A351581 Four-column table read by rows giving quadruples of integers [w,x,y,z] with w > 0, x > 1, y > 1 and z > 0 such that y^2 - y - x*z = 0 and x^2 = w*y, sorted by y then by x.
2, 2, 2, 1, 3, 3, 3, 2, 12, 6, 3, 1, 1, 2, 4, 6, 4, 4, 4, 3, 9, 6, 4, 2, 36, 12, 4, 1, 5, 5, 5, 4, 20, 10, 5, 2, 80, 20, 5, 1, 6, 6, 6, 5, 150, 30, 6, 1, 7, 7, 7, 6, 28, 14, 7, 3, 63, 21, 7, 2, 252, 42, 7, 1, 2, 4, 8, 14, 8, 8, 8, 7, 98, 28, 8, 2, 392, 56, 8, 1
Offset: 1
Examples
Table begins: [ w, x, y, z] ----------------- [ 2, 2, 2, 1]; [ 3, 3, 3, 2]; [ 12, 6, 3, 1]; [ 1, 2, 4, 6]; [ 4, 4, 4, 3]; [ 9, 6, 4, 2]; [ 36, 12, 4, 1]; [ 5, 5, 5, 4]; [ 20, 10, 5, 2]; [ 80, 20, 5, 1]; [ 6, 6, 6, 5]; [150, 30, 6, 1]; [ 7, 7, 7, 6]; [ 28, 14, 7, 3]; [ 63, 21, 7, 2]; [252, 42, 7, 1]; [ 2, 4, 8, 14]; [ 8, 8, 8, 7]; [ 98, 28, 8, 2]; [392, 56, 8, 1]; [ 1, 3, 9, 24]; [ 4, 6, 9, 12]; [ 9, 9, 9, 8]; [ 16, 12, 9, 6]; [ 36, 18, 9, 4]; [ 64, 24, 9, 3]; [144, 36, 9, 2]; [576, 72, 9, 1]; ... For row [1, 2, 4, 6], f(a,b,c) = a*b*c + 2*(a*b + a*c + b*c) + 4*(a+b+c) + 6. Theta = -2; id_1 = -1, id_2 = -3. The associative function f(a,b) = a*b + 2*(a+b) + 2 has theta = -2 and id = -1; f(f(a,b),c) = f(a,b,c). Another associative function g(a,b) = -a*b - 2*(a+b) - 6 with theta = -2 and id = -3 likewise gives g(g(a,b),c) = f(a,b,c).
Links
- David Lovler, Table of n, a(n) for n = 1..2720
- David Lovler, The first 680 quadruples for y up to 100.
Programs
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PARI
{ my(y); for (y = 2, 9, fordiv (y^2-y, x, if(type(w = x^2/y) == "t_INT", print([w, x, y, (y^2-y)/x]) )) ) }
Formula
Looking at A336013, if [X,Y,Z] is a row and f(a,b) = X*a*b + Y*(a+b) + Z is the corresponding associative function with id = -Z/Y and theta = -Y/X, then the composition f(f(a,b),c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z = f(a,b,c) gives the quadruple [w,x,y,z]. f(a,b,c) has the same theta as f(a,b); the two identity elements for f(a,b,c) are id and 2*theta - id.
If theta and the identity elements are computed from a quadruple, f(a,b,c) can be written as (a*b*c - theta*(a*b + a*c + b*c) + theta^2*(a+b+c) - theta^3)/(id-theta)^2 + theta. The square in the denominator ensures that f(a,b,c) is the same for either id.
Two parameters are sufficient to describe a row. For n*s > 1, rows are [w,x,y,z] = [n, n*s, n*s^2, (n^2*s^4-n*s^2)/(n*s)] = [n, n*s, n*s^2, n*s^3 - s]. In terms of n and s, theta = -s and id = s*(-1 +- 1/sqrt(n)). Rows with s=1 stand out as having w=x=y; theta = -1 and id = -1 +- 1/sqrt(w).
Comments