A106216 -id:A106216 - cp's OEIS frontend

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

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.

A106218 Positions of 2's in A106216.

Original entry on oeis.org

6, 9, 12, 21, 27, 33, 69, 75, 84, 87, 93, 96, 111, 123, 126, 162, 165, 192, 225, 228, 231, 234, 237, 240, 264, 270, 273, 276, 288, 300, 306, 318, 321, 339, 345, 354, 357, 360, 378, 381, 384, 390, 417, 426, 429, 432, 438, 441, 447, 453, 468, 471, 474, 477, 483

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Cf. A106216, A106217, A106219.

PARI

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

A083349 Least positive integers not appearing previously such that the self-convolution cube-root of this sequence consists entirely of integers.

Original entry on oeis.org

1, 3, 6, 4, 9, 12, 7, 15, 18, 2, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 5, 54, 57, 10, 60, 63, 8, 66, 69, 72, 75, 78, 13, 81, 84, 87, 90, 93, 96, 99, 102, 16, 105, 108, 19, 111, 114, 11, 117, 120, 14, 123, 126, 22, 129, 132, 135, 138, 141, 25, 144, 147, 150, 153, 156, 28

Offset: 0

Paul D. Hanna, Apr 25 2003; revised May 01 2005

nonn

A permutation of the positive integers. Positive integers congruent to 1 (mod 3) appear in ascending order at positions given by A106213. Positive integers congruent to 2 (mod 3) appear in ascending order at positions given by A106214. The self-convolution cube-root is A083350.

			The self-convolution cube of A083350 equals this sequence: {1, 1, 1, -1, 3, 0, -6, 17, -17, -19, 114, ...}^3 = {1, 3, 6, 4, 9, 12, 7, 15, 18, ...}.
A083350(x)^3 = A(x) = 1 + 3x + 6x^2 + 4x^3 + 9x^4 + 12x^5 + 7x^6 + ...

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A106213, A106214, A083350, A106216.

a[n_] := a[n] = Module[{A, P, t}, A = 1+3x; P = Table[0, 3(n+1)]; P[[1]] = 1; P[[3]] = 2; For[j = 2, j <= n, j++, For[k = 2, k <= 3(n+1), k++, If[P[[k]] == 0, t = Coefficient[(A + k x^j + x^2 O[x]^j)^(1/3), x, j]; If[Denominator[t] == 1, P[[k]] = j+1; A = A + k*x^j; Break[]]]]]; Coefficient[A + x O[x]^n, x, n]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 66}] (* Jean-François Alcover, Jul 25 2018, translated from PARI *)

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

A106220 Coefficients of g.f. A(x) where 0 <= a(n) <= 3 for all n>1, with initial terms {1,4}, such that A(x)^(1/4) consists entirely of integer coefficients.

Original entry on oeis.org

1, 4, 2, 0, 3, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 3, 0, 2, 0, 3, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 2, 0, 1, 0, 2, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Equals the self-convolution 4th power of A106221. What is the frequency of occurrence of the nonzero digits?

			A(x) = 1 + 4*x + 2*x^2 + 3*x^4 + 2*x^6 + x^8 + 2*x^14 +...
A(x)^(1/4) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 10*x^5 - 26*x^6 +-...
A106221 = {1,1,-1,2,-4,10,-26,71,-199,569,-1652,...}.

Antti Karttunen, Table of n, a(n) for n = 0..1200

Cf. A106221, A106216, A106222, A106224, A106226.

{a(n)=local(A=1+4*x);if(n==0,1, for(j=1,n, for(k=0,3,t=polcoeff((A+k*x^j+x*O(x^j))^(1/4),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff(A+x*O(x^n),n)))}

A(z)=0 at z=-0.30239090673234876830066191989552890839853849934485...

A106222 Coefficients of g.f. A(x) where 0 <= a(n) <= 4 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

Original entry on oeis.org

1, 5, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Equals the self-convolution 5th power of A106223. What is the frequency of occurrence of the nonzero digits?

			A(x) = 1 + 5*x + x^5 + 3*x^10 + x^15 + 4*x^20 + x^35 +...
A(x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 +-...
A106223 = {1,1,-2,6,-21,80,-320,1326,-5637,24434,...}.

Cf. A106223, A106216, A106220, A106224, A106226.

{a(n)=local(A=1+5*x);if(n==0,1, for(j=1,n, for(k=0,4,t=polcoeff((A+k*x^j+x*O(x^j))^(1/5),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff(A+x*O(x^n),n)))}

A(z)=0 at z=-0.1999361633111821182995648612577212067...

A106224 Coefficients of g.f. A(x) where 0 <= a(n) <= 5 for all n>1, with initial terms {1,6}, such that A(x)^(1/6) consists entirely of integer coefficients.

