

A296269


Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n2)*b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.


2



2, 3, 10, 37, 82, 167, 312, 567, 987, 1697, 2852, 4744, 7820, 12819, 20927, 34069, 55356, 89824, 145620, 235927, 382075, 618577, 1001276, 1620528, 2622532, 4243843, 6867215, 11111957, 17980132, 29093112, 47074332, 76168599, 123244155, 199414084, 322659643
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OFFSET

0,1


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). See A296245 for a guide to related sequences.


LINKS



EXAMPLE

a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(0)*b(2) = 10;
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, ...)


MATHEMATICA

a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n  1] + a[n  2] + b[n  2] b[n];
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}]; (* A296269 *)
Table[b[n], {n, 0, 20}] (* complement *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



