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.

A281181 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^3 dx ).

Original entry on oeis.org

1, 1, 13, 493, 37369, 4732249, 901188997, 240798388357, 85948640603761, 39504564917358001, 22726779729476308093, 15998009117983994065693, 13526765851190230940840809, 13528070218935445806530640649, 15795819619923464298050697616117, 21294937666865806704402646632389557, 32828500597549179599563478551377297121, 57385924456400269824204023290894357442401, 112904615348383588847189789579363784912180973
Offset: 0

Views

Author

Paul D. Hanna, Jan 16 2017

Keywords

Comments

From Paul Curtz, Jan 20 2017: (Start)
a(n) mod 10 = periodic sequence of length 8: repeat [1, 1, 3, 3, 9, 9, 7, 7] = duplicated A001148(n).
a(n) mod 9 = 1, followed by period 3: repeat [1, 4, 7]. See A100402. See also A281280, A281182, A281183, A281184 (1, followed by 3's).
a(n+p) - a(n) is a multiple of 12. (End)

Examples

			E.g.f.: C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! + 39504564917358001*x^18/18! + 22726779729476308093*x^20/20! +...
such that
(1) C(x) = cosh( Integral C(x)^3 dx ),
(2) C(x)^2 - S(x)^2 = 1, and
(3) C(x) = 1 + Integral C(x)^3*S(x) dx,
where S(x) begins:
S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! + 1000749157519458304*x^19/19! + 635146072839001735168*x^21/21! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
RELATED SERIES.
As power series with reduced fractional coefficients, S(x) and C(x) begin:
S(x) = x + 2/3*x^3 + 11/15*x^5 + 292/315*x^7 + 3548/2835*x^9 + 273766/155925*x^11 + 15360178/6081075*x^13 + 214706776/58046625*x^15 +...
C(x) = 1 + 1/2*x^2 + 13/24*x^4 + 493/720*x^6 + 37369/40320*x^8 + 4732249/3628800*x^10 + 901188997/479001600*x^12 + 240798388357/87178291200*x^14 +...
Related powers of series C(x) are given as follows.
C(x)^2 = 1 + 2*x^2/2! + 32*x^4/4! + 1376*x^6/6! + 114176*x^8/8! + 15519488*x^10/10! + 3132551168*x^12/12! + 879422726144*x^14/14! + 327670676455424*x^16/16! + 156439068819587072*x^18/18! +...+ A281183(n)*x^(2*n)/(2*n)! +...
where C(x)^2 = 1 + S(x)^2.
C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...+ A281184(n)*x^(2*n)/(2*n)! +...
where C(x)^3 = d/dx log( C(x) + S(x) ).
Also, C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
C(x)^4 = 1 + 4*x^2/2! + 88*x^4/4! + 4672*x^6/6! + 454144*x^8/8! + 70084096*x^10/10! + 15728822272*x^12/12! + 4836914249728*x^14/14! + 1952137912385536*x^16/16! + 1000749157519458304*x^18/18! +...+ A281180(n+1)*x^(2*n)/(2*n)! +...
where C(x)^4 = d/dx S(x).

Links

Crossrefs

Cf. A281180 (S), A281182 (C+S), A281183 (C^2), A281184 (C^3), A001148, A100402, A122553.

Programs

  • Mathematica
    nMax = 30; m = maxExponent = 2*nMax; a[n_] := Module[{S = x, C = 1}, For[i = 1, i <= n, i++, S = Integrate[C^4 + x*O[x]^m // Normal, x] + O[x]^m // Normal; C = 1 + Integrate[S*C^3 + O[x]^m // Normal, x]] + O[x]^m // Normal; (2*n)!*Coefficient[C, x, 2*n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nMax}] (* Jean-François Alcover, Jan 20 2017, adapted from PARI *)
nmax = 20; Table[(CoefficientList[Sqrt[D[InverseSeries[Series[(2*x + Sin[2*x])/4, {x, 0, 2*nmax - 1}], x], x]], x] * Range[0, 2*nmax - 2]!)[[2*n - 1]], {n, 1, nmax}] (* Vaclav Kotesovec, Sep 02 2017 *)
  • PARI
    {a(n) = my(S=x,C=1); for(i=0,n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C,2*n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
E.g.f. C(x) = d/dx Series_Reversion( ( x*sqrt(1 - x^2) + asin(x) )/2 ).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral cos(x)^2 dx ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( (2*x + sin(2*x))/4 ) )^(1/2).
E.g.f. C(x) = ( d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) )^(1/3).
E.g.f. C(x) = ( d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) )^(1/3).
E.g.f. C(x) and related series S(x) (e.g.f. of A281180) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^4 dx.
(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
(3.b) C(x) = cosh( Integral C(x)^3 dx ).
(3.c) S(x) = sinh( Integral C(x)^3 dx ).
Derivatives.
(4.a) S'(x) = C(x)^4.
(4.b) C'(x) = C(x)^3*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
(5.f) C(x)^4 = d/dx Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.g) C(x)^5 = d/dx Series_Reversion( Integral C(i*x)^5 dx ).

