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A295947 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), 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. 5
2, 4, 11, 21, 39, 68, 116, 194, 322, 529, 865, 1409, 2290, 3716, 6024, 9759, 15803, 25584, 41410, 67018, 108453, 175497, 283977, 459502, 743508, 1203040, 1946579, 3149651, 5096263, 8245948, 13342246, 21588230, 34930513, 56518781, 91449334, 147968156 (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) = 4, b(0) = 1, b(1) = 3, b(2) = 5, so that

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

a(2) = a(1) + a(0) + b(2) = 11;

Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)

MATHEMATICA

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

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

j = 1; While[j < 16, 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}];  (* A295947 *)

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

CROSSREFS

Cf. A001622, A000045, A295862.

Sequence in context: A152597 A320679 A126972 * A018774 A102608 A290439

Adjacent sequences:  A295944 A295945 A295946 * A295948 A295949 A295950

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 08 2017

STATUS

approved

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Last modified September 23 07:26 EDT 2021. Contains 347609 sequences. (Running on oeis4.)