A196017 -id:A196017 - cp's OEIS frontend

A197019 Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) and to the positive x and y axes.

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

Offset: 0

Clark Kimberling, Oct 08 2011

Let (x,y) denote the point of tangency.  Then
x=0.33861718723736417045737960551501765846156681578...
y=0.21464425212782002883052365316387247038020190838...
slope=-0.332183120530610097233795968342303024088179...
(The Mathematica program includes a graph.)

			radius=0.171994517348101690739024865448714954394...

Cf. A197016, A196017, A196018, A197020.

r = .172; c = 4;
Show[Plot[Cos[c*x], {x, 0, Pi}],
 ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]    (* x coordinate of tangency point *)
y = Cos[c*x1] (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197019 *)
slope = -Sin[x1]   (* slope at tangency point *)

A196012 Number of n X 3 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 1, 2, 7, 13, 16, 32, 65, 125, 211, 390, 732, 1368, 2478, 4555, 8417, 15564, 28583, 52598, 96921, 178684, 328956, 605723, 1115678, 2055336, 3785356, 6971537, 12840292, 23650713, 43560381, 80229677, 147768685, 272167135, 501285666

Offset: 1

R. H. Hardin, Sep 26 2011

nonn

Column 3 of A196017.

			All solutions for n=4:
..2..2..0....2..2..2....0..2..2....2..3..2....2..2..2....2..3..2....2..2..2
..3..3..2....2..0..2....2..3..3....3..4..3....2..0..2....3..3..3....2..0..2
..2..0..2....3..3..3....2..0..2....3..4..3....2..3..3....2..0..2....3..3..2
..2..2..2....2..3..2....2..2..2....2..3..2....0..2..2....2..2..2....2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A196017.

Empirical: a(n) = a(n-1) +a(n-2) +3*a(n-4) -a(n-5) -2*a(n-6).

Empirical g.f.: x*(1 - x)*(1 + x + x^2 + 5*x^3 + 6*x^4) / (1 - x - x^2 - 3*x^4 + x^5 + 2*x^6). - Colin Barker, May 08 2018

A196013 Number of n X 4 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 1, 7, 25, 27, 91, 231, 455, 1135, 2693, 6041, 14257, 33349, 76941, 179553, 418345, 971503, 2261807, 5264419, 12243481, 28490729, 66295633, 154236017, 358872825, 835013523, 1942797397, 4520363539, 10517663443, 24471508585, 56938341019

Offset: 1

R. H. Hardin, Sep 26 2011

nonn

Column 4 of A196017.

			Some solutions for n=5:
..2..3..3..2....2..2..3..2....2..3..2..2....2..2..2..0....2..2..3..2
..3..4..4..3....2..0..3..3....2..3..0..2....2..0..3..2....2..0..3..3
..3..4..3..3....3..3..3..3....0..3..2..3....3..3..4..3....3..3..3..3
..3..3..0..2....2..3..0..2....2..3..0..2....3..4..4..3....3..3..0..2
..2..3..2..2....0..2..2..2....2..3..2..2....2..3..3..2....2..3..2..2

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A196017.

Empirical: a(n) = a(n-1) +2*a(n-2) +4*a(n-3) -a(n-4) -4*a(n-5) -2*a(n-6) -3*a(n-7) -3*a(n-8) +a(n-10) for n>14.

Empirical g.f.: x*(1 + 4*x^2 + 12*x^3 - 15*x^4 - 9*x^5 - x^6 - 8*x^7 + x^8 - 3*x^9 + 2*x^10+ 6*x^11 + 2*x^12 - 2*x^13) / (1 - x - 2*x^2 - 4*x^3 + x^4 + 4*x^5 + 2*x^6 + 3*x^7 + 3*x^8 - x^10). - Colin Barker, May 08 2018

A196014 Number of nX5 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 3, 13, 27, 106, 316, 835, 2350, 7232, 21846, 63985, 186029, 548387, 1625398, 4800819, 14129048, 41608725, 122724222, 362055916, 1067475979, 3146535362, 9276448225, 27352573231, 80649512515, 237779505381, 701041117758

Offset: 1

R. H. Hardin Sep 26 2011

nonn

Column 5 of A196017

			Some solutions for n=4
..2..3..3..3..2....2..3..3..2..2....2..2..3..2..0....0..2..3..3..2
..3..4..3..4..3....3..4..3..0..2....2..0..3..3..2....2..3..4..4..3
..3..3..0..3..3....3..4..4..3..2....2..3..3..0..2....2..0..3..4..3
..2..3..2..3..2....2..3..3..2..0....0..2..3..2..2....2..2..3..3..2

