A248968 -id:A248968

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A248965

Decimal expansion of area bounded by x = 1, y = 1/x, and y = sin(1/x).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

FORMULA

Let u,v,w be the constants at A248965, A248967, and A248968, respectively. Then u = w - v.

EXAMPLE

area = 0.0812827268084612325963682078235506606...

MAPLE

evalf(int(1/x-sin(1/x), x = 1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

MATHEMATICA

v = Integrate[1/x - Sin[1/x], {x, 1, Infinity}]

N[v, 1000]

s = Join[{0}, RealDigits[N[v, 1000]][[1]] ](* A248965 *)

Plot[{1/x, Sin[1/x]}, {x, 1, 4}]

CROSSREFS

Cf. A248966, A248967, A248968.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Oct 18 2014

STATUS

approved

A248966

Decimal expansion of area bounded by x = 1, y = Pi/x, and y = sin(Pi/x).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

EXAMPLE

area = 2.0366242672406191211213948107665304350...

MAPLE

evalf(int(Pi/x-sin(Pi/x), x = 1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

MATHEMATICA

v = Integrate[Pi/x - Sin[Pi/x], {x, 1, Infinity}]

N[v, 1000]

s = RealDigits[N[v, 1000]][[1]] (* A248966 *)

Plot[{Pi/x, Sin[Pi/x]}, {x, 1, 2*Pi}]

CROSSREFS

Cf. A248965, A248967, A248968.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Oct 18 2014

STATUS

approved

A248967

Decimal expansion of the area bounded by x = 1, y = 1/x, and y = tan(1/x).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..300

FORMULA

Let u,v,w be the constants at A248965, A248967, and A248968, respectively. Then v = w - u.

EXAMPLE

area = 0.21307590546539230747963807682...

MAPLE

evalf(int(tan(1/x)-1/x, x = 1..infinity), 120); # Vaclav Kotesovec, Oct 20 2014

MATHEMATICA

v = Integrate[Tan[1/x] - 1/x, {x, 1, Infinity}]

N[v, 100]

s = RealDigits[N[v, 200]][[1]] (* A248967 *)

Plot[{Tan[1/x], 1/x}, {x, 1, 4}]

CROSSREFS

Cf. A248965, A248966, A248968.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Oct 18 2014

STATUS

approved

A248974

Floor( 1/(n*sinh(1/n) + n*sin(1/n) - 2) ).

+10
1

59, 959, 4859, 15359, 37499, 77759, 144059, 245759, 393659, 599999, 878459, 1244159, 1713659, 2304959, 3037499, 3932159, 5011259, 6298559, 7819259, 9599999, 11668859, 14055359, 16790459, 19906559, 23437499, 27418559, 31886459, 36879359, 42436859, 48599999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

When the numbers k*sinh[1/k] - 1 and 1 - k*sin[1/k], for k >=1, are jointly ranked, the former occupy positions 1,3,5,7,... and the latter occupy positions 2,4,6,8,... The difference between neighbors is n*Sinh[1/n] + n*Sin[1/n] - 2, so that A248968 represents the closeness between neighbors. All the terms end in 9.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = 60*n^4 - 1.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Colin Barker, Oct 22 2014

G.f.: x*(x^4-64*x^3-654*x^2-664*x-59) / (x-1)^5. - Colin Barker, Oct 22 2014

MATHEMATICA

Table[Floor[1/(n*Sinh[1/n] + n*Sin[1/n] - 2)], {n, 1, 60}]

PROG

(PARI) Vec(x*(x^4-64*x^3-654*x^2-664*x-59)/(x-1)^5 + O(x^100)) \\ Colin Barker, Oct 22 2014

(Magma) [Floor(1/(n*Sinh(1/n) + n*Sin(1/n) - 2)): n in [1..30]]; // Vincenzo Librandi, Oct 23 2014