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 1-6 of 6 results.

A272823 G.f. satisfies: A(x) = 1 + 4*x*AGM(A(x), A(x)^3).

Original entry on oeis.org

1, 4, 32, 336, 4032, 52336, 715392, 10144192, 147836416, 2200709040, 33319564288, 511496462656, 7942988228608, 124551530359360, 1969386732874752, 31364967043386112, 502686338657607680, 8101474649157519536, 131212844750426749696, 2134554132316280052288, 34862936239396076532736, 571454628433830080180288, 9397626191659208856570880, 155006334509119865698297600, 2563700952357088703495372800
Offset: 0

Views

Author

Paul D. Hanna, May 07 2016

Keywords

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 336*x^3 + 4032*x^4 + 52336*x^5 + 715392*x^6 + 10144192*x^7 + 147836416*x^8 + 2200709040*x^9 + 33319564288*x^10 +...
where A(x) = 1 + 4*x*AGM(A(x), A(x)^3).
RELATED SERIES.
A(x)^2 = 1 + 8*x + 80*x^2 + 928*x^3 + 11776*x^4 + 158432*x^5 + 2220416*x^6 + 32070528*x^7 + 474038272*x^8 + 7136118624*x^9 + 109031206528*x^10 +...
A(x)^3 = 1 + 12*x + 144*x^2 + 1840*x^3 + 24768*x^4 + 346704*x^5 + 4999424*x^6 + 73774656*x^7 + 1108876800*x^8 + 16918514448*x^9 + 261355433856*x^10 +...
(A(x) + A(x)^3)/2 = 1 + 8*x + 88*x^2 + 1088*x^3 + 14400*x^4 + 199520*x^5 + 2857408*x^6 + 41959424*x^7 + 628356608*x^8 + 9559611744*x^9 + 147337499072*x^10 +...
where A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
AGM(1+4*x, (1+4*x)^3) = 1 + 8*x + 20*x^2 + 16*x^3 - 4*x^4 + 16*x^6 - 64*x^7 + 172*x^8 - 352*x^9 + 560*x^10 - 832*x^11 + 2512*x^12 - 13568*x^13 + 65984*x^14 +...
where A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3) ).

Links

Crossrefs

Cf. A171454, A272822.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)*(1 + s^2) / (2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 - 7*s^4 + 4*s^5))]} /. FindRoot[{2*Pi*r*s^3 / EllipticK[1 - 1/s^4] == s - 1, Pi*r*(1 + s + s^2 + s^3 - 2*s^4) == (1 - s)*s * EllipticE[1 - 1/s^4]}, {r, 1/18}, {s, 3/2}, WorkingPrecision -> 80] (* Vaclav Kotesovec, Nov 15 2023 *)
  • PARI
    /* From definition: A(x) = 1 + 4*x*AGM(A(x), A(x)^3) */
{a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A, A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2) */
{a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, (A + A^3)/2)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1+4*x, (1+4*x)^3)) */
{a(n) = my(A=1); A = 1 + 4*serreverse(x/agm(1+4*x,(1+4*x)^3 +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
(2) A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3) ).
a(n) ~ c * d^n / n^(3/2), where d = 17.6088646774568498919315031912184045773663297219819943809841685080399155... and c = 0.3922200012562096239034743054558268956365939170567699740621520897631... - Vaclav Kotesovec, Nov 15 2023

A369536 Expansion of g.f. A(x) satisfying A(x) = 1 + 4*x * AGM(1, A(x)^4).

Original entry on oeis.org

1, 4, 32, 384, 5376, 81920, 1318912, 22071296, 380084224, 6691479552, 119890509824, 2178958163968, 40073602269184, 744399420391424, 13946358907011072, 263220821247393792, 5000085343337185280, 95520905055747178496, 1834027221478623150080, 35372549509799248658432
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2024

Keywords

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 4*x + 32*x^2 + 384*x^3 + 5376*x^4 + 81920*x^5 + 1318912*x^6 + 22071296*x^7 + 380084224*x^8 + 6691479552*x^9 + 119890509824*x^10 + ...
RELATED SERIES.
x/AGM(1, (1 + 4*x)^4) = x - 8*x^2 + 32*x^3 - 64*x^4 + 3584*x^7 - 22528*x^8 + 34816*x^9 + 245760*x^10 - 1163264*x^11 - 3211264*x^12 + ...
where A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.

