A296293 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 13, 33, 74, 147, 275, 492, 855, 1455, 2450, 4070, 6712, 11003, 17967, 29255, 47542, 77154, 125092, 202683, 328255, 531463, 860290, 1392374, 2253336, 3646435, 5900551, 9547823, 15449270, 24998079, 40448399, 65447594, 105897177, 171346025, 277244528

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 A296245 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + 2*b(2) = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
j = 1; While[j < 10, 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}]; (* A296293 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A100512 A051474 A062708 * A190816 A084910 A124024

Adjacent sequences: A296290 A296291 A296292 * A296294 A296295 A296296

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved