A229526 The c coefficients of the transform ax^2 + (4a/k - b)x + 4a/k^2 + 2b/k + c= 0 for a,b,c = 1,-1,-1, k = 1,2,3...

5, 1, 1, -1, -11, -5, -31, -11, -59, -19, -95, -29, -139, -41, -191, -55, -251, -71, -319, -89, -395, -109, -479, -131, -571, -155, -671, -181, -779, -209, -895, -239, -1019, -271, -1151, -305, -1291, -341, -1439, -379, -1595, -419, -1759, -461, -1931, -505

Offset: 1

Author

Russell Walsmith, Sep 27 2013

Keywords

• sign
• easy

Comments

The positive/negative roots of ax^2 + bx + c = 0 combine with the negative/positive roots of (ck^2 - bk + c)x^2 +(2ck - b)x + c = 0 to define a point on the hyperbola kxy + x + y = 0. To shift such points (roots) to the hyperbola’s other line, put the coefficients of these equations into the formula ax^2 + (4a/k - b)x + 4a/k^2 + 2b/k + c = 0. Let a,b,c = 1,-1,-1 and k = 1,2,3... Then the coefficients given by this last equation are the sequence 1,5,5; 1,3,1; 1,7/3,1/9... Clearing fractions, the c coefficients are the sequence above.

The n-th term = the (positive) n-4th term of A229525.

Examples

			For k = 5, the coefficients are 1, 9/5, -11/25. Clearing fractions gives 25, 45, -11 and -11 = a[5].
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Russell Walsmith, CL-Chemy III: Hyper-Quadratics

Crossrefs

The a coefficients are A168077, b coefficients are A171621, the sum of a, b and c coefficients is A229525.

Programs

PARI

Vec(-x*(x^5+x^4-4*x^3-14*x^2+x+5)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Nov 02 2014

Formula

ax^2 + (4a/k - b)x + 4a/k^2 + 2b/k + c; a,b,c = 1,-1,-1, k = 1,2,3..n.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: -x*(x^5+x^4-4*x^3-14*x^2+x+5) / ((x-1)^3*(x+1)^3). - Colin Barker, Nov 02 2014

a(n) = (-5+3*(-1)^n)*(-4-2*n+n^2)/8. - Colin Barker, Nov 03 2014

Extensions

More terms from Colin Barker, Nov 02 2014