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A295962 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, 11, 20, 37, 64, 109, 182, 302, 496, 811, 1321, 2147, 3484, 5648, 9150, 14818, 23989, 38829, 62841, 101694, 164560, 266280, 430867, 697175, 1128071, 1825276, 2953378, 4778686, 7732097, 12510817, 20242949, 32753803, 52996790, 85750632, 138747462 (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 = 11

Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

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}];  (* A295962 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A025079 A222770 A036652 * A097072 A049977 A000677

Adjacent sequences:  A295959 A295960 A295961 * A295963 A295964 A295965

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified July 16 23:49 EDT 2019. Contains 325092 sequences. (Running on oeis4.)