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A294540 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n - 1, where a(0) = 1, a(1) = 2, b(0) = 3. 2
1, 2, 9, 20, 41, 76, 135, 232, 392, 652, 1075, 1761, 2873, 4674, 7590, 12310, 19949, 32311, 52316, 84686, 137064, 221815, 358947, 580833, 939854, 1520764, 2460698, 3981545, 6442329, 10423963, 16866384, 27290442, 44156924, 71447467, 115604495, 187052069 (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.  See A294532 for a guide to related sequences.  Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622)..

LINKS

Table of n, a(n) for n=0..35.

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

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, so that

b(1) = 4 (least "new number")

a(2)  = a(1) + a(0) + b(0) + 3 = 9

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...)

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

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

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

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 40}]  (* A294540 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A173102 A090398 A091941 * A248435 A272211 A259035

Adjacent sequences:  A294537 A294538 A294539 * A294541 A294542 A294543

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 03 2017

STATUS

approved

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Last modified May 16 00:49 EDT 2021. Contains 343937 sequences. (Running on oeis4.)