Links

Crossrefs

Cf. A171454, A272823, A369537, A369538, A369539.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, Sqrt[s*(1 - s - s^8 + s^9)/(2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 + 2*s^4 + 2*s^5 + 2*s^6 - 14*s^7 + 9*s^8))]} /. FindRoot[{1 + 2*Pi*r*s^4/EllipticK[1 - 1/s^8] == s, (s^8 - 1)/(s - 1) + 2*(s - 1)*s^3 * EllipticE[1 - 1/s^8]/(Pi*r) == 4*s^7}, {r, 1/20}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
  • PARI
    /* From definition: A(x) = 1 + 4*x*AGM(1, A(x)^4) */
{a(n)=my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(1, A^4)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (1 + A(x)^4)/2) */
{a(n)=my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, (1 + A^4)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1, (1+4*x)^4)) */
{a(n) = my(A=1); A = 1 + 4*serreverse(x/agm(1, (1+4*x)^4 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 4*x * AGM(1, A(x)^4).
(2) A(x) = 1 + 4*x * AGM(A(x)^2, (1 + A(x)^4)/2).
(3) A(x) = 1 + 4 * Series_Reversion( x / AGM(1, (1 + 4*x)^4) ).
(4) A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.
a(n) ~ c * d^n / n^(3/2), where d = 20.8911293747878758394214491571395886690885608604807120892771607914028... and c = 0.2494539611266913248489641272521896595054291412784920760863145867198... - Vaclav Kotesovec, Jan 29 2024

A369537 Expansion of g.f. A(x) satisfying A(x) = 1 + 4*x * AGM(A(x)^2, A(x)^4).

Original entry on oeis.org

1, 4, 48, 784, 14784, 302960, 6554624, 147336384, 3407207936, 80538522544, 1937217000576, 47262640993344, 1166745699940352, 29090562313367104, 731508300407392256, 18530124876627212032, 472416442490053386240, 12112314681652019632304, 312110730162591314249088
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2024

Keywords

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Examples

			G.f.: A(X) = 1 + 4*x + 48*x^2 + 784*x^3 + 14784*x^4 + 302960*x^5 + 6554624*x^6 + 147336384*x^7 + 3407207936*x^8 + 80538522544*x^9 + 1937217000576*x^10 + ...
RELATED SERIES.
x / AGM((1 + 4*x)^2, (1 + 4*x)^4) = x - 12*x^2 + 92*x^3 - 576*x^4 + 3220*x^5 - 16784*x^6 + 83536*x^7 - 402560*x^8 + 1894308*x^9 - 8751600*x^10 + ...
where A( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.
A(x)^2 = 1 + 8*x + 112*x^2 + 1952*x^3 + 38144*x^4 + 799456*x^5 + 17566848*x^6 + 399375232*x^7 + 9315958784*x^8 + 221714573152*x^9 + ...
A(x)^3 = 1 + 12*x + 192*x^2 + 3568*x^3 + 72384*x^4 + 1554768*x^5 + 34760064*x^6 + 800484672*x^7 + 18858757632*x^8 + 452388579088*x^9 + ...
A(x)^4 = 1 + 16*x + 288*x^2 + 5696*x^3 + 120064*x^4 + 2646464*x^5 + 60279552*x^6 + 1407812352*x^7 + 33532936192*x^8 + 811514412736*x^9 + ...
(A(x)^2 + A(x)^4)/2 = 1 + 12*x + 200*x^2 + 3824*x^3 + 79104*x^4 + 1722960*x^5 + 38923200*x^6 + 903593792*x^7 + 21424447488*x^8 + 516614492944*x^9 + ...

