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A296286 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 3, 8, 23, 51, 104, 197, 364, 641, 1104, 1865, 3112, 5145, 8452, 13821, 22528, 36637, 59488, 96485, 156372, 253297, 410173, 664020, 1074791, 1739459, 2814950, 4555163, 7370923, 11926954, 19298805, 31226749, 50526608, 81754477, 132282273, 214038008 (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 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) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5

a(2) = a(0) + a(1) + 2*b(0) = 8

Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, ...)

MATHEMATICA

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

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

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}]; (* A296286 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A183930 A183922 A340493 * A068602 A255834 A027212

Adjacent sequences:  A296283 A296284 A296285 * A296287 A296288 A296289

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 14 2017

STATUS

approved

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Last modified October 25 20:31 EDT 2021. Contains 348256 sequences. (Running on oeis4.)