A069981 Hermite's problem: number of positive integral solutions to x + y + z = n subject to x <= y + z, y <= z + x and z <= x + y.

0, 0, 0, 1, 3, 3, 7, 6, 12, 10, 18, 15, 25, 21, 33, 28, 42, 36, 52, 45, 63, 55, 75, 66, 88, 78, 102, 91, 117, 105, 133, 120, 150, 136, 168, 153, 187, 171, 207, 190, 228, 210, 250, 231, 273, 253, 297, 276, 322, 300, 348, 325, 375, 351, 403, 378, 432

Offset

0,5

References

G. Pólya and G. Szegő, Problems and Theorems in Analysis I, Springer-Verlag, Part I, Chap. 1, Problem 31.

Links

William A. Tedeschi, Table of n, a(n) for n = 0..10000

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 16.

G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis: The Omega Package, Europ. J. Combin., 22 (2001), 887-904.

Formula

G.f.: x^3*(1 + 2*x - 2*x^2)/(1 - x)/(1 - x^2)^2.

a(n) = (n+8)*(n-2)/8 for n even, (n^2-1)/8 for n odd.

a(n) = (2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16 for n>0. - Luce ETIENNE, Jun 29 2015

E.g.f.: (16 + (x^2 + x - 16)*cosh(x) + (x^2 + 7*x - 1)*sinh(x))/8. - Stefano Spezia, May 09 2022

Mathematica

f[n_]:=If[EvenQ[n], ((n+8)(n-2))/8, (n^2-1)/8]; Join[{0}, Array[f, 60]] (* Harvey P. Dale, Jul 26 2011 *)

Prog

(PARI) for(n=0, 60, print1(if(n==0, 0, (2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16), ", ")) \\ G. C. Greubel, Jun 10 2018
(Magma) [0] cat [(2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16: n in [1..60]]; // G. C. Greubel, Jun 10 2018

Crossrefs

Cf. A005044.

Sequence in context: A146970 A078708 A096273 * A000199 A243099 A324877

Adjacent sequences: A069978 A069979 A069980 * A069982 A069983 A069984

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, May 06 2002

Status

approved