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

A281277 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 347", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 11, 1010, 101, 110111, 1100, 11101001, 10111, 1111011100, 110111, 111110101000, 1010011, 11111101110100, 11001001, 1111111010000011, 101111100, 111111110111000101, 1101110011, 11111111101010011010, 10100000101, 1111111111011101110011, 110010001000
Offset: 0

Views

Author

Robert Price, Jan 18 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A281278, A281279, A281280.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 347; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]

A281278 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 347", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 110, 101, 10100, 111011, 11000, 10010111, 111010000, 11101111, 11101100000, 101011111, 1100101000000, 101110111111, 100100110000000, 1100000101111111, 111110100000000, 101000111011111111, 1100111011000000000, 1011001010111111111, 101000001010000000000
Offset: 0

Views

Author

Robert Price, Jan 18 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A281277, A281279, A281280.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 347; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

A281279 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 347", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 3, 10, 5, 55, 12, 233, 23, 988, 55, 4008, 83, 16244, 201, 65155, 380, 261573, 883, 1047194, 1285, 4192115, 3208, 16771111, 6116, 67099657, 14323, 268412948, 22525, 1073708039, 50160, 4294875663, 91360, 17179730463, 211408, 68719103095, 361224
Offset: 0

Views

Author

Robert Price, Jan 18 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A281277, A281278, A281280.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 347; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
Showing 1-4 of 4 results.

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