

A294555


Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n1) + b(n2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.


2



1, 2, 13, 27, 54, 97, 169, 286, 477, 787, 1290, 2106, 3428, 5568, 9032, 14638, 23710, 38390, 62144, 100580, 162772, 263402, 426226, 689682, 1115965, 1805707, 2921734, 4727505, 7649305, 12376878, 20026253, 32403203, 52429530, 84832809, 137262417, 222095306
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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(n1) > (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), 113.


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(1) + b(0) + 3 = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, ...)


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  1] + b[n  2] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
Table[a[n], {n, 0, 40}] (* A294555 *)
Table[b[n], {n, 0, 10}]


CROSSREFS

Cf. A001622, A294532.
Sequence in context: A274869 A018400 A091052 * A031090 A358296 A294556
Adjacent sequences: A294552 A294553 A294554 * A294556 A294557 A294558


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 15 2017


EXTENSIONS

Definition corrected by Georg Fischer, Sep 27 2020


STATUS

approved



