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 1-7 of 7 results.

A072224 Continued fraction expansion of constant A072223.

Original entry on oeis.org

0, 1, 1, 9, 1, 1, 5, 11, 1, 2, 1, 1, 6, 1, 1, 1, 5, 2, 5, 1, 1, 2, 2, 1, 1, 23, 1, 21, 1, 1, 4, 1, 6, 1, 1, 2, 2, 30, 3, 1, 6, 3, 3, 2, 4, 1, 3, 2, 3, 1, 13, 43, 24, 1, 6, 2, 6, 1, 2, 3, 81, 4, 8, 6, 1, 4, 5, 2, 1, 1, 1, 2, 26, 1, 2, 8, 2, 13, 1, 10, 3, 1, 2, 1, 6, 1, 7, 1, 1, 1, 11, 1, 2, 16, 1, 3, 1
Offset: 0

Views

Author

Michele Dondi (bik.mido(AT)tiscalinet.it), Jul 04 2002

Keywords

Crossrefs

Cf. A072223, A072176.

Programs

  • Mathematica
    ContinuedFraction[ x /. FindRoot[ (1-x^2)^2==x, {x, 0.5}, WorkingPrecision->200 ], 100 ]

A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 17, 27, 46, 86, 165, 308, 558, 1006, 1841, 3421, 6383, 11863, 21966, 40697, 75662, 141099, 263429, 491778, 918104, 1715259, 3208078, 6005818, 11250198, 21082487, 39524241, 74135187, 139128897, 261228200, 490682127, 922015964, 1733127107, 3258939997, 6130162494, 11534742080
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2014

Keywords

Comments

Limit a(n)/a(n+1) = t^2 = 0.524888598656404... (A072223) where t is the positive real root of 1 - x - x^4 = 0.
Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^4). - Seiichi Manyama, Apr 29 2025

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 17*x^7 +...
where, by definition,
A(x) = 1 + x*(1 + x^3) + x^2*(1 + 2^2*x^3 + x^6)
+ x^3*(1 + 3^2*x^3 + 3^2*x^6 + x^9)
+ x^4*(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)
+ x^5*(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^4) + 2*x^4/(1-x+x^4)^3 + 6*x^8/(1-x+x^4)^5 + 20*x^12/(1-x+x^4)^7 + 70*x^16/(1-x+x^4)^9 + 252*x^20/(1-x+x^4)^11 + 924*x^24/(1-x+x^4)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^3) + x^2*(1 + 6*x^3 + x^6)/2
+ x^3*(1 + 15*x^3 + 15*x^6 + x^9)/3
+ x^4*(1 + 28*x^3 + 70*x^6 + 28*x^9 + x^12)/4
+ x^5*(1 + 45*x^3 + 210*x^6 + 210*x^9 + 45*x^12 + x^15)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 16*x^5/5 + 31*x^6/6 + 50*x^7/7 + 77*x^8/8 + 145*x^9/9 + 306*x^10/10 + 628*x^11/11 + 1199*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+4*x^3+5*x^4-4*x^7)/((1-x+2*x^2+x^4)*(1-x-2*x^2+x^4)).

Links

Crossrefs

Cf. A072223, A181665, A246840, A246884, A248193.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[(1 - x + x^4)^2 - 4 x^4], {x, 0, 40}], x] (* Michael De Vlieger, Sep 10 2021 *)
  • PARI
    /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x + x^4)^2 - 4*x^4 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(4*m) / (1 - x + x^4 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^3)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(3*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
  • PARI
    /* From formula for a(n): */
{a(n)=sum(k=0, n\3, binomial(n-3*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(4*n) / (1 - x + x^4)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(3*k)] * (1-x^3)^(2*n+1).
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(3*k) ).
G.f.: 1 / sqrt((1 - x + 2*x^2 + x^4)*(1 - x - 2*x^2 + x^4)).
G.f.: 1 / sqrt((1 - x + x^4)^2 - 4*x^4).
G.f.: 1 / sqrt((1 - x - x^4)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/3]} C(n-3*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-2)*a(n-4) + (2*n-5)*a(n-5) - (n-4)*a(n-8). - Seiichi Manyama, Aug 10 2024

A376658 Decimal expansion of a constant related to the asymptotics of A376624 and A376625.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 01 2024

Keywords

Examples

			8.46018724425296480975230009888917594335470635951014367622821158904321498...

Crossrefs

Cf. A072223, A333198, A376621, A376624, A376625, A376659, A376660, A377075.
Cf. A356032.

