Search: a115365 -id:a115365
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A243108
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Decimal expansion of the 5th du Bois-Reymond constant.
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+10
8
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0, 0, 1, 0, 5, 6, 1, 0, 2, 1, 0, 7, 3, 3, 4, 8, 0, 9, 2, 0, 5, 6, 2, 1, 9, 9, 1, 5, 8, 2, 1, 0, 7, 8, 1, 1, 7, 6, 7, 4, 4, 6, 0, 8, 0, 6, 1, 0, 2, 5, 6, 8, 0, 7, 3, 3, 9, 4, 4, 5, 4, 4, 5, 6, 7, 4, 4, 1, 1, 5, 3, 9, 9, 6, 2, 9, 1, 6, 1, 7, 4, 0, 1, 9, 7, 8, 4, 4, 8, 1, 8, 7, 6, 8, 3, 3, 1, 3, 3, 5, 6, 2, 6, 4, 1, 9
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OFFSET
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0,5
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COMMENTS
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The k-th Du Bois Reymond constant c_k is asymptotic to 2/(1+r^2)^(k/2), where r = A115365 = 4.4934094579090641753... is smallest positive root of the equation tan(r) = r. - Vaclav Kotesovec, Aug 20 2014
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.
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LINKS
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EXAMPLE
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0.00105610210733480920562199158210781176744608061...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A055133
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Matrix inverse of A008459 (squares of entries of Pascal's triangle).
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+10
7
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1, -1, 1, 3, -4, 1, -19, 27, -9, 1, 211, -304, 108, -16, 1, -3651, 5275, -1900, 300, -25, 1, 90921, -131436, 47475, -7600, 675, -36, 1, -3081513, 4455129, -1610091, 258475, -23275, 1323, -49, 1, 136407699, -197216832, 71282064, -11449536, 1033900, -59584, 2352, -64, 1
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OFFSET
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0,4
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COMMENTS
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Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013
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LINKS
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FORMULA
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T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y + (3 - 4*x + x^2)*y^2/2!^2 + (-19 + 27*x - 9*x^2 + x^3)*y^3/3!^2 + ....
The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A008459 has a generating function E(y)*E(x*y).
Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(n,k)^2*R(k,x) with initial value R(0,x) = 1.
There appears to be a connection between the zeros of the Bessel function E(x) and the real zeros of the row polynomials R(n,x). Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -1.44579 64907 ... ( = -(A115365/2)^2). It appears that the real zeros of R(n,x) approach zeros of E(alpha*x) as n increases. A numerical example is given below. Indeed, it may be the case that lim_{n -> inf} R(n,x)/R(n,0) = E(alpha*x) for arbitrary complex x. (End)
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EXAMPLE
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Table T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1;
-1, 1;
3, -4, 1;
-19, 27, -9, 1;
211, -304, 108, -16, 1;
-3651, 5275, -1900, 300, -25, 1;
90921, -131436, 47475, -7600, 675, -36, 1;
Function | Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x) | 1, 5.40649, 7.23983
R(10,x) | 1, 5.26894, 12.97405, 18.53109
R(15,x) | 1, 5.26894, 12.94909, 24.04769, 33.87883
R(20,x) | 1, 5.26894, 12.94909, 24.04216, 38.54959, 53.32419
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 5.26894, 12.94909, 24.04216, 38.54835, 56.46772, ...
where alpha = -1.44579 64907 ... ( = -(A115365/2)^2).
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( (BesselJZeros(0,n)/BesselJZeros(0,1))^2 ). (End)
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MAPLE
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T:= proc(n) local M;
M:= Matrix(n+1, (i, j)-> binomial(i-1, j-1)^2)^(-1);
seq(M[n+1, i], i=1..n+1)
end:
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MATHEMATICA
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T[n_] := Module[{M}, M = Table[Binomial[i-1, j-1]^2, {i, 1, n+1}, {j, 1, n+1}] // Inverse; Table[M[[n+1, i]], {i, 1, n+1}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 28 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A147862
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Decimal expansion of smallest positive solution to x^2 = tan x.
