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A295950 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 5
1, 4, 10, 20, 37, 65, 111, 187, 310, 510, 834, 1359, 2209, 3585, 5812, 9416, 15249, 24687, 39959, 64670, 104654, 169350, 274031, 443409, 717469, 1160908, 1878408, 3039348, 4917789, 7957171, 12874995, 20832202, 33707235, 54539476, 88246751, 142786268 (list; graph; refs; listen; history; text; internal format)
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).
See A295862 for a guide to related sequences.
LINKS
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) = 4, b(0) = 2, b(1) = 3, b(2) = 5;
b(3) = 6 (least "new number");
a(2) = a(1) + a(0) + b(2) = 10;
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, ...)
MATHEMATICA
a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; 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}] (* A295950 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A008058 A301170 A188280 * A038420 A008254 A301178
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 08 2017
STATUS
approved

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Last modified March 1 06:17 EST 2024. Contains 370430 sequences. (Running on oeis4.)