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A296297 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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. 2
2, 4, 16, 38, 82, 160, 296, 526, 910, 1544, 2584, 4282, 7046, 11549, 18847, 30681, 49848, 80886, 131130, 212453, 344063, 557041, 901676, 1459338, 2361686, 3821749, 6184215, 10006801, 16191912, 26199670, 42392602, 68593357, 110987111, 179581689, 290570126 (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) = 4, b(0) = 1, b(1) = 3, b(2) = 5

a(2) = a(0) + a(1) + 2*b(2) = 16

Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, ...)

MATHEMATICA

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; 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}]; (* A296297 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A000216 A110998 A051861 * A290265 A223093 A333022

Adjacent sequences:  A296294 A296295 A296296 * A296298 A296299 A296300

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 14 2017

STATUS

approved

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Last modified September 24 06:24 EDT 2020. Contains 337317 sequences. (Running on oeis4.)