A110630 -id:A110630 - cp's OEIS frontend

A112570 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110630, which consists entirely of numbers 1 through 4.

1, 1, 1, 1, -1, 2, 0, 1, -2, 5, -5, 4, -6, 18, -30, 35, -43, 84, -167, 261, -352, 545, -1010, 1790, -2783, 4207, -7025, 12464, -21071, 33567, -54154, 92317, -159366, 266150, -435285, 725260, -1239404, 2112351, -3535532, 5894852, -9964767, 17008752, -28880694, 48645873

Offset: 0

Paul D. Hanna, Sep 14 2005

sign

A110630 is formed from every 2nd term of A083954, which also consists entirely of numbers 1 through 4.

			A(x) = 1 + x + x^2 + x^3 - x^4 + 2*x^5 + x^7 - 2*x^8 + 5*x^9 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + x^4 + 4*x^5 + 3*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 27*x^4 + 36*x^5 +...
A(x)^4 (mod 8) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
G(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 + 4*x^6 +...
where G(x) is the g.f. of A083954.

Cf. A110630, A083954.

{a(n)=local(d=2,m=4,A=1+m*x); for(j=2,d*n, for(k=1,m,t=polcoeff((A+k*x^j+x*O(x^j))^(1/m),j); if(denominator(t)==1,A=A+k*x^j;break))); polcoeff(Ser(vector(n+1,i,polcoeff(A,d*(i-1))))^(1/2),n)}

G.f. A(x) satisfies: A(x)^4 (mod 8) = g.f. of A083954.

A110629 Every 4th term of A083954 such that the self-convolution 4th power is congruent modulo 8 to A083954, which consists entirely of numbers 1 through 4.

Original entry on oeis.org

1, 3, 1, 3, 3, 2, 4, 3, 2, 3, 3, 4, 2, 2, 2, 1, 1, 4, 1, 3, 4, 3, 2, 2, 1, 1, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 4, 1, 4, 4, 1, 4, 2, 3, 1, 2, 1, 4, 3, 3, 1, 4, 3, 3, 2, 3, 4, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 1, 4, 2, 2, 3, 1, 4, 3, 2, 1, 4, 3, 4, 4, 2, 1, 4, 1, 4, 4, 2, 4, 4, 1, 3, 3, 4, 1, 1, 1, 4, 3, 2, 1, 3, 1, 2, 2

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 09 2005

nonn

			A(x) = 1 + 3*x + x^2 + 3*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + ...
A(x)^4 = 1 + 12*x + 58*x^2 + 156*x^3 + 315*x^4 + 620*x^5 +...
A(x)^4 (mod 8) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
G083954(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
where G083954(x) is the g.f. of A083954.

Cf. A083954, A110630.

{a(n)=local(d=4,m=4,A=1+m*x); for(j=2,d*n, for(k=1,m,t=polcoeff((A+k*x^j+x*O(x^j))^(1/m),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,d*n)}

a(n) = A083954(4*n) for n>=0.

cp's OEIS Frontend

A112570 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110630, which consists entirely of numbers 1 through 4.

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