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A295954 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 4, 12, 23, 43, 75, 128, 214, 354, 582, 951, 1549, 2517, 4084, 6620, 10724, 17365, 28111, 45499, 73635, 119160, 192822, 312010, 504861, 816901, 1321793, 2138726, 3460552, 5599312, 9059899, 14659247, 23719183, 38378468, 62097690, 100476198, 162573929 (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) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5

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

a(2) = a(1) + a(0) + b(2) + 1 = 12

Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, ...)

MATHEMATICA

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; 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}];  (* A295954 *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A173650 A303030 A291404 * A052416 A059322 A163908

Adjacent sequences:  A295951 A295952 A295953 * A295955 A295956 A295957

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)