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 1-8 of 8 results.

A107087 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x.

Original entry on oeis.org

1, 2, -1, 2, -5, 12, -30, 82, -233, 668, -1949, 5802, -17503, 53302, -163783, 507418, -1582869, 4966790, -15667573, 49658264, -158059506, 505013014, -1619144976, 5207596574, -16797286048, 54323516786, -176113786590, 572236347200, -1863213388504, 6078422747600, -19865661574555
Offset: 0

Views

Author

Paul D. Hanna, May 11 2005

Keywords

Comments

Self-convolution of A107086. Self-convolution yields A107088.

Examples

			A(x)^2 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
A(x^2) = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...

Crossrefs

Cf. A107086, A107088.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(subst(A,x,x^2)+4*x+x*O(x^n))^(1/2)); polcoeff(A,n,x)}
  • PARI
    {a(n)=local(m,A); if(n<0,0, A=1+O(x); m=1; while(m<=n, A=sqrt(4*x+subst(A,x,x^2));m*=2); polcoeff(A,n))} /* Michael Somos, May 15 2005 */

Formula

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4 + 4*w - 2*u^2*v - 3*v^2. - Michael Somos, May 15 2005

A223026 G.f. A(x) satisfies: A(x)^8 = A(x^2)^4 + 8*x.

Original entry on oeis.org

1, 1, -3, 14, -76, 441, -2678, 16813, -108093, 707451, -4696017, 31530792, -213715953, 1460072247, -10042361784, 69473047716, -483046768116, 3373552141194, -23653214175084, 166422650191122, -1174621198245837, 8314055808436788, -58998774106863513
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2013

Keywords

Comments

The limit a(n+1)/a(n) seems to be near -7.46...

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 76*x^4 + 441*x^5 - 2678*x^6 +-...
where
A(x)^8 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +-...
A(x^2)^4 = 1 + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +-...
A(x)^2 = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +-...

Links

Crossrefs

Cf. A107086, A107089, A228711, A228712.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^4+8*x+x*O(x^n))^(1/8)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))

Formula

Self-convolution yields A228711.

A223142 G.f. satisfies: A(x)^2 = A(x^2)^2 + 4*x.

Original entry on oeis.org

1, 2, 0, 0, 2, -4, 8, -16, 32, -56, 88, -112, 64, 240, -1264, 4064, -10814, 25500, -54200, 102832, -166020, 190808, 22304, -1058880, 4412424, -13496544, 35306480, -82326496, 172081840, -315115328, 464910368, -363016000, -871587808, 5713552456, -20289991016
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^4 - 4*x^5 + 8*x^6 - 16*x^7 + 32*x^8 - 56*x^9 +...
where
A(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 4*x^8 + 4*x^16 + 4*x^32 +...+ 4*x^(2^n) +...

Links

Crossrefs

Cf. A223143, A107086, A223026.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1,#binary(n), A=(subst(A, x, x^2)^2+4*x+x*O(x^n))^(1/2)); polcoeff(A, n, x)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: A(x) = sqrt( 1 + Sum_{n>=0} 4*x^(2^n) ).

A223143 G.f. satisfies: A(x)^3 = A(x^2)^3 + 9*x.

Original entry on oeis.org

1, 3, -6, 27, -141, 819, -5022, 31968, -209202, 1398420, -9505854, 65499759, -456410943, 3210397173, -22763553876, 162524220984, -1167359075781, 8429107868541, -61148608627518, 445450238075655, -3257116365714831, 23896262127268719, -175854177039133998
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2013

Keywords

Examples

			G.f.: A(x) = 1 + 3*x - 6*x^2 + 27*x^3 - 141*x^4 + 819*x^5 - 5022*x^6 +...
where
A(x)^3 = 1 + 9*x + 9*x^2 + 9*x^4 + 9*x^8 + 9*x^16 + 9*x^32 +...+ 9*x^(2^n) +...

Links

Crossrefs

Cf. A223142, A107086, A223026.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^3+9*x+x*O(x^n))^(1/3)); polcoeff(A, n, x)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: A(x) = ( 1 + Sum_{n>=0} 9*x^(2^n) )^(1/3).

A107088 G.f. A(x) satisfies: A(x) = A(x^2)^(1/2) + 4*x.

Original entry on oeis.org

1, 4, 2, 0, -1, 0, 2, 0, -5, 0, 12, 0, -30, 0, 82, 0, -233, 0, 668, 0, -1949, 0, 5802, 0, -17503, 0, 53302, 0, -163783, 0, 507418, 0, -1582869, 0, 4966790, 0, -15667573, 0, 49658264, 0, -158059506, 0, 505013014, 0, -1619144976, 0, 5207596574, 0, -16797286048, 0, 54323516786, 0, -176113786590, 0
Offset: 0

Views

Author

Paul D. Hanna, May 11 2005

Keywords

Comments

Self-convolution 4th power of A107086. Self-convolution of A107087.