Links

Crossrefs

Cf. A171454, A272823, A369536, A369538, A369539.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, s*(s - 1) * Sqrt[(1 + s + s^2 + s^3)/(2*Pi*(4 + s + 2*s^2 + 2*s^3 - 14*s^4 + 9*s^5))]} /. FindRoot[{1 + 2*Pi*r*s^4 / EllipticK[1 - 1/s^4] == s, 2*Pi*r*(1 - 2*s^4) + (-1 + s) * EllipticE[1 - 1/s^4] + (-1 + s^4)*Pi*r*s/(-1 + s) == 0}, {r, 1/30}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
  • PARI
    /* From definition: A(x) = 1 + 4*x*AGM(A(x)^2, A(x)^4) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, A^4)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula: A(x) = 1 + 4*x*AGM(A(x)^3, (A(x)^2 + A(x)^4)/2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^3, (A^2 + A^4)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From A(x) = 1 + 4*Series_Reversion(x/AGM((1+4*x)^2, (1+4*x)^4)) */
{a(n) = my(A=1); A = 1 + 4*serreverse(x/agm((1+4*x)^2, (1+4*x)^4 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 4*x * AGM(A(x)^2, A(x)^4).
(2) A(x) = 1 + 4*x * AGM(A(x)^3, (A(x)^2 + A(x)^4)/2).
(3) A(x) = 1 + 4 * Series_Reversion( x / AGM((1 + 4*x)^2, (1 + 4*x)^4) ).
(4) A( x/AGM((1 + 4*x)^2, (1 + 4*x)^4) ) = 1 + 4*x.
a(n) ~ c * d^n / n^(3/2), where d = 28.0338265004083388867842940412535265992903265132288705384671366058202... and c = 0.21370406929731394715730174119301970236922500578435406822814969355660... - Vaclav Kotesovec, Jan 29 2024

A369538 Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x), A(x)^2).

Original entry on oeis.org

1, 8, 96, 1376, 21760, 366176, 6431488, 116551040, 2163118080, 40907835232, 785471061760, 15272052137856, 300077039734784, 5949171298710144, 118858435514103808, 2390669459946235392, 48369365721497534464, 983759515642369327456, 20101539919939043645184, 412461687626131640565632
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2024

Keywords

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 8*x + 96*x^2 + 1376*x^3 + 21760*x^4 + 366176*x^5 + 6431488*x^6 + 116551040*x^7 + 2163118080*x^8 + 40907835232*x^9 + 785471061760*x^10 + ...
RELATED SERIES.
x / AGM(1 + 8*x, (1 + 8*x)^2) = x - 12*x^2 + 116*x^3 - 1040*x^4 + 8996*x^5 - 76272*x^6 + 638672*x^7 - 5303616*x^8 + 43782436*x^9 - 359852592*x^10 + ...
where
A( x/AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.

Links

Crossrefs

Cf. A171454, A272823, A369536, A369537, A369539.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)/(2*Pi*(1 + 2*s - 2*s^2 + s^3))]} /. FindRoot[{1 + 4*Pi*r*s^2/EllipticK[1 - 1/s^2] == s, 4*Pi*r*(-1 - s + s^2) == (s - 1)*EllipticE[1 - 1/s^2]}, {r, 1/25}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
  • PARI
    /* From definition: A(x) = 1 + 8*x*AGM(A(x), A(x)^2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A, A^2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(3/2), (A(x) + A(x)^2)/2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^(3/2), (A + A^2)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From A(x) = 1 + 8*Series_Reversion(x/AGM(1+8*x, (1+8*x)^2)) */
{a(n) = my(A=1); A = 1 + 8*serreverse(x/agm(1+8*x, (1+8*x)^2 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 8*x * AGM(A(x), A(x)^2).
(2) A(x) = 1 + 8*x * AGM(A(x)^(3/2), (A(x) + A(x)^2)/2).
(3) A(x) = 1 + 8 * Series_Reversion( x / AGM(1 + 8*x, (1 + 8*x)^2) ).
(4) A( x / AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.
a(n) ~ c * d^n / n^(3/2), where d = 22.1630051344803196287731245642346070282303059361700001080950958441256... and c = 0.99743551254261758609104583646696482831141906954702821438454764216307... - Vaclav Kotesovec, Jan 29 2024

A369539 Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x)^2, A(x)^3).