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+10
6
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4, 6, 6, 6, 4, 9, 9, 5, 6, 3, 4, 4, 4, 6, 4, 4, 2, 7, 6, 9, 4, 3, 2, 8, 3, 0, 1, 4, 6, 0, 1, 7, 9, 4, 0, 3, 0, 2, 1, 1, 1, 3, 6, 2, 6, 8, 7, 7, 2, 8, 5, 9, 7, 7, 6, 9, 2, 4, 0, 6, 1, 4, 3, 0, 9, 1, 4, 2, 6, 0, 5, 4, 2, 2, 0, 9, 9, 8, 6, 8, 9, 7, 1, 5, 1, 4, 6, 4, 1, 9, 7, 6, 0, 4, 1, 9, 3, 0, 9, 7
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OFFSET
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1,1
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LINKS
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EXAMPLE
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4.66649956344464427694328301460179403021113626877285...
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MATHEMATICA
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RealDigits[x /. FindRoot[Tan[x] == x^2, {x, 4.5}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008, Nov 19 2008
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STATUS
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approved
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A147868
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Decimal expansion of smallest positive solution to x^8 = tan x.
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+10
6
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1, 0, 8, 2, 1, 8, 4, 2, 0, 5, 8, 6, 8, 1, 9, 5, 4, 4, 5, 6, 1, 9, 5, 8, 7, 1, 0, 6, 7, 1, 9, 9, 8, 6, 1, 8, 7, 6, 8, 3, 9, 6, 1, 5, 2, 5, 9, 0, 5, 9, 4, 1, 1, 4, 0, 3, 4, 0, 8, 4, 2, 6, 7, 0, 9, 8, 7, 6, 3, 4, 7, 3, 9, 8, 4, 2, 6, 9, 8, 1, 9, 5, 3, 1, 1, 0, 7, 2, 6, 4, 2, 4, 9, 3, 8, 8, 5, 0, 3, 4
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OFFSET
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1,3
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LINKS
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MATHEMATICA
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RealDigits[x/.FindRoot[x^8==Tan[x], {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 04 2016 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
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STATUS
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approved
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A147864
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Decimal expansion of smallest positive solution to x^4 = tan x.
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+10
4
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4, 7, 1, 0, 3, 5, 7, 6, 3, 4, 5, 2, 3, 1, 4, 9, 5, 0, 5, 0, 4, 5, 3, 6, 6, 4, 3, 0, 4, 4, 2, 4, 3, 6, 3, 7, 3, 0, 6, 0, 7, 0, 3, 3, 8, 1, 8, 1, 1, 8, 4, 1, 1, 5, 2, 4, 9, 9, 8, 1, 0, 4, 4, 8, 8, 8, 6, 9, 8, 5, 9, 9, 6, 8, 6, 1, 7, 6, 9, 7, 2, 3, 3, 1, 1, 9, 8, 9, 4, 5, 5, 0, 2, 9, 0, 4, 3, 2, 6, 0
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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RealDigits[x/.FindRoot[x^4==Tan[x], {x, 4.7, 4.72}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Dec 15 2011 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
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STATUS
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approved
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A147867
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Decimal expansion of smallest positive solution to x^7 = tan x.
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+10
4
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1, 1, 0, 2, 1, 1, 3, 0, 6, 7, 2, 0, 5, 0, 1, 1, 0, 1, 1, 8, 0, 4, 0, 9, 2, 4, 3, 2, 6, 4, 2, 0, 4, 7, 5, 7, 9, 9, 9, 9, 5, 6, 0, 5, 4, 0, 4, 2, 7, 0, 8, 5, 3, 3, 8, 4, 1, 3, 0, 4, 1, 3, 2, 3, 1, 7, 4, 9, 6, 7, 2, 5, 7, 9, 8, 3, 8, 5, 8, 3, 8, 0, 4, 3, 9, 2, 2, 4, 8, 2, 4, 7, 0, 5, 5, 7, 8, 4, 7, 3
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OFFSET
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1,4
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LINKS
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EXAMPLE
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1.10211306720501101180409243264204757999956054042708...
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MATHEMATICA
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RealDigits[x /. FindRoot[Tan[x] == x^7, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
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STATUS
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approved
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A147863
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Decimal expansion of smallest positive solution to x^3 = tan x.