Programs

  • Mathematica
    RealDigits[E^(Sqrt[2*Log[r]^2 + 4*PolyLog[2, Sqrt[r]]]) /. r -> 1/(2*Sqrt[3/(4 + ((155 - 3*Sqrt[849])/2)^(1/3) + ((155 + 3*Sqrt[849])/2)^(1/3))]) - Sqrt[8/3 - ((155 - 3*Sqrt[849])/2)^(1/3)/3 - ((155 + 3*Sqrt[849])/2)^(1/3)/3 + 2*Sqrt[3/(4 + ((155 - 3*Sqrt[849])/2)^(1/3) + ((155 + 3*Sqrt[849])/2)^(1/3))]]/2, 10, 105][[1]]

Formula

Equals exp(sqrt(2*(log(r)^2 + 2*polylog(2, sqrt(r))))), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.
Equals limit_{n->infinity} A376624(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376625(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A377075(n)^(1/sqrt(n)).
Equals exp(2*sqrt(2*log(A356032)^2 + polylog(2, A356032))).

A298813 Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x + 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 13 2018

Keywords

Comments

Let (d(n)) = (1,0,1,0,1,0,1,...), s(n) = sqrt(s(n-1) + d(n)) for n > 0, and s(0) = 1.
Then s(2n) -> 1.49021612009995..., as in A298813;
and s(2n+1) -> 1.22074408..., as in A060007.
Let (e(n)) = (0,1,0,1,0,1,0,...), t(n) = sqrt(t(n-1) + e(n)) for n > 0, and t(0) = 1.
Then t(2n) -> 1.22074408..., as in A060007;
and t(2n+1) -> 1.49021612009995..., as in A298813.
The four solutions are: x1, this one; x2, the least A072223; and the two complex ones x3=-1.007552359378... + 0.513115795597...*i and x4, its complex conjugate; Re(x3) = Re(x4) = -(x1+x2)/2; Im(x3) = -Im(x4) = sqrt(1/(x1*x2) - Re(x3)^2). - Andrea Pinos, Sep 20 2023

Examples

			1.49021612009995...

Links

Crossrefs

Cf. A060007, A072223, A298814, A298815.

Programs

  • Mathematica
    r = x /. NSolve[x^4 - 2 x^2 - x + 1 == 0, x, 100][[4]];
RealDigits[r][[1]]; (* A298813 *)
RealDigits[Root[x^4-2x^2-x+1,2],10,120][[1]] (* Harvey P. Dale, May 02 2022 *)
  • PARI
    solve(x=1, 2, x^4 - 2*x^2 - x + 1) \\ Michel Marcus, Nov 05 2018

Formula

Equals sqrt((1 + 2*cos(arccos(155/128)/3))/3) + sqrt(2/3 - 2*cos(arccos(155/128)/3)/3 + sqrt(3/(1 + 2*cos(arccos(155/128)/3)))/4). - Vaclav Kotesovec, Sep 21 2023
Equals sqrt(1/3 + s/9 + 1/s) + sqrt(2/3 - s/9 - 1/s + 1 / (4 * sqrt(1/3 + s/9 + 1/s))) where s = (4185/128 + sqrt(5570289/16384))^(1/3). - Michal Paulovic, Dec 30 2023

A376624 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 18, 23, 33, 44, 57, 77, 99, 125, 163, 207, 259, 328, 407, 503, 626, 769, 938, 1149, 1397, 1687, 2044, 2458, 2943, 3531, 4213, 5011, 5957, 7055, 8334, 9838, 11580, 13594, 15948, 18661, 21790, 25425, 29593, 34386, 39918, 46250, 53501, 61824, 71325
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A053282, A072223, A376542, A376622, A376581, A376658.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j-1))^2.
a(n) ~ (r^(3/4)/sqrt(8*(1 + 3*r^2))) * A376658^sqrt(n) / sqrt(n), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

A356032 Decimal expansion of the positive real root of x^4 + x - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Aug 27 2022

Keywords

Comments

The other real (negative) root is -A060007.
One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

Examples

			r = 0.724491959000515611588372282187036565786494481350011017270...

Crossrefs

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

Programs

  • Mathematica
    First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)
  • PARI
    solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022
  • PARI
    polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

Formula

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).
Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025

A376625 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 3, 0, 5, 1, 9, 2, 13, 6, 20, 12, 27, 23, 39, 40, 51, 69, 70, 108, 92, 169, 125, 252, 166, 370, 227, 527, 307, 743, 425, 1021, 586, 1393, 816, 1867, 1132, 2481, 1577, 3256, 2184, 4247, 3019, 5479, 4149, 7036, 5670, 8966, 7698, 11377, 10386, 14356, 13915, 18060
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000700, A072223, A179049, A340647, A376542, A376658.

Programs

  • Mathematica
    nmax=80; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j))^2.
a(n) ~ (r^(1/4) * sqrt(log(r)^2 + 2*polylog(2, sqrt(r))) / (2*Pi*sqrt(1 + 3*r^2))) * A376658^sqrt(n) / n, where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.
Showing 1-7 of 7 results.

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