cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A106223 Self-convolution 5th power equals A106222, which consists entirely of digits {0,1,2,3,4} after the initial terms {1,5}.

Original entry on oeis.org

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107541, 479192, -2157027, 9792618, -44780207, 206053429, -953296364, 4431418833, -20686477329, 96930426941, -455717114981, 2149060994827, -10162417338993, 48176297258115, -228910042632050, 1089957826522693, -5199911987465160
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Examples

			A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 +-...
A(x)^5 = 1 + 5*x + x^5 + 3*x^10 + x^15 + 4*x^20 + x^35 +...
A106222 = {1,5,0,0,0,1,0,0,0,0,3,0,0,0,0,1,0,0,0,0,4,...}.

Crossrefs

Cf. A106222, A106219, A106221, A106225.

Programs

  • PARI
    {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))^(1/5),n)))}

Formula

Limit a(n+1)/a(n) = -5.001596426773442826534115368782519...

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

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

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

Examples

			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,...}.

Links

Crossrefs

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

Programs

  • PARI
    {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)))}

Formula

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

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

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

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

Examples

			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,...}.

Crossrefs

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

Programs

  • PARI
    {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)))}

Formula

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

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

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

Examples

			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,...}.

Crossrefs

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

Programs

  • PARI
    {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)))}

A196344 Coefficients of g.f. A(x) where -2 <= a(n) <= 2 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, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Oct 01 2011

Keywords

Comments

A(z) = 0 at z = -0.1999360614 0060333411 6048445601 7524456163 6606200034 1004259693 ...

Examples

			G.f.: A(x) = 1 + 5*x + x^5 - 2*x^10 + x^15 - x^20 + x^35 - 2*x^40 - x^45 - 2*x^50 + x^55 + 2*x^65 - x^70 + 2*x^80 +...
where the following series consists entirely of integer coefficients:
A(x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 +...+ A196345(n)*x^n +...

Links

Crossrefs

Cf. A196345 (5th root), A196346 (quintisection), A106222 (variant).

Programs

  • PARI
    {a(n)=local(A=1+5*x); if(n==0, 1, if(n%5==0,for(j=1, n, for(k=-2, 2, t=polcoeff((A+k*x^j+x*O(x^j))^(1/5), j);
if(denominator(t)==1, A=A+k*x^j; break)))); polcoeff(A+x*O(x^n), n))}
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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