Examples

			A(x) = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
A(x^2)^(1/2) = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...

Crossrefs

Cf. A107086, A107087.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=subst(A,x,x^2)^(1/2)+4*x+x*O(x^n)); polcoeff(A,n,x)}

A187814 G.f. A(x) satisfies: 1/A(x)^2 + 4*x*A(x)^2 = 1/A(x^2) + 2*x*A(x^2).

Original entry on oeis.org

1, 1, 6, 41, 334, 2901, 26651, 253709, 2483395, 24829132, 252506507, 2603798287, 27161758393, 286118173600, 3039211373800, 32517513415886, 350122302629869, 3790909121211262, 41249405668333107, 450832515809731316, 4947009705400704588, 54479711308604703264, 601933495810972446631
Offset: 0

Views

Author

Paul D. Hanna, Aug 30 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 41*x^3 + 334*x^4 + 2901*x^5 + 26651*x^6 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^2 + 4*x*A(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
1/A(x^2) + 2*x*A(x^2) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 13*x^2 + 94*x^3 + 786*x^4 + 6962*x^5 + 64793*x^6 +...
A(x)^4 = 1 + 4*x + 30*x^2 + 240*x^3 + 2117*x^4 + 19512*x^5 + 186706*x^6 +...
1/A(x) = 1 - x - 5*x^2 - 30*x^3 - 233*x^4 - 1949*x^5 - 17503*x^6 +...
1/A(x)^2 = 1 - 2*x - 9*x^2 - 50*x^3 - 381*x^4 - 3132*x^5 - 27878*x^6 +...
The g.f. of A107086 begins:
F(x) = 1 + x - x^2 + 2*x^3 - 5*x^4 + 13*x^5 - 35*x^6 + 99*x^7 - 289*x^8 +...
where F(x)^4 = F(x^2)^2 + 4*x:
F(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 +...
F(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 +...

Crossrefs

Cf. A107086.
Cf. variants: A228712, A228928.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1/subst(A, x, x^2) + 2*x*subst(A, x, x^2) - 4*x*A^2 +x*O(x^n))^(1/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1/A(x)^2 + 4*x*A(x)^2 = F(x)^2,
(2) 1/A(x^2) + 2*x*A(x^2) = F(x)^2,
(3) A(x) = ( (F(x)^2 - sqrt(F(x)^4 - 16*x)) / (8*x) )^(1/2),
(4) A(x^2) = (F(x)^2 - sqrt(F(x)^4 - 8*x)) / (4*x),
where F(x) = (F(x^2)^2 + 4*x)^(1/4) is the g.f. of A107086.

A228711 G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 8*x.

Original entry on oeis.org

1, 2, -5, 22, -115, 646, -3822, 23496, -148368, 955822, -6256273, 41480668, -277954706, 1879118354, -12800031737, 87758481546, -605091552753, 4192829686338, -29180958305391, 203887504096188, -1429568781831693, 10055261467844862, -70929518958227340
Offset: 0

Views

Author

Paul D. Hanna, Aug 30 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +...
where A(x)^4 = A(x^2)^2 + 8*x as demonstrated by:
A(x)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 +...
A(x)^4 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...
The g.f. of A228712 begins:
G(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...
and satisfies: sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2) = A(x).

Crossrefs

Cf. A228712, A107086, A223026.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+8*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2),
(2) sqrt(A(x^2)^2 + 4*x) = 1/G(x^4) + 2*x*G(x^4),
where G(x) is the g.f. of A228712.
Self-convolution of A223026.

A228927 G.f. A(x) satisfies: A(x)^16 = A(x^2)^8 + 16*x.

Original entry on oeis.org

1, 1, -7, 70, -798, 9737, -124124, 1631041, -21911758, 299371219, -4144898772, 58007463920, -819038646307, 11650826921489, -166786290656152, 2400680788969898, -34719393978035312, 504223005531434252, -7349846348644213981, 107489242662154350550
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2013

Keywords

Examples

			G.f.: A(x) = 1 + x - 7*x^2 + 70*x^3 - 798*x^4 + 9737*x^5 - 124124*x^6 +...
where
A(x)^16 = 1 + 16*x + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 -+...
A(x^2)^8 = 1 + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 -+...

Crossrefs

Cf. A228928.
Cf. variants: A107086, A223026.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^8+16*x+x*O(x^n))^(1/16)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-8 of 8 results.

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