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A295951 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 5
2, 3, 10, 19, 36, 63, 108, 182, 302, 497, 813, 1325, 2154, 3496, 5668, 9184, 14873, 24079, 38975, 63078, 102078, 165182, 267287, 432497, 699813, 1132340, 1832184, 2964556, 4796773, 7761363, 12558171, 20319571, 32877780, 53197390, 86075210, 139272641 (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.

FORMULA

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2) + f(n-2)*b(3) + ... + f(2)*b(n-1) + f(1)*b(n), where f(n) = A000045(n), the n-th Fibonacci number.

EXAMPLE

a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5;

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

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

Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17,  ...)

MATHEMATICA

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

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

j = 1; While[j < 5, 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}]  (* A295951 *)

Table[b[n], {n, 0, 20}]  (* complement *)

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A175569 A275020 A122822 * A083944 A306106 A165550

Adjacent sequences:  A295948 A295949 A295950 * A295952 A295953 A295954

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)