

A295757


Solution of the complementary equation a(n) = a(n1) + a(n3) + a(n4) + b(n1), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences.


1



1, 2, 3, 4, 15, 29, 46, 76, 132, 220, 356, 580, 949, 1543, 2498, 4047, 6560, 10623, 17191, 27822, 45030, 72870, 117910, 190790, 308720, 499531, 808263, 1307806, 2116091, 3423920, 5540025, 8963959, 14504008, 23467992, 37972016, 61440024, 99412066, 160852117
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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.
a(n)/a(n1) > (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).


LINKS



EXAMPLE

a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that
b(4) = 9 (least "new number")
a(4) = a(3) + a(1) + a(0) + b(3) = 15
Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...)


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; a[3] = 4;
b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8;
a[n_] := a[n] = a[n  1] + a[n  3] + a[n  4] + b[n  1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
z = 36; Table[a[n], {n, 0, z}] (* A295757 *)
Table[b[n], {n, 0, 20}] (*complement *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



