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.

A112574 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110649, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 3, 0, 2, 0, 3, -8, 30, -90, 290, -930, 3000, -9696, 31461, -102420, 334467, -1095510, 3598464, -11852026, 39136629, -129548493, 429817733, -1429178703, 4761992751, -15898024868, 53174651133, -178168302693, 597971203902, -2010093276240, 6767100270918
Offset: 0

Views

Author

Paul D. Hanna, Sep 14 2005

Keywords

Comments

A110649 is formed from every 2nd term of A084067, which also consists entirely of numbers 1 through 12.
Why are so many a(n) divisible by 3, i.e., 22 of the first 28? - Jonathan Vos Post, Sep 14 2005

Examples

			A(x) = 1 + 3*x + 2*x^3 + 3*x^5 - 8*x^6 + 30*x^7 - 90*x^8 +..
A(x)^2 = 1 + 6*x + 9*x^2 + 4*x^3 + 12*x^4 + 6*x^5 +...
A(x)^4 = 1 + 12*x + 54*x^2 + 116*x^3 + 153*x^4 + 228*x^5 +..
A(x)^4 (mod 8) = 1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 4*x^5 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +..
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A110649, A084067.

Programs

  • PARI
    {a(n)=local(d=2,m=12,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)}

Formula

G.f. A(x) satisfies: A(x)^4 (mod 8) = g.f. of A084067.

A110645 Every 12th term of A084067.

Original entry on oeis.org

1, 6, 12, 2, 8, 9, 2, 7, 12, 2, 3, 4, 9, 10, 12, 9, 2, 5, 12, 9, 8, 10, 1, 9, 7, 2, 9, 4, 11, 1, 10, 5, 4, 1, 12, 2, 9, 4, 5, 11, 12, 12, 9, 5, 12, 5, 3, 12, 1, 4, 6, 2, 10, 11, 5, 1, 6, 10, 12, 9, 11, 9, 9, 5, 3, 3, 7, 2, 6, 7, 5, 9, 5, 10, 12, 5, 5, 4, 4, 2, 1, 8, 7, 11, 7, 6, 1, 2, 5, 10, 9, 8, 9, 9
Offset: 0

Views

Author

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

Keywords

Crossrefs

Cf. A084067, A110646, A110647, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=12,m=12,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)}

A110646 Every 6th term of A084067.

Original entry on oeis.org

1, 4, 6, 8, 12, 10, 2, 12, 8, 4, 9, 4, 2, 6, 7, 8, 12, 12, 2, 2, 3, 4, 4, 2, 9, 6, 10, 6, 12, 2, 9, 10, 2, 2, 5, 6, 12, 2, 9, 8, 8, 8, 10, 8, 1, 4, 9, 10, 7, 10, 2, 6, 9, 10, 4, 8, 11, 2, 1, 4, 10, 2, 5, 2, 4, 2, 1, 8, 12, 2, 2, 12, 9, 2, 4, 4, 5, 8, 11, 4, 12, 4, 12, 6, 9, 12, 5, 6, 12, 10, 5, 12, 3
Offset: 0

Views

Author

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

Keywords

Crossrefs

Cf. A084067, A110645, A110647, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=6,m=12,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)}

A110647 Every 4th term of A084067 where the self-convolution 4th power is congruent modulo 8 to A084067, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 9, 12, 6, 12, 9, 12, 6, 6, 2, 6, 12, 8, 3, 12, 9, 6, 12, 2, 3, 3, 7, 9, 9, 12, 3, 3, 2, 12, 6, 3, 9, 3, 4, 6, 3, 9, 6, 3, 10, 6, 9, 12, 9, 12, 9, 9, 6, 2, 9, 12, 5, 3, 6, 12, 9, 6, 9, 12, 6, 8, 6, 12, 10, 9, 12, 1, 9, 3, 9, 12, 6, 7, 12, 12, 2, 9, 3, 9, 12, 12, 4, 9, 9, 11, 6, 6, 1, 9, 6, 10, 3, 12
Offset: 0

Views

Author

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

Keywords

Examples

			A(x) = 1 + 9*x + 12*x^2 + 6*x^3 + 12*x^4 + 9*x^5 +...
A(x)^4 = 1 + 36*x + 534*x^2 + 4236*x^3 + 19785*x^4 +...
A(x)^4 (mod 8) = 1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 4*x^5 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +...
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A084067, A110645, A110646, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=4,m=12,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)}

A110648 Every third term of A084067 where the self-convolution third power is congruent modulo 9 to A084067, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 4, 4, 8, 6, 12, 8, 12, 12, 12, 10, 12, 2, 8, 12, 4, 8, 8, 4, 4, 9, 4, 4, 12, 2, 12, 6, 8, 7, 4, 8, 12, 12, 8, 12, 8, 2, 8, 2, 8, 3, 12, 4, 12, 4, 12, 2, 12, 9, 12, 6, 12, 10, 8, 6, 12, 12, 12, 2, 8, 9, 12, 10, 12, 2, 8, 2, 4, 5, 4, 6, 12, 12, 8, 2, 12, 9, 4, 8, 4, 8, 12, 8, 4, 10, 8, 8, 12, 1, 12
Offset: 0

Views

Author

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

Keywords

Examples

			A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 6*x^4 + 12*x^5 +...
A(x)^3 = 1 + 12*x + 60*x^2 + 184*x^3 + 450*x^4 + 948*x^5 +...
A(x)^3 (mod 9) = 1 + 3*x + 6*x^2 + 4*x^3 + 3*x^5 + 4*x^6 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +...
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A084067, A110645, A110646, A110647, A110649.

Programs

  • PARI
    {a(n)=local(d=3,m=12,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)}
Showing 1-5 of 5 results.

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