Extensions

Name simplified by Paul D. Hanna, Jan 22 2017

A281180 E.g.f. S(x) satisfies: S(x) = Integral (1 + S(x)^2)^2 dx.

Original entry on oeis.org

1, 4, 88, 4672, 454144, 70084096, 15728822272, 4836914249728, 1952137912385536, 1000749157519458304, 635146072839001735168, 488855521088102855606272, 448599416591747486039670784, 483861305506660094099058589696, 606050665000453965359938841608192, 872366179652871528356910686198038528, 1430068361869553198039835379199635357696, 2648687881942689612933392158083076801429504, 5503854158077547090902251582359116752300802048
Offset: 1

Views

Author

Paul D. Hanna, Jan 16 2017

Keywords

Examples

			E.g.f.: S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! + 1000749157519458304*x^19/19! + 635146072839001735168*x^21/21! +...
such that
(1) C(x)^2 - S(x)^2 = 1, and
(2) S'(x) = C(x)^4,
where C(x) begins:
C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! + 39504564917358001*x^18/18! + 22726779729476308093*x^20/20! +...+ A281181(n)*x^(2*n)/(2*n)! +...
RELATED SERIES.
As power series with reduced fractional coefficients, S(x) and C(x) begin:
S(x) = x + 2/3*x^3 + 11/15*x^5 + 292/315*x^7 + 3548/2835*x^9 + 273766/155925*x^11 + 15360178/6081075*x^13 + 214706776/58046625*x^15 +...
C(x) = 1 + 1/2*x^2 + 13/24*x^4 + 493/720*x^6 + 37369/40320*x^8 + 4732249/3628800*x^10 + 901188997/479001600*x^12 + 240798388357/87178291200*x^14 +...
The series reversion of the e.g.f. begins:
Series_Reversion(S(x)) = x - 2/3*x^3 + 3/5*x^5 - 4/7*x^7 + 5/9*x^9 - 6/11*x^11 + 7/13*x^13 - 8/15*x^15 +...
which equals ( x/(1+x^2) + atan(x) )/2.
Related powers of series C(x) are given as follows.
C(x)^2 = 1 + 2*x^2/2! + 32*x^4/4! + 1376*x^6/6! + 114176*x^8/8! + 15519488*x^10/10! + 3132551168*x^12/12! + 879422726144*x^14/14! + 327670676455424*x^16/16! + 156439068819587072*x^18/18! +...+ A281183(n)*x^(2*n)/(2*n)! +...
where C(x)^2 = 1 + S(x)^2.
C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...+ A281184(n)*x^(2*n)/(2*n)! +...
where C(x)^3 = d/dx log( C(x) + S(x) ).
C(x)^4 = 1 + 4*x^2/2! + 88*x^4/4! + 4672*x^6/6! + 454144*x^8/8! + 70084096*x^10/10! + 15728822272*x^12/12! + 4836914249728*x^14/14! + 1952137912385536*x^16/16! + 1000749157519458304*x^18/18! +...
where C(x)^4 = d/dx S(x).

Links

Crossrefs

Cf. A281181 (C), A281182 (C+S), A281183 (C^2), A281184 (C^3).

Programs

  • Mathematica
    nMax = 30; m = maxExponent = 2*nMax; a[n_] := Module[{S = x, C = 1}, For[i = 1, i <= n, i++, S = Integrate[C^4 + x*O[x]^m // Normal, x] + O[x]^m // Normal; C = 1 + Integrate[S*C^3 + O[x]^m // Normal, x]] + O[x]^m // Normal; (2*n - 1)!*Coefficient[S, x, 2*n - 1]]; Table[an = a[n]; Print[ "a(", n, ") = ", an]; an, {n, 1, nMax}] (* Jean-François Alcover, Jan 20 2017, adapted from first PARI program *)
nmax = 20; Table[(CoefficientList[InverseSeries[Series[(x/(1 + x^2) + ArcTan[x])/2, {x, 0, 2*nmax - 1}], x], x] * Range[0, 2*nmax - 1]!)[[2*n]], {n, 1, nmax}] (* Vaclav Kotesovec, Sep 02 2017 *)
  • PARI
    {a(n) = my(S=x,C=1); for(i=1,n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,30,print1(a(n),", "))
  • PARI
    /* S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ) */
{a(n) = my(S=x); S = serreverse( intformal( 1/(1 + x^2 +x*O(x^(2*n)))^2)); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
E.g.f. S(x) = Series_Reversion( ( x/(1+x^2) + atan(x) )/2 ).
E.g.f. S(x) and related series C(x) (e.g.f. of A281181) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^4 dx.
(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
(3.b) C(x) = cosh( Integral C(x)^3 dx ).
(3.c) S(x) = sinh( Integral C(x)^3 dx ).
Derivatives.
(4.a) S'(x) = C(x)^4.
(4.b) C'(x) = C(x)^3*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).

