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A296287 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 3, 7, 22, 49, 101, 198, 362, 640, 1101, 1861, 3105, 5134, 8434, 13792, 22481, 36561, 59365, 96286, 156050, 252796, 409350, 662696, 1072644, 1735988, 2809332, 4546074, 7356216, 11903158, 19260302, 31164450, 50425806, 81591376, 132018370, 213611004 (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(0) = 7

Complement: (b(n)) = (1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ...)

MATHEMATICA

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

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}]; (* A296287 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A109456 A155745 A067738 * A187014 A053966 A010738

Adjacent sequences:  A296284 A296285 A296286 * A296288 A296289 A296290

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 14 2017

STATUS

approved

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Last modified April 15 07:40 EDT 2021. Contains 342975 sequences. (Running on oeis4.)