A124261 Sequence terms are generated by solving the n x n linear algebra problem [H]x = b, where b is the unit vector. Only xn, the last unknown is used.

1, 1, 2, 3, 12, 20, 300, 525, 1960, 49392, 1481760, 5821200, 164656800, 336370320, 3741185448

Offset: 1

Author

L. Van Warren (van(AT)wdv.com), Oct 23 2006

Keywords

• eigen
• frac
• hard
• nonn

Comments

These matrices are notorious for being ill-conditioned, they are solved using rational arithmetic. To reproduce these numbers, or larger ones, one can edit the program above with a text editor and then drag it into the Maxima window. The last term required approx. 10 minutes to generate on an Intel dual-core 6600 clocked at 2.4 GHz with 2 Gig of RAM. To change the order of the Hilbert matrix, just change the order variable at the top of the file. It is currently 4, which runs quickly. So a point of curiosity really. The numbers become extremely expensive to find, growing at least as the cube of n times a large constant. So to me they are a kind of symbolic gold. The 100th number for example, is probably not knowable at the current time, but that is speculation on my part. Perhaps you will observe some functional relationship that allows their simple generation thus "Cracking the Hilbert-Warren Sequence Code". For example 12 = 4 * 4 - 4, 20 = 5 * 5 - 5, 300 = 20 * 20 - 100, 525 = 25 * 25 - 100 and so on.

Examples

			Triangularization of [H] is performed, followed by inversion or back substitution. The terms are the denominators of the last solution xn to the Hilbert matrices of order 1, 2, ... 15, respectively. The numerator is unity in all cases.
For order 3 problem the solutions are: ([1/6],[ -1/3],[1/2]), thus a() = 2, since we just take the denominator.
		

Links

• William Schelter, DOE Maxima - A Symbolic Algebra Program
• Gilbert Strang, Linear Algebra and Its Applications
• L. V. Warren, Cracking the Hilbert-Warren Code

Crossrefs

Cf. A124264.

Programs

Maxima

load("eigen"); order : 4; X : columnvector(makelist(concat(x,i), i, 1, order)); h[i,j] := 1/(i + j -1); Unity[i,j] := 1; A : genmatrix(h, order, order); A . X; B : genmatrix(Unity, 1, order); A . X = B; Ap : triangularize(A); Ap . X = B; App : invert(Ap); App . B;

Formula

Solve the n x n linear algebra problem [H]x = b, where b is the unit vector and n is the order. Only xn, the last unknown is used.

Extensions

The author said that this version of the submission was unsatisfactory and set me a better version in email, which may replace this. - N. J. A. Sloane, Oct 24 2006