OFFSET
1,1
COMMENTS
Aside from the first term, this appears to be a subset of A060215.
Same as A071296 if you drop a(0) and replace each repeated pair x,x with 0,x (credit to Daniel Suteu for pointing this out).
These are also the least a(n) such that the continued fraction expansion for sqrt(a(n) - floor(a(n))) begins with (n-1) 1's.
FORMULA
Conjectures from Colin Barker, Aug 26 2019: (Start)
G.f.: x*(2 + x + 4*x^2 - 42*x^3 - 15*x^4 - 39*x^5 + 100*x^6 + x^7 - 61*x^8 + 172*x^9 + 31*x^10 - 17*x^11 + 26*x^12 - 2*x^13 + x^14 - 2*x^15) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x + x^2)*(1 - x - x^2)*(1 + x + 2*x^2 - x^3 + x^4)*(1 + 3*x + 8*x^2 + 3*x^3 + x^4)).
a(n) = a(n-1) + 21*a(n-3) - 21*a(n-4) - 50*a(n-6) + 50*a(n-7) - 86*a(n-9) + 86*a(n-10) - 13*a(n-12) + 13*a(n-13) + a(n-15) - a(n-16) for n>16.
(End)
EXAMPLE
For n = 5 the convergents of sqrt(13) are 3/1, 4/1, 7/2, 11/3, 18/5, 119/33, ... and the first five denominators are 1, 1, 2, 3, 5, which match the first five Fibonacci numbers. Since 13 is the first number with this property, then a(5) = 13.
MATHEMATICA
c = 1;
n = 2;
F = Table[Fibonacci[n], {n, 20}];
While[c <= 14,
If[! IntegerQ[Sqrt[n]]
&&
Denominator[Convergents[Sqrt[n], c]] == F[[1 ;; c]],
Print[n, " ", Denominator[Convergents[Sqrt[n], c]]];
c++; n--];
n++
]
CROSSREFS
KEYWORD
nonn
AUTHOR
Greg Dresden, Aug 11 2019
STATUS
approved