OFFSET
1,1
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 Q = ax^2 + (4a/k - b)x + 4a/k^2 + 2b/k + c = 0. For a,b,c = 1,-1,-1 and k = 1,2,3..., the coefficients given by Q are the sequence 1,5,5; 1,3,1; 1,7/3,1/9... Clearing fractions and summing a+b+c gives the sequence.
The negative of the n-th term is the n+4th term of the c coefficient sequence A229526.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Russell Walsmith, CL-Chemy III: Hyper-Quadratics
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).
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^3-2*x^2+5*x+11) / ((x-1)^3*(x+1)^3). - Colin Barker, Nov 02 2014
a(n) = -(-5+3*(-1)^n)*(4+6*n+n^2)/8. - Colin Barker, Nov 03 2014
EXAMPLE
For k = 5, the coefficients are 1, 9/5, -11/25. Clearing fractions, 25, 45, -11 and 25 + 45 -11 = 59 = a[5].
PROG
(PARI) Vec(-x*(x^5-x^4-4*x^3-2*x^2+5*x+11)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Nov 02 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Russell Walsmith, Sep 26 2013
EXTENSIONS
More terms from Colin Barker, Nov 02 2014
STATUS
approved