 A213850 A336529

Adjacent sequences:  A295950 A295951 A295952 * A295954 A295955 A295956

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified July 28 00:54 EDT 2021. Contains 346316 sequences. (Running on oeis4.)