Extensions

Name simplified by Paul D. Hanna, Jan 22 2017

A281182 E.g.f. C(x) + S(x) = exp( Integral C(x)^3 dx ) where C(x) and S(x) are described by A281181 and A281180, respectively.

Original entry on oeis.org

1, 1, 1, 4, 13, 88, 493, 4672, 37369, 454144, 4732249, 70084096, 901188997, 15728822272, 240798388357, 4836914249728, 85948640603761, 1952137912385536, 39504564917358001, 1000749157519458304, 22726779729476308093, 635146072839001735168, 15998009117983994065693, 488855521088102855606272, 13526765851190230940840809, 448599416591747486039670784, 13528070218935445806530640649
Offset: 0

Views

Author

Paul D. Hanna, Jan 16 2017

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4672*x^7/7! + 37369*x^8/8! + 454144*x^9/9! + 4732249*x^10/10! + 70084096*x^11/11! + 901188997*x^12/12! +...
where A(x) = C(x) + S(x) and the series for C(x) and S(x) begin:
C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! +...+ A281181(n)*x^(2*n)/(2*n)! +...
S(x) = S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
such that C(s) + S(x) = exp( Integral C(x)^3 dx ).
The logarithm of the e.g.f. begins:
log(C(x) + S(x)) = x + 3*x^3/3! + 57*x^5/5! + 2739*x^7/7! + 246801*x^9/9! + 35822307*x^11/11! + 7636142793*x^13/13! + 2246286827091*x^15/15! +...
which equals Integral C(x)^3 dx.
Also, log(C(x) + S(x)) = Series_Reversion( Integral 1/cosh(x)^3 dx ).

Links

Crossrefs

Cf. A281180 (S), A281181 (C), A281183 (C^2), A281184 (C^3).

Programs

  • Mathematica
    CoefficientList[Exp[InverseSeries[Series[(Sinh[x]/Cosh[x]^2 + ArcTan[Sinh[x]])/2, {x, 0, 30}], x]], x] * Range[0, 30]! (* Vaclav Kotesovec, Sep 02 2017 *)
  • PARI
    {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^n)); C = 1 + intformal( S*C^3 ) ); n!*polcoeff(C + S, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* From S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ) */
{a(n) = my(S=x); S = serreverse( intformal( 1/(1 + x^2 +x*O(x^n))^2)); n!*polcoeff(sqrt(1+S^2) + S, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
E.g.f. exp( Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ) ).
E.g.f. C(x) + S(x) where related series S(x) and C(x) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^4 dx.
(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
(3.b) C(x) = cosh( Integral C(x)^3 dx ).
(3.c) S(x) = sinh( Integral C(x)^3 dx ).
Derivatives.
(4.a) S'(x) = C(x)^4.
(4.b) C'(x) = C(x)^3*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).

A281184 E.g.f. C(x)^3 = d/dx log(C(x) + S(x)), where C(x) and S(x) are described by A281181 and A281180, respectively.

Original entry on oeis.org

1, 3, 57, 2739, 246801, 35822307, 7636142793, 2246286827091, 871869519033249, 431649452286233283, 265466419357802436057, 198541440131880248161779, 177448471205103040365902001, 186781461066456338787698757027, 228695537454759099917373077023593, 322272887805877963568678968978370451, 517868815187736150011294497645677002049
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2017

Keywords

Examples

			E.g.f.: C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...
where related series C(x) and S(x) begin:
C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! +...+ A281181(n)*x^(2*n)/(2*n)! +...
S(x) = S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
Also, the logarithm of C(x) + S(x) begins:
log(C(x) + S(x)) = x + 3*x^3/3! + 57*x^5/5! + 2739*x^7/7! + 246801*x^9/9! + 35822307*x^11/11! + 7636142793*x^13/13! + 2246286827091*x^15/15! +...
which equals Integral C(x)^3 dx.

Links

Crossrefs

Cf. A281180 (S), A281181 (C), A281182 (C+S), A281183 (C^2).

Programs

  • PARI
    {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C^3, 2*n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* E.g.f. d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) */
{a(n) = my(C3=1); C3 = deriv( serreverse( intformal( 1/cosh(x +x*O(x^(2*n)))^3 ) ) ); (2*n)!*polcoeff(C3, 2*n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
E.g.f. C(x)^3 = d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ).
E.g.f. C(x)^3 = d/dx log(C(x) + S(x)) where C(x) and S(x) satisfy:
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
Integrals.
(2.a) S(x) = Integral C(x)^4 dx.
(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
Exponential.
(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
(3.b) C(x) = cosh( Integral C(x)^3 dx ).
(3.c) S(x) = sinh( Integral C(x)^3 dx ).
Derivatives.
(4.a) S'(x) = C(x)^4.
(4.b) C'(x) = C(x)^3*S(x).
(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
(4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
Explicit Solutions.
(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
Showing 1-4 of 4 results.

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