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+10
3
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4, 7, 0, 2, 7, 7, 4, 5, 3, 7, 0, 5, 7, 4, 1, 3, 6, 2, 5, 4, 3, 3, 7, 9, 0, 8, 1, 1, 0, 4, 8, 9, 2, 6, 6, 9, 6, 8, 5, 6, 9, 7, 8, 6, 1, 6, 7, 5, 3, 4, 5, 7, 4, 0, 6, 9, 3, 7, 8, 0, 5, 2, 8, 8, 9, 7, 7, 4, 7, 1, 2, 5, 9, 7, 3, 1, 1, 6, 6, 8, 6, 5, 0, 0, 2, 0, 3, 9, 0, 2, 2, 8, 1, 7, 3, 7, 5, 7, 8, 4
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OFFSET
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1,1
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LINKS
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EXAMPLE
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4.70277453705741362543379081104892669685697861675345...
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MATHEMATICA
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RealDigits[x /. FindRoot[Tan[x] == x^3, {x, 4.7}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
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STATUS
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approved
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A147865
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Decimal expansion of smallest positive solution to x^5 = tan x.
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+10
3
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1, 2, 3, 1, 8, 1, 8, 0, 4, 3, 1, 4, 8, 8, 8, 9, 7, 8, 0, 1, 8, 2, 4, 3, 2, 9, 5, 7, 6, 7, 2, 7, 9, 2, 3, 6, 3, 4, 0, 1, 5, 1, 1, 3, 8, 1, 3, 9, 7, 8, 4, 2, 0, 7, 9, 1, 3, 1, 2, 2, 5, 1, 8, 6, 1, 1, 3, 9, 7, 7, 1, 8, 7, 5, 9, 1, 4, 1, 5, 8, 4, 1, 1, 6, 5, 7, 5, 7, 2, 4, 7, 8, 7, 4, 0, 1, 5, 5, 0, 4
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OFFSET
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1,2
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LINKS
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EXAMPLE
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1.23181804314888978018243295767279236340151138139784...
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MATHEMATICA
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RealDigits[x /. FindRoot[Tan[x] == x^5, {x, 1.2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
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STATUS
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approved
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A213053
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Decimal expansion of the absolute minimum of sinc(x) = sin(x)/x (negated).
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+10
3
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2, 1, 7, 2, 3, 3, 6, 2, 8, 2, 1, 1, 2, 2, 1, 6, 5, 7, 4, 0, 8, 2, 7, 9, 3, 2, 5, 5, 6, 2, 4, 7, 0, 7, 3, 4, 2, 2, 3, 0, 4, 4, 9, 1, 5, 4, 3, 5, 5, 8, 7, 4, 8, 2, 3, 6, 5, 4, 4, 9, 0, 2, 7, 7, 1, 4, 5, 0, 5, 3, 4, 3, 5, 8, 9, 0, 6, 3, 2, 2, 9, 1, 8, 5, 5, 6, 8, 0, 5, 0, 6, 5, 3, 9, 2, 3, 5, 4, 9, 5, 1, 5, 2, 0, 1
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OFFSET
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0,1
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COMMENTS
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Minimum value of the first negative lobe of sinc(x), attained for abs(x) = A115365.
The involute of the unit circle which starts at (1,0) crosses the x-axis for the first time at x = 1/a. - Álvar Ibeas, Jul 28 2017
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LINKS
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FORMULA
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EXAMPLE
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min[real x](sinc(x)) = -0.2172336282112216574082...
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MATHEMATICA
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digits = 105; NMinimize[ Sinc[x], x, WorkingPrecision -> digits+5] // First // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A102015
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Continued fraction expansion of smallest positive root of tan(x) = x.
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+10
1
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4, 2, 37, 2, 3, 4, 5, 1, 2, 1, 1, 2, 1, 5, 3, 3, 3, 5, 52, 1, 40, 2, 2, 20, 3, 2, 3, 12, 1, 19, 18, 1, 1, 24, 1, 8, 3, 2, 1, 2, 2, 4, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 2, 17, 4, 3, 1, 2, 2, 3, 1, 7, 1, 1, 6, 31, 13, 13, 3, 5, 2, 2, 1, 1, 1, 1, 1, 27, 2, 2, 9, 1, 6, 1, 1, 1, 2, 3, 2, 2, 1, 3, 1, 4, 3, 1
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OFFSET
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0,1
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LINKS
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MATHEMATICA
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ContinuedFraction[BesselJZero[3/2, 1], 100] (* Amiram Eldar, Jun 09 2021 *)
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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