I don't know if I'd agree with NJA Sloane the evidence for it is "overwhelming" but it is pretty good. I find http://oeis.org/A007299 unconvincing since it has so few entries. And I have strong doubts this sequence is monotonic. You know Laplace said the probability the sun rises tomorrow based on fact it was true N previous days, is (N+1)/(N+2). The probability the Hadamard conjecture is true is in my view about 99%. It previously was conjectured that Williamson-form Hadamard matrices always existed. True for n=1,2,3,...,34. But false for n=35 (no 140x140 Hadamard of Williamson form exists, 140=4*35). So this should serve as a cautionary tale. I suggest you solve my recently posted question about highly cancelling polynomials. If you do I will be able with probability 99% to make very strong progress on the Hadamard conjecture. (In fact in unpublished work I already did make strong progress on the Hadamard conjecture...)