

A229525


Sum of 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...


2



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, 2111, 551, 2299, 599, 2495, 649, 2699, 701, 2911, 755
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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 nth 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, CLChemy III: HyperQuadratics
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(n2)3*a(n4)+a(n6). G.f.: x*(x^5x^44*x^32*x^2+5*x+11) / ((x1)^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^5x^44*x^32*x^2+5*x+11)/((x1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Nov 02 2014


CROSSREFS

The a coefficients are A168077, b coefficients are A171621, c coefficients are A229526.
Sequence in context: A168206 A120831 A253254 * A174103 A038319 A002547
Adjacent sequences: A229522 A229523 A229524 * A229526 A229527 A229528


KEYWORD

nonn,easy


AUTHOR

Russell Walsmith, Sep 26 2013


EXTENSIONS

More terms from Colin Barker, Nov 02 2014


STATUS

approved



