A106218 -id:A106218 - cp's OEIS frontend

A106216 Coefficients of g.f. A(x) where 0 <= a(n) <= 2 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

1, 3, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, May 01 2005

nonn

The self-convolution cube-root equals A106219. Positions of 1's is given by A106217. Positions of 2's is given by A106218. What is the frequency of occurrence of the 1's and 2's?

			A(x)^(1/3) = 1 + 1x - 1x^2 + 2x^3 - 4x^4 + 9x^5 - 21x^6 + 53x^6 -+...

Cf. A106217, A106218, A106219, A083953.

{a(n)=local(A=1+3*x);if(n==0,1,if(n==1,3, for(j=2,n, for(k=0,2,t=polcoeff((A+k*x^j+x*O(x^j))^(1/3),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,n)))}

A(z)=0 at z=-0.322846893915891638743032676733152456643928599...

A106219 Self-convolution cube-root of A106216, which consists entirely of digits {0,1,2} after the initial terms {1,3}.

Original entry on oeis.org

1, 1, -1, 2, -4, 9, -21, 53, -137, 362, -971, 2642, -7272, 20211, -56631, 159795, -453650, 1294797, -3713100, 10693036, -30910440, 89657680, -260860962, 761114168, -2226409022, 6528039545, -19182376302, 56479676608, -166605140314, 492304708589, -1457061274821, 4318906269671

Offset: 0

Paul D. Hanna, May 01 2005

sign

			A(x) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 21*x^6 + 53*x^7 -+...
A(x)^3 = 1 + 3*x + x^3 + 2*x^6 + 2*x^9 + 2*x^12 + 2*x^21 + x^24 +...
A106216 = {1,3,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,...}.

Cf. A106216, A106217, A106218.

{a(n)=local(A=1+3*x);if(n==0,1, for(j=1,n, for(k=0,2,t=polcoeff((A+k*x^j+x*O(x^j))^(1/3),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff((A+x*O(x^n))^(1/3),n)))}

Limit a(n+1)/a(n) = -3.09744345956297443415996844224370585278444314...

A106217 Positions of 1's in A106216.

Original entry on oeis.org

0, 3, 24, 30, 48, 57, 63, 72, 81, 99, 105, 114, 120, 129, 132, 135, 141, 147, 156, 159, 168, 177, 180, 195, 198, 201, 204, 207, 210, 222, 243, 249, 252, 261, 267, 279, 282, 285, 297, 309, 312, 315, 327, 333, 342, 351, 375, 387, 393, 399, 402, 408, 411, 414

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Cf. A106216, A106218, A106219.

PARI

a(n) = 0 (mod 3) for all n.

cp's OEIS Frontend

A106216 Coefficients of g.f. A(x) where 0 <= a(n) <= 2 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

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A106219 Self-convolution cube-root of A106216, which consists entirely of digits {0,1,2} after the initial terms {1,3}.

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A106217 Positions of 1's in A106216.

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