A295862 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 9, 18, 34, 60, 104, 175, 291, 479, 784, 1278, 2078, 3373, 5470, 8863, 14354, 23239, 37616, 60879, 98520, 159425, 257972, 417425, 675426, 1092881, 1768338, 2861251, 4629622, 7490908, 12120566, 19611511, 31732115, 51343665, 83075820, 134419526, 217495388

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). Following is a guide to related sequences:

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Complementary equation: a(n) = a(n-1) + a(n-2) + b(n); initial values (a(0), a(1); b(0), b(1), b(2)):

A295862: (1,3; 2,4,5)

A295947: (2,4; 1,3,5)

A295948: (3,4; 1,2,5)

A295949: (1,2; 3,4,5)

A295950: (1,4; 2,3,5)

A295951: (2,3; 1,4,5)

A295952: (1,5; 2,3,4)

Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) + 1; initial values (a(0), a(1); b(0), b(1), b(2)):

A295953: (1,3; 2,4,5)

A295954: (2,4; 1,3,5)

A295955: (3,4; 1,2,5)

A295956: (1,2; 3,4,5)

A295957: (1,4; 2,3,5)

A295958: (2,3; 1,4,5)

A295959: (1,5; 2,3,4)

Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) - 1; initial values (a(0), a(1); b(0), b(1), b(2)):

A295860: (1,3; 2,4,5)

A295961: (2,4; 1,3,5)

A295962: (3,4; 1,2,5)

A295963: (1,2; 3,4,5)

A295964: (1,4; 2,3,5)

A295965: (2,3; 1,4,5)

A295966: (1,5; 2,3,4)

Links

Clark Kimberling, Table of n, a(n) for n = 0..3000

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) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, so that
b(3) = 6 (least "new number");
a(2) = a(1) + a(0) + b(2) = 9;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, ...)

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
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}]  (*A295862*)
Table[b[n], {n, 0, 20}] (*complement*)

Crossrefs

Cf. A001622, A000045, A295947.

Sequence in context: A256524 A210970 A293406 * A246695 A132920 A127645

Adjacent sequences: A295859 A295860 A295861 * A295863 A295864 A295865

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved