A062545 -id:A062545 - cp's OEIS frontend

A144396 The odd numbers greater than 1.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A062546 Decimal expansion of the 2nd Du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jason Earls, Jun 26 2001

			0.194528049465325113...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 238.
Francois Le Lionnais, Les nombres remarquables, Paris, Hermann, 1983, p. 23.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Simon Plouffe, Du Bois-Reymond constant to 1024 digits.
Simon Plouffe, The Du Bois-Reymond constant to 1024 digits.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062545, A224196 (c3), A207528 (c4), A243108 (c5), A245333 (c6).

Cf. A073747 (has a similar continued fraction).

RealDigits[(E^2 - 7)/2, 10, 105][[1]] (* Jean-François Alcover, Jun 12 2018 *)

(exp(2)-7)/2 \\ Stefano Spezia, Dec 10 2024

Equals (exp(2)-7)/2.
Continued fraction: [0, 5, 7, 9, 11, 13, ...]. - Gerald McGarvey, Oct 15 2005
Equals exp(1)*cosh(1) - 4 = BesselI(1/2, 1)/BesselI(3/2, 1) - 3. - Terry D. Grant, Jul 20 2018

A085466 a(n) is the denominator of the polynomial in e^2 giving the (2n)th du Bois Reymond constant.

Original entry on oeis.org

2, 8, 32, 384, 1536, 10240, 368640, 10321920, 4587520, 297271296, 29727129600, 435997900800, 15695924428800, 116598295756800, 1523551064555520, 1371195958099968000, 5484783832399872000, 41440588955910144000

Offset: 1

Eric W. Weisstein, Jul 01 2003

nonn

			{(-7 + e^2)/2, (-25 - 4*e^2 + e^4)/8, (-98 + 3*e^2 - 6*e^4 + e^6)/32}

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A085467.

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a := proc(n) local r ; r := residue(x^2/(1+x^2)^n/(tan(x)-x),x=I) ; r := -3-2*subs(tanh(1)=(x-1/x)/(x+1/x),%) ; r := taylor(r,x=0,16*n+2) ; cf := 1 ; for p from 0 to 2*n by 2 do cf := lcm(cf,denom(coeftayl(r,x=0,p))) ; od ; r := simplify(convert(r*cf,polynom)) ; RETURN([cf,r]) ; end: A085466 := proc() # n = 1 invalid formula printf("2, ") ; for n from 2 to 14 do a085467 := a(n)[1] : printf("%d, ",a085467) ; od : end: A085466() ; # R. J. Mathar, Apr 05 2007

a = {}; Do[p = FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x, I}]]]; AppendTo[a, Denominator[p]], {n, 1, 9}]; a (* Artur Jasinski, Mar 26 2008 *)

More terms from R. J. Mathar, Apr 05 2007

Extended by Max Alekseyev, Sep 15 2009

A138730 Continued fraction for 4th Du Bois Reymond constant.

Original entry on oeis.org

0, 190, 1, 4, 2, 1, 1, 1, 6, 10, 6, 6, 1, 3, 2, 9, 67, 2, 3, 1, 7, 1, 2, 1, 1, 1, 2, 1, 1, 4, 1, 68, 5, 6, 1, 1, 1, 4, 1, 1, 5, 1, 4, 8, 2, 5, 5, 1, 1, 3, 1, 2, 2, 6, 1, 1, 1, 9, 2, 1, 1, 1, 17, 5, 21, 2, 9, 1, 3, 1, 2, 4, 1, 3, 5, 1, 56, 1, 4, 14, 5, 17, 4, 2, 34, 1, 18, 1, 1, 4, 1, 5, 16, 1, 3, 27, 1, 11

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A207528, A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^2), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(4)-4*exp(2)-25)/8) \\ Charles R Greathouse IV, Feb 24 2012

A138729 a(n) = -A085466(n) times the free coefficient of the 2n-th Du Bois-Reymond polynomial in e^2.

Original entry on oeis.org

7, 25, 98, 1167, 4650, 30930, 1111860, 31100895, 13812610, 894570642, 89419472100, 1311049104750, 47185076099700, 350440263072900, 4578242813103960, 4119778157653533375, 16476927824724617250, 124478128839848033250, 40326914169455544130500

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a = {7}; Do[p = FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x, I}]]]; AppendTo[a, -First[p[[2]]]], {n, 2, 9}]; a (*Artur Jasinski*)

Edited and extended by Max Alekseyev, Sep 15 2009

A138731 Continued fraction for 6th Du Bois Reymond constant.

