%I #35 Dec 03 2015 14:00:00

%S 1,2,3,5,6,7,10,14,15

%N Numbers from the Conway-Schneeberger 15-theorem.

%C The 15-theorem asserts that a positive definite integral quadratic form represents all numbers iff it represents the numbers in this sequence. "Integral" here means that the quadratic form equals x^T M x, where x is an integer vector and M is an integer matrix. - _T. D. Noe_, Mar 30 2006

%C Union of the first five triangular numbers {1, 3, 6, 10, 15} and their Möbius transform {1, 2, 5, 7, 14}, in ascending order. - _Daniel Forgues_, Feb 24 2015

%D Manjul Bhargava, On the Conway-Schneeberger fifteen theorem, Contemporary Mathematics 272 (1999), 27-37.

%D J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 141.

%D J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 23-26.

%D J. H. Conway and W. A. Schneeberger, personal communication.

%H Manjul Bhargava, <a href="http://hofprints.hofstra.edu/24/01/Bhargava,_Manjul_%282001%29_The_fifteen_theorem,_and_generalizations._In_Proceedings_Robert_J._Bumcrot_Festschrift,_Hofstra_University.pdf">The Fifteen Theorem and Generalizations</a>

%H Ivars Peterson, <a href="http://www.sciencenews.org/articles/20060311/bob9.asp">All Square: Science News Online</a> (subscription required)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/15_and_290_theorems">15 and 290 theorems</a>.

%F From _Daniel Forgues_, Feb 24 & 26 2015: (Start)

%F a(2n-1) = t_n = n*(n+1)/2 = A000217(n), 1 <= n <= 5;

%F a(2n) = Sum{d|(n+1)} mu(d) t_{(n+1)/d} = A007438(n+1), 1 <= n <= 4. (End)

%e a(2*1) = Sum{d|(1+1)} mu(d) t_{(1+1)/d} = 1 * t_2 + (-1) * t_1 = 3 - 1 = 2;

%e a(2*2) = Sum{d|(2+1)} mu(d) t_{(2+1)/d} = 1 * t_3 + (-1) * t_1 = 6 - 1 = 5;

%e a(2*3) = Sum{d|(3+1)} mu(d) t_{(3+1)/d} = 1 * t_4 + (-1) * t_2 + 0 * t_1 = 10 - 3 = 7;

%e a(2*4) = Sum{d|(4+1)} mu(d) t_{(4+1)/d} = 1 * t_5 + (-1) * t_1 = 15 - 1 = 14.

%t a[n_] := If[OddQ[n], (n+1)*(n+3)/8, DivisorSum[n/2+1, MoebiusMu[#]*(n+2#+2)*(n+2)/(8#^2) &]]; Array[a, 9] (* _Jean-François Alcover_, Dec 03 2015 *)

%Y Cf. A030051, A116582, A154363.

%K nonn,fini,full,nice

%O 1,2

%A _N. J. A. Sloane_