

A030050


Numbers from the ConwaySchneeberger 15theorem.


4




OFFSET

1,2


COMMENTS

The 15theorem 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
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


REFERENCES

Manjul Bhargava, On the ConwaySchneeberger fifteen theorem, Contemporary Mathematics 272 (1999), 2737.
J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 141.
J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 2326.
J. H. Conway and W. A. Schneeberger, personal communication.


LINKS

Table of n, a(n) for n=1..9.
Manjul Bhargava, The Fifteen Theorem and Generalizations
Ivars Peterson, All Square: Science News Online (subscription required)
Wikipedia, 15 and 290 theorems.


FORMULA

From Daniel Forgues, Feb 24 & 26 2015: (Start)
a(2n1) = t_n = n*(n+1)/2 = A000217(n), 1 <= n <= 5;
a(2n) = Sum{d(n+1)} mu(d) t_{(n+1)/d} = A007438(n+1), 1 <= n <= 4. (End)


EXAMPLE

a(2*1) = Sum{d(1+1)} mu(d) t_{(1+1)/d} = 1 * t_2 + (1) * t_1 = 3  1 = 2;
a(2*2) = Sum{d(2+1)} mu(d) t_{(2+1)/d} = 1 * t_3 + (1) * t_1 = 6  1 = 5;
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;
a(2*4) = Sum{d(4+1)} mu(d) t_{(4+1)/d} = 1 * t_5 + (1) * t_1 = 15  1 = 14.


MATHEMATICA

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] (* JeanFrançois Alcover, Dec 03 2015 *)


CROSSREFS

Cf. A030051, A116582, A154363.
Sequence in context: A139826 A182048 A028722 * A018336 A194359 A220355
Adjacent sequences: A030047 A030048 A030049 * A030051 A030052 A030053


KEYWORD

nonn,fini,full,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