Original entry on oeis.org

0, 4531, 1, 1, 3, 1, 8, 3, 2, 1, 3, 1, 1, 9, 1, 2, 9, 1, 1, 5, 3, 2, 1, 1, 5, 1, 1, 3, 5, 3, 2, 1, 6, 1, 7, 2, 4, 1, 3, 1, 23, 2, 8, 2, 1, 19, 1, 23, 1, 3, 1, 18, 1, 2, 1, 1, 3, 24, 1, 4, 1, 1, 5, 1, 1, 2, 1, 1, 89, 1, 1, 1, 2, 1, 8, 1, 52, 1, 1, 1, 1, 1, 4, 1, 15, 1, 1, 1, 5, 3, 1, 2, 1, 4, 1, 4, 24, 5, 1, 5

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^3), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(6)-6*exp(4)+3*exp(2)-98)/32) \\ Charles R Greathouse IV, Feb 24 2012

A138732 Continued fraction for 8th Du Bois Reymond constant.

Original entry on oeis.org

0, 99232, 2, 6, 1, 3, 4, 2, 1, 2, 1, 16, 8, 2, 57, 13, 2, 1, 16, 1, 1, 6, 5, 5, 1, 1, 6, 1, 1, 3, 1, 1, 2, 9, 18, 1, 8, 15, 2, 2, 1, 2, 1, 2, 1, 5, 2, 3, 5, 1, 3, 4, 17, 11, 1, 2, 1, 6, 1, 2, 3, 15, 3, 12, 1, 8, 6, 1, 1, 1, 2, 4, 29, 44, 1, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 22, 1, 2, 5, 8, 2, 2, 5, 2

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^4), {x, I}]]], 100] (* Artur Jasinski *)

contfrac((3*exp(8) - 24*exp(6) + 36*exp(4) - 8*exp(2)  - 1167)/384) \\ Michel Marcus, Sep 09 2013

A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

Original entry on oeis.org

5, 190, 4531, 99232, 2125044, 45190209, 958768567, 20325471335, 430773893366, 9128872855695, 193450867955197, 4099389985205820, 86869246502331992, 1840823999333339814, 39008411877876819180, 826616742911186406242

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a = {}; Do[AppendTo[a,Last[ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x,I}]]], 2]]], {n, 1, 9}]; a (*Artur Jasinski*)

a(n) = floor(1/C(2n)), where C(2n) is the 2n-th Du Bois Reymond constant. [From Max Alekseyev, Sep 15 2009]

Extended by Max Alekseyev, Sep 15 2009

A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

Original entry on oeis.org

1, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Robert Price, Jun 12 2013

nonn

Appears to be essentially the same as A176271, A140139, A130773, A062545. - R. J. Mathar, Aug 23 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945, A222947, A223133.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

Original entry on oeis.org

0, 1, 0, 1, -1, -1, -1, 0, 0, 3, 1, -5, 18, -13, -7, -11, 70, -135, 65, -10, 45, 111, -609, 1215, -1350, 1275, -621, -141, -1009, 6188, -16758, 27335, -26845, 12474, -2548, 1883, 10977, -81353, 270004, -511791, 584710, -420287, 216468, -70169, -3599, -146691, 1248210, -4715217, 10303461, -14439411

Offset: 0

Gerald McGarvey, Mar 02 2005

For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].

Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.

Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]

For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).

For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).

Added the keyword tabl Gerald McGarvey, Aug 20 2009

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A144396 The odd numbers greater than 1.

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A062546 Decimal expansion of the 2nd Du Bois-Reymond constant.

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A085466 a(n) is the denominator of the polynomial in e^2 giving the (2n)th du Bois Reymond constant.

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A138730 Continued fraction for 4th Du Bois Reymond constant.

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A138729 a(n) = -A085466(n) times the free coefficient of the 2n-th Du Bois-Reymond polynomial in e^2.

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A138731 Continued fraction for 6th Du Bois Reymond constant.

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A138732 Continued fraction for 8th Du Bois Reymond constant.

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A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).