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A296296 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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. 2
2, 3, 15, 36, 79, 155, 288, 513, 889, 1510, 2529, 4193, 6914, 11328, 18494, 30107, 48921, 79385, 128702, 208524, 337706, 546755, 885033, 1432409, 2318114, 3751248, 6070142, 9822227, 15893265, 25716449, 41610734, 67328268, 108940186, 176269708, 285211220 (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 A296245 for a guide to related sequences.

LINKS

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

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

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

a(2) = a(0) + a(1) + 2*b(2) = 15

Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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] + 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}]; (* A296296 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A342867 A060753 A241198 * A143880 A037388 A298370

Adjacent sequences:  A296293 A296294 A296295 * A296297 A296298 A296299

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 14 2017

STATUS

approved

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Last modified May 19 18:35 EDT 2022. Contains 353847 sequences. (Running on oeis4.)