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Search: a115365 -id:a115365
Displaying 1-10 of 16 results found. page 1 2
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A243108 Decimal expansion of the 5th du Bois-Reymond constant. +10
8
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..450
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.
EXAMPLE
0.00105610210733480920562199158210781176744608061...
CROSSREFS
Cf. A062546 (c2), A224196 (c3), A207528 (c4), A245333 (c6).
Cf. A115365, A079330, A088989.
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Aug 19 2014
STATUS
approved
A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle). +10
7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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
LINKS
Alois P. Heinz, Rows n = 0..99, flattened
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498; see p. 493 with beta_{n,k} = |T(n,k)|.
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, 308(24) (2008), 6466-6500.
FORMULA
T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.
From Peter Bala, Jul 24 2013: (Start)
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)
EXAMPLE
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;
... [edited by Petros Hadjicostas, Aug 24 2019]
From Peter Bala, Jul 24 2013: (Start)
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)
MAPLE
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:
seq(T(n), n=0..10); # Alois P. Heinz, Mar 14 2013
MATHEMATICA
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 *)
CROSSREFS
Cf. A000275, A008459 (matrix inverse), A115365.
KEYWORD
sign,tabl
AUTHOR
Christian G. Bower, Apr 25 2000
STATUS
approved
A147862 Decimal expansion of smallest positive solution to x^2 = tan x. +10
6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..100.
EXAMPLE
4.66649956344464427694328301460179403021113626877285...
MATHEMATICA
RealDigits[x /. FindRoot[Tan[x] == x^2, {x, 4.5}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
CROSSREFS
Cf. A115365, A147863, A147864, A147865, A147866, A147867, A147868.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008, Nov 19 2008
STATUS
approved
A147868 Decimal expansion of smallest positive solution to x^8 = tan x. +10
6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..100.
MATHEMATICA
RealDigits[x/.FindRoot[x^8==Tan[x], {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 04 2016 *)
CROSSREFS
Cf. A115365, A147862-A147867.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
STATUS
approved
A147864 Decimal expansion of smallest positive solution to x^4 = tan x. +10
4
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..100.
MATHEMATICA
RealDigits[x/.FindRoot[x^4==Tan[x], {x, 4.7, 4.72}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Dec 15 2011 *)
CROSSREFS
Cf. A115365, A147862-A147868.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
STATUS
approved
A147867 Decimal expansion of smallest positive solution to x^7 = tan x. +10
4
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..100.
EXAMPLE
1.10211306720501101180409243264204757999956054042708...
MATHEMATICA
RealDigits[x /. FindRoot[Tan[x] == x^7, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
CROSSREFS
Cf. A115365, A147862, A147863, A147864, A147865, A147866, A147868.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
STATUS
approved
A147863 Decimal expansion of smallest positive solution to x^3 = tan x. +10
3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..100.
EXAMPLE
4.70277453705741362543379081104892669685697861675345...
MATHEMATICA
RealDigits[x /. FindRoot[Tan[x] == x^3, {x, 4.7}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
CROSSREFS
Cf. A115365, A147862, A147864, A147865, A147866, A147867, A147868.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
STATUS
approved
A147865 Decimal expansion of smallest positive solution to x^5 = tan x. +10
3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..100.
EXAMPLE
1.23181804314888978018243295767279236340151138139784...
MATHEMATICA
RealDigits[x /. FindRoot[Tan[x] == x^5, {x, 1.2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 09 2021 *)
CROSSREFS
Cf. A115365, A147862, A147863, A147864, A147866, A147867, A147868.
KEYWORD
nonn,cons
AUTHOR
Grover Hughes (ghughes(AT)magtel.com), Nov 16 2008
STATUS
approved
A213053 Decimal expansion of the absolute minimum of sinc(x) = sin(x)/x (negated). +10
3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
Table of n, a(n) for n=0..104.
FORMULA
Equals -1 / sqrt(1 + A115365^2) = cos(A115365). - Álvar Ibeas, Jul 28 2017
EXAMPLE
min[real x](sinc(x)) = -0.2172336282112216574082...
MATHEMATICA
digits = 105; NMinimize[ Sinc[x], x, WorkingPrecision -> digits+5] // First // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
RealDigits[Sinc[BesselJZero[3/2, 1]], 10, 100][[1]] (* Vladimir Reshetnikov, May 13 2016 *)
PROG
(PARI) y=solve(x=4, 4.5, tan(x)-x); -sin(y)/y \\ Charles R Greathouse IV, Jun 10 2012
CROSSREFS
Cf. A115365.
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, Jun 09 2012
STATUS
approved
A102015 Continued fraction expansion of smallest positive root of tan(x) = x. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..98.
MATHEMATICA
ContinuedFraction[BesselJZero[3/2, 1], 100] (* Amiram Eldar, Jun 09 2021 *)
CROSSREFS
Cf. A115365 (decimal expansion).
KEYWORD
nonn,cofr
AUTHOR
N. J. A. Sloane, Jun 19 2007
EXTENSIONS
Offset changed by Andrew Howroyd, Aug 04 2024
STATUS
approved
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Last modified August 22 04:22 EDT 2024. Contains 375356 sequences. (Running on oeis4.)