Original entry on oeis.org

1, 8, 160, 4192, 125184, 4039264, 137183488, 4831873408, 174884458496, 6464875435872, 243049515606272, 9264347436276608, 357204831146577920, 13906950967902306944, 545951685104975276032, 21587442538147647608320, 858975581766808512823296, 34369283236381014527279456
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2024

Keywords

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 8*x + 160*x^2 + 4192*x^3 + 125184*x^4 + 4039264*x^5 + 137183488*x^6 + 4831873408*x^7 + 174884458496*x^8 + 6464875435872*x^9 + 243049515606272*x^10 + ...
RELATED SERIES.
x/AGM((1 + 8*x)^2, (1 + 8*x)^3) = x - 20*x^2 + 276*x^3 - 3248*x^4 + 34980*x^5 - 356112*x^6 + 3487568*x^7 - 33204160*x^8 + 309415716*x^9 - 2835178320*x^10 + ...
where A( x/AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.

Links

Crossrefs

Cf. A171454, A272823, A369536, A369537, A369538.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)/(2*Pi*(4 + s - 7*s^2 + 4*s^3))]} /. FindRoot[{1 + 4*Pi*r*s^3/EllipticK[1 - 1/s^2] == s, 4*Pi*r*s*(2 + s - 2*s^2) + (-1 + s)*EllipticE[1 - 1/s^2] == 0}, {r, 1/50}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
  • PARI
    /* From definition: A(x) = 1 + 8*x*AGM(A(x)^2, A(x)^3) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^2, A^3)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^(5/2), (A^2 + A^3)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From A(x) = 1 + 8*Series_Reversion(x/AGM((1+8*x)^2, (1+8*x)^3)) */
{a(n) = my(A=1); A = 1 + 8*serreverse(x/agm((1+8*x)^2, (1+8*x)^3 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 8*x * AGM(A(x)^2, A(x)^3).
(2) A(x) = 1 + 8*x * AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2).
(3) A(x) = 1 + 8 * Series_Reversion( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ).
(4) A( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.
a(n) ~ c * d^n / n^(3/2), where d = 43.7139872016060880921082193574226064477439580563964019841877818207326... and c = 0.32250297108028000960144303111184352981179935271075437927423118550208... - Vaclav Kotesovec, Jan 29 2024
A(1/d) = 1.6405711647668295617017794194853407... where d is given above. - Paul D. Hanna, Jan 29 2024

A171455 G.f. satisfies: A(x) = 1 + 2*x*AGM(1, A(x)^4).

Original entry on oeis.org

1, 2, 8, 48, 336, 2560, 20608, 172432, 1484704, 13069296, 117080576, 1063944416, 9783594304, 90869069872, 851218195008, 8032861976544, 76295247548480, 728766670652368, 6996258626856320, 67467783946608064, 653254749175955584, 6348266152788407648, 61896814517299122560
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 336*x^4 + 2560*x^5 + ...
A(x)^2 = 1 + 4*x + 20*x^2 + 128*x^3 + 928*x^4 + 7232*x^5 + ...
A(x)^4 = 1 + 8*x + 56*x^2 + 416*x^3 + 3280*x^4 + 27008*x^5 + ...
AGM(1, A(x)^4) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10304*x^5 + ...

Links

Crossrefs

Cf. A171454.

Programs

  • Mathematica
    (* Calculation of constants {d,c}: *) {1/r, Sqrt[s*(1 - s - s^8 + s^9) / (2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 + 2*s^4 + 2*s^5 + 2*s^6 - 14*s^7 + 9*s^8))]} /. FindRoot[{Pi*r*s^4 / EllipticK[1 - 1/s^8] == s - 1, -4*Pi*r*s^7 + Pi*r*(-1 + s^8)/(-1 + s) + 4*(-1 + s)*s^3 * EllipticE[1 - 1/s^8] == 0}, {r, 1/10}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 15 2023 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+2*x*agm(1,A^4));polcoeff(A,n)}

Formula

a(n) ~ c * d^n / n^(3/2), where d = 10.4455646873939379197107245785697943345442804302403560446385803957... and c = 0.249453961126691324848964127252189659505429141278492076086314586719... - Vaclav Kotesovec, Nov 15 2023

Extensions

More terms from Jinyuan Wang, Feb 25 2020
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