Years ago Paul Leopardi told me (to my surprise) "I think the Hadamard matrix conjecture is wrong". Here is another such phenomenon (a type of object where the apparent initial growth is misleading): Costas arrays See https://oeis.org/A001441 and other sequences found via https://oeis.org/search?q=name%3A%22costas+array%22&sort=&language=&go=Searc... Best regards, jj * Veit Elser <ve10@cornell.edu> [Sep 07. 2016 18:02]:
The Hadamard matrix conjecture holds that such matrices exist for all orders that are divisible by 4. After surveying what’s been done on the classification/enumeration of Hadamard matrices (e.g. http://neilsloane.com/hadamard/) I’ve felt that what’s humbling about the conjecture is that we lack the knowledge to prove the existence of even a single Hadamard (at each possible order) when the evidence points to a very rapid growth in their number. Now I’m not so sure.
Let x = log_2(N), N = order of Hadamard (a multiple of 4), and y = (1/N^2)log_2(num(N)), where num(N) is the number of Hadamard matrices of order N. Sequence A206711 gives num(N) for N = 1, … , 32. If you plot y vs. x you get a very straight line: y = 0.78785 - 0.09458 x. Taking this literally, there should be a maximum in the number of Hadamard matrices at order N = 196, and beyond that the number plummets, vanishing at around N = 322. The available constructions (beyond this number) would then represent isolated points.
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