Original entry on oeis.org

1, 6, 3, 2, 3, 0, 0, 0, 3, 4, 3, 0, 0, 0, 3, 2, 0, 0, 0, 0, 3, 2, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 4, 3, 0, 2, 0, 0, 4, 0, 0, 5, 0, 3, 2, 0, 0, 3, 0, 0, 0, 3, 0, 3, 0, 3, 0, 3, 0, 2, 0, 3, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 5, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Equals the self-convolution 6th power of A106225. What is the frequency of occurrence of the nonzero digits?

			A(x) = 1 + 6*x + 3*x^2 + 2*x^3 + 3*x^4 + 3*x^8 + 4*x^9 + 3*x^10 +...
A(x)^(1/6) = 1 + x - 2*x^2 + 7*x^3 - 27*x^4 + 114*x^5 - 506*x^6 +-...
A106225 = {1,1,-2,7,-27,114,-506,2322,-10919,52316,-254369,...}.

Cf. A106225, A106216, A106220, A106222, A106226.

{a(n)=local(A=1+6*x);if(n==0,1, for(j=1,n, for(k=0,5,t=polcoeff((A+k*x^j+x*O(x^j))^(1/6),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff(A+x*O(x^n),n)))}

A(z)=0 at z=-0.18172379526003557530948965401615522817...

A106226 Coefficients of g.f. A(x) where 0 <= a(n) <= 6 for all n>1, with initial terms {1,7}, such that A(x)^(1/7) consists entirely of integer coefficients.

Original entry on oeis.org

1, 7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, May 01 2005

nonn

Equals the self-convolution 7th power of A106227. What is the frequency of occurrence of the nonzero digits?

			A(x) = 1 + 7*x + x^7 + 4*x^14 + 6*x^21 + 5*x^28 + x^35 + 6*x^42 +...
A(x)^(1/7) = 1 + x - 3*x^2 + 13*x^3 - 65*x^4 + 351*x^5 - 1989*x^6 +-...
A106227 = {1,1,-3,13,-65,351,-1989,11650,-69900,427167,...}.

Cf. A106227, A106216, A106220, A106222, A106224.

{a(n)=local(A=1+7*x);if(n==0,1, for(j=1,n, for(k=0,6,t=polcoeff((A+k*x^j+x*O(x^j))^(1/7),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff(A+x*O(x^n),n)))}

A196306 Coefficients of g.f. A(x) where -1 <= a(n) <= 1 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

Original entry on oeis.org

1, 3, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0

Offset: 0

Paul D. Hanna, Oct 01 2011

sign

A(z) = 0 at z = -0.3218604126 5330206348 1666946119 2701743677 6909817084 3826086189 5315539535 3583969883 ...

			G.f.: A(x) = 1 + 3*x + x^3 - x^6 - x^9 - x^12 - x^18 + x^21 - x^24 - x^30 - x^33 + x^39 - x^42 - x^45 + x^48 + x^54 - x^60 - x^66 - x^69 + x^72 - x^75 + x^84 - x^87 - x^90 - x^93 - x^96 - x^102 + x^108 - x^114 - x^126 + x^132 - x^135 - x^138 + x^141 - x^144 - x^147 - x^153 - x^159 - x^162 + x^165 - x^168 - x^171 + x^174 - x^180 - x^183 + x^186 - x^189 - x^192 + x^195 +...
where
A(x)^(1/3) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 22*x^6 + 55*x^7 - 142*x^8 + 375*x^9 - 1009*x^10 + 2753*x^11 - 7599*x^12 + 21178*x^13 - 59509*x^14 + 168401*x^15 - 479477*x^16 + 1372536*x^17 - 3947678*x^18 + 11402376*x^19 - 33059314*x^20 + 96177750*x^21 +...+ A196307(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1001

Cf. A196307 (cube-root), A196308 (trisection); variants: A106216, A083953.

{a(n)=local(A=1+3*x); if(n==0, 1, if(n%3==0,for(j=1, n, for(k=-1, 1, 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+x*O(x^n), n))}

cp's OEIS Frontend

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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PARI

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A106217 Positions of 1's in A106216.

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A106218 Positions of 2's in A106216.

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A083349 Least positive integers not appearing previously such that the self-convolution cube-root of this sequence consists entirely of integers.

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Mathematica

PARI

A106220 Coefficients of g.f. A(x) where 0 <= a(n) <= 3 for all n>1, with initial terms {1,4}, such that A(x)^(1/4) consists entirely of integer coefficients.

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A106222 Coefficients of g.f. A(x) where 0 <= a(n) <= 4 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

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A106224 Coefficients of g.f. A(x) where 0 <= a(n) <= 5 for all n>1, with initial terms {1,6}, such that A(x)^(1/6) consists entirely of integer coefficients.

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A106226 Coefficients of g.f. A(x) where 0 <= a(n) <= 6 for all n>1, with initial terms {1,7}, such that A(x)^(1/7) consists entirely of integer coefficients.

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A196306 Coefficients of g.f. A(x) where -1 <= a(n) <= 1 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

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