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A251659
G.f. satisfies: A(x) = A(x^2 - x^3)/(1-x).
1
1, 1, 2, 1, 3, -1, 2, -1, 5, -8, 9, 2, -3, -5, 19, -13, 1, -16, 83, -104, -40, 263, -198, -332, 998, -1276, 999, -272, -799, 1991, -2301, -321, 7291, -15610, 17173, -5673, -15387, 35638, -46540, 33430, 47130, -245231, 498292, -507754, -145879, 1531208, -2818709, 2246844, 1870729, -9062799, 14639125
OFFSET
0,3
COMMENTS
The eratic behavior of this sequence is likely due to the fact that the roots of 1-x+x^2 = 0 are complex: (1 + I*sqrt(3))/2 and (1 - I*sqrt(3))/2.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = A( x^4 * (1-x)^2 * (1-x^2+x^3) ) / ( (1-x) * (1-x^2+x^3) ).
(2) A(x) = A( x^8 * (1-x)^4 * (1-x^2+x^3)^2 * (1 - x^4*(1-x)^2*(1-x^2+x^3)) ) / ( (1-x) * (1-x^2+x^3) * (1 - x^4*(1-x)^2*(1-x^2+x^3)) ).
(3) A(x) = Sum_{n>=0} a(n) * x^(2*n) * (1-x)^(n-1).
(4) A(x) = Sum_{n>=0} a(n) * x^(4*n) * (1-x)^(2*n-1) * (1-x^2+x^3)^(n-1).
(5) A(x) = Sum_{n>=0} a(n) * x^(8*n) * (1-x)^(4*n-1) * (1-x^2+x^3)^(2*n-1) * (1 - x^4*(1-x)^2*(1-x^2+x^3))^(n-1).
Recurrence/asymptotic formulas:
a(n) = 1 + Sum_{k=[(n+1)/3]..[n/2]} (-1)^n * C(k-1, n-2*k) * a(k), for n>1 with a(0)=1, a(1)=1.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 - x^5 + 2*x^6 - x^7 + 5*x^8 +...
such that A(x) = A(x^2 - x^3)/(1-x), where:
A(x^2-x^3) = 1 + x^2 - x^3 + 2*x^4 - 4*x^5 + 3*x^6 - 3*x^7 + 6*x^8 - 13*x^9 + 17*x^10 - 7*x^11 - 5*x^12 - 2*x^13 + 24*x^14 - 32*x^15 + 14*x^16 - 17*x^17 + 99*x^18 - 187*x^19 + 64*x^20 + 303*x^21 - 461*x^22 - 134*x^23 + 1330*x^24 +...
Also, the g.f. satisfies:
A(x) = 1/(1-x) + a(1)*x^2 + a(2)*x^4*(1-x) + a(3)*x^6*(1-x)^2 + a(4)*x^8*(1-x)^3 + a(5)*x^10*(1-x)^4 + a(6)*x^12*(1-x)^5 + a(7)*x^14*(1-x)^6 +...
Illustration of Recurrence:
a(0) = 1;
a(1) = 1;
a(2) = 1 + 1*a(1);
a(3) = 1 + 0*a(1);
a(4) = 1 + 0*a(1) + 1*a(2);
a(5) = 1 + 0*a(1) + -1*a(2);
a(6) = 1 + 0*a(1) + 0*a(2) + 1*a(3);
a(7) = 1 + 0*a(1) + 0*a(2) + -2*a(3);
a(8) = 1 + 0*a(1) + 0*a(2) + 1*a(3) + 1*a(4);
a(9) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + -3*a(4);
a(10) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + 3*a(4) + 1*a(5);
a(11) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + -1*a(4) + -4*a(5);
a(12) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + 0*a(4) + 6*a(5) + 1*a(6);
a(13) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + 0*a(4) + -4*a(5) + -5*a(6);
a(14) = 1 + 0*a(1) + 0*a(2) + 0*a(3) + 0*a(4) + 1*a(5) + 10*a(6) + 1*a(7); ...
which may be written as:
a(n) = 1 + Sum_{k=[(n+1)/3]..[n/2]} (-1)^n * C(k-1, n-2*k) * a(k), for n>1.
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, #binary(n)+1, A=subst(A, x, x^2-x^3)/(1-x +x*O(x^n))); polcoeff(A, n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) /* Recurrence: */
{a(n) = 1 + if(n>1, (-1)^n * sum(k=(n+1)\3, n\2, binomial(k-1, n-2*k) * a(k) ))}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
Cf. A251572.
Sequence in context: A355758 A206778 A101872 * A174892 A317656 A319136
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 10 2015
STATUS
approved