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 2*a(n-1) +a(n-2) +a(n-3) +19*a(n-4) -7*a(n-5) -16*a(n-6) -26*a(n-7) -90*a(n-8) +26*a(n-10) +41*a(n-11) +143*a(n-12) +3*a(n-13) -26*a(n-14) -17*a(n-15) -40*a(n-16) -4*a(n-17) +36*a(n-18) -18*a(n-19) -24*a(n-20) +11*a(n-21) +8*a(n-22) -4*a(n-23) for n>26

A196015 Number of n X 6 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 5, 16, 91, 316, 708, 3572, 14607, 54710, 196832, 782926, 3062632, 11714725, 44732791, 173085686, 668753382, 2573295221, 9902406595, 38178807169, 147173375542, 566944981911, 2184078533126, 8416292695545, 32431165832934

Offset: 1

R. H. Hardin, Sep 26 2011

nonn

			Some solutions for n=4:
..2..2..3..3..2..2....2..2..3..2..2..0....2..3..2..3..2..2....0..2..3..2..3..2
..2..0..3..2..0..2....2..0..2..0..2..2....3..2..0..2..0..2....2..3..3..0..2..3
..3..3..3..0..2..3....3..3..3..2..0..2....2..0..2..3..3..3....2..0..3..2..0..2
..2..3..3..2..3..2....2..2..0..2..2..2....2..2..2..0..2..2....2..2..3..3..2..2

R. H. Hardin, Table of n, a(n) for n = 1..200

Column 6 of A196017.

Empirical: a(n) = a(n-1) +7*a(n-2) +11*a(n-3) +44*a(n-4) -24*a(n-5) -237*a(n-6) -250*a(n-7) -219*a(n-8) +211*a(n-9) +584*a(n-10) -127*a(n-11) +1087*a(n-12) +2905*a(n-13) +2083*a(n-14) +1536*a(n-15) +22*a(n-16) -2414*a(n-17) -3256*a(n-18) -2665*a(n-19) -3346*a(n-20) +8268*a(n-21) +11410*a(n-22) -14170*a(n-23) +13403*a(n-24) -38028*a(n-25) -27873*a(n-26) +9341*a(n-27) -22367*a(n-28) +21475*a(n-29) -13978*a(n-30) -7695*a(n-31) +48470*a(n-32) -3495*a(n-33) -2854*a(n-34) +6746*a(n-35) -24008*a(n-36) -2108*a(n-37) -6882*a(n-38) -5945*a(n-39) +15575*a(n-40) +8171*a(n-41) +80*a(n-42) +1201*a(n-43) -813*a(n-44) -403*a(n-45) -488*a(n-46) -598*a(n-47) +10*a(n-48) -143*a(n-49) -14*a(n-50) +3*a(n-51) -4*a(n-52) +6*a(n-53) +a(n-54) for n>59.

A196016 Number of n X 7 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 9, 32, 231, 835, 3572, 22711, 100094, 517009, 2740712, 14291156, 72486659, 375604408, 1948549100, 10070403094, 52010772351, 269069912146, 1391421035119, 7193724490586, 37198028475134, 192356730926735, 994618459574872, 5142944193784753, 26593708572196934

Offset: 1

R. H. Hardin, Sep 26 2011

nonn

			Some solutions for n=4:
..2..3..3..3..3..2..2....2..2..3..3..2..3..2....2..3..3..3..3..3..2
..3..3..4..4..3..0..2....2..0..2..3..0..2..3....3..3..4..4..4..3..3
..2..0..3..4..4..3..3....3..2..0..3..2..0..2....2..0..3..3..3..0..2
..2..2..3..3..3..3..2....2..3..2..3..3..2..2....2..2..2..0..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..226

Column 7 of A196017.

A196011 Number of n X n 0..4 arrays with each element equal to the number of its horizontal and vertical neighbors within one of itself.

Original entry on oeis.org

1, 1, 2, 25, 106, 708, 22711, 670449, 44864449, 5122273171, 1106078073333, 426788297174150

Offset: 1

R. H. Hardin Sep 26 2011

nonn

Diagonal of A196017

			Some solutions for n=4
..2..3..2..2....2..3..2..0....0..2..3..2....0..2..2..2....2..2..3..2
..3..3..0..2....3..4..3..2....2..3..4..3....2..3..0..2....2..0..3..3
..3..4..3..2....2..3..0..2....2..0..3..2....3..4..3..2....2..3..4..3
..2..3..2..0....0..2..2..2....2..2..2..0....2..3..2..0....0..2..3..2

cp's OEIS Frontend

A197019 Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) and to the positive x and y axes.

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Mathematica

A196012 Number of n X 3 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

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A196013 Number of n X 4 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

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A196014 Number of nX5 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

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A196015 Number of n X 6 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

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A196016 Number of n X 7 0..4 arrays with each element equal to the number its horizontal and vertical neighbors within one of itself.

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Crossrefs

A196011 Number of n X n 0..4 arrays with each element equal to the number of its horizontal and vertical neighbors within one of itself.

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