OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
FORMULA
G.f.: A(x) = Series_Reversion( (sqrt(1+4*x-4*x^2)-1)/2 + x^2 ).
G.f. satisfies: A(x) = x + ( x - A(x)^2 )^2.
G.f. satisfies: [A(x)^3 - A(-x)^3]/2 = x*[A(x) + A(-x)]/2.
Recurrence: 27*(n-2)*(n-1)*n*(16*n^3 - 160*n^2 + 521*n - 552)*a(n) = 18*(n-2)*(n-1)*(32*n - 105)*a(n-1) - 12*(n-2)*(384*n^5 - 5376*n^4 + 29288*n^3 - 77560*n^2 + 99709*n - 49665)*a(n-2) + 48*(2*n - 7)*(32*n^3 - 248*n^2 + 623*n - 505)*a(n-3) + 64*(n-4)*(2*n - 9)*(2*n - 7)*(16*n^3 - 112*n^2 + 249*n - 175)*a(n-4). - Vaclav Kotesovec, Dec 27 2013
Limit n->infinity |a(n)|^(1/n) = 4/3*sqrt(3+2*sqrt(3)) = 3.3899463424498833... - Vaclav Kotesovec, Dec 27 2013
EXAMPLE
A(x) = x + x^2 - 2*x^3 - 3*x^4 + 10*x^5 + 18*x^6 - 68*x^7 - 131*x^8 +...
A(x)^2 = x^2 + 2*x^3 - 3*x^4 - 10*x^5 + 18*x^6 + 68*x^7 - 131*x^8 - 530*x^9 +...
A(x)^3 = x^3 + 3*x^4 - 3*x^5 - 20*x^6 + 18*x^7 + 153*x^8 - 131*x^9 -++-...
Note that a bisection of A^3 equals a bisection of A.
MATHEMATICA
Rest[CoefficientList[1 + InverseSeries[Series[(Sqrt[1+4*x-4*x^2]-1)/2 + x^2, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Dec 27 2013 *)
PROG
(PARI) {a(n)=if(n<1, 0, polcoeff(serreverse((sqrt(1+4*x-4*x^2 +x*O(x^n))-1)/2 + x^2), n))}
(PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=x+subst(A, x, (-x+x*O(x^n)))^2); polcoeff(A, n)}
(PARI) {a(n)=local(A=x+x*O(x^n)); for(i=0, n, A = x + x^2 - 2*x*A^2 + A^4); polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 19 2008, Jul 20 2008
STATUS
approved