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A295956 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, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 9, 18, 35, 62, 108, 182, 303, 499, 817, 1332, 2166, 3516, 5702, 9239, 14963, 24225, 39212, 63462, 102700, 166189, 268917, 435135, 704082, 1139248, 1843362, 2982643, 4826039, 7808717, 12634793, 20443548, 33078380, 53521968, 86600389, 140122399 (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) = 2, b(0) = 3, b(1) = 4, b(2) = 5

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

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

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

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; 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}];  (* A295956 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A028881 A294535 A294543 * A296843 A200085 A083708

Adjacent sequences:  A295953 A295954 A295955 * A295957 A295958 A295959

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified May 10 23:31 EDT 2021. Contains 343784 sequences. (Running on oeis4.)