

A296248


Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n)^2, 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, 30, 69, 148, 281, 510, 891, 1522, 2557, 4248, 7001, 11474, 18731, 30494, 49549, 80404, 130353, 211198, 342035, 553762, 896373, 1450760, 2347809, 3799298, 6147891, 9948030, 16096882, 26045936, 42143907, 68190999, 110336131, 178528426, 288865926
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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



FORMULA

a(n) = H + R, where H = f(n1)*a(0) + f(n)*a(1) and R = f(n1)*b(2)^2 + f(n2)*b(3)^2 + ... + f(2)*b(n1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the nth Fibonacci number.


EXAMPLE

a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(2) = 30
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)


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;
j = 1; While[j < 6 , 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}] (* A296248 *)
Table[b[n], {n, 0, 20}] (* complement *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



