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 1-4 of 4 results.

A106221 Self-convolution 4th power equals A106220, which consists entirely of digits {0,1,2,3} after the initial terms {1,4}.

Original entry on oeis.org

1, 1, -1, 2, -4, 10, -26, 71, -199, 569, -1652, 4855, -14413, 43153, -130143, 394967, -1205268, 3695771, -11381215, 35183209, -109138163, 339599993, -1059702401, 3315256789, -10396158911, 32671424776, -102879610571, 324557399534, -1025643986057, 3246330348415, -10290418283163
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Examples

			A(x) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 10*x^5 - 26*x^6 + 71*x^7 -+...
A(x)^4 = 1 + 4*x + 2*x^2 + 3*x^4 + 2*x^6 + x^8 + 2*x^14 +...
A106220 = {1,4,2,0,3,0,2,0,1,0,0,0,0,0,2,0,0,0,2,...}.

Crossrefs

Cf. A106220, A106219, A106223, A106225.

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))^(1/4),n)))}

Formula

Limit a(n+1)/a(n) = -3.30697774878897620974321728382452592372871...

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

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

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

Examples

			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,...}.

Crossrefs

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

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),n)))}

Formula

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

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)))}
Showing 1-4 of 4 results.

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