

A294423


Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n1)  b(n2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.


2



1, 3, 8, 15, 28, 49, 85, 142, 236, 388, 635, 1035, 1684, 2733, 4432, 7181, 11630, 18829, 30478, 49327, 79826, 129175, 209024, 338223, 547273, 885522, 1432822, 2318372, 3751223, 6069625, 9820879, 15890536, 25711448, 41602018, 67313501, 108915555, 176229093
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OFFSET

0,2


COMMENTS

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 for a guide to related sequences.
Conjecture: a(n)/a(n1) > (1 + sqrt(5))/2, the golden ratio.


LINKS



EXAMPLE

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1)  b(0) + 2 = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 11, 13, ...)


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n  1] + a[n  2] + b[n  1]  b[n  2] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
Table[a[n], {n, 0, 40}] (* A294423 *)
Table[b[n], {n, 0, 10}]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



