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A295955 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
3, 4, 13, 24, 45, 78, 133, 222, 367, 602, 984, 1602, 2603, 4223, 6845, 11088, 17954, 29064, 47041, 76129, 123196, 199352, 322576, 521957, 844563, 1366551, 2211146, 3577730, 5788910, 9366675, 15155621, 24522333, 39677992, 64200364, 103878396, 168078801 (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) -> (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) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5

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

a(2) = a(1) + a(0) + b(2) + 1 = 13

Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, ...)

MATHEMATICA

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; 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}];  (* A295955 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A182691 A026700 A187775 * A151521 A142860 A111954

Adjacent sequences:  A295952 A295953 A295954 * A295956 A295957 A295958

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

EXTENSIONS

Definition corrected by Georg Fischer, Sep 27 2020

STATUS

approved

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Last modified June 16 19:49 EDT 2021. Contains 345068 sequences. (Running on oeis4.)