A073747 -id:A073747 - cp's OEIS frontend

A005408 The odd numbers: a(n) = 2*n + 1.

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

Offset: 0

N. J. A. Sloane

Leibniz's series: Pi/4 = Sum_{n>=0} (-1)^n/(2n+1) (cf. A072172).
Beginning of the ordering of the natural numbers used in Sharkovski's theorem - see the Cielsielski-Pogoda paper.
The Sharkovski ordering begins with the odd numbers >= 3, then twice these numbers, then 4 times them, then 8 times them, etc., ending with the powers of 2 in decreasing order, ending with 2^0 = 1.
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(6).
Also continued fraction for coth(1) (A073747 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
a(1) = 1; a(n) is the smallest number such that a(n) + a(i) is composite for all i = 1 to n-1. - Amarnath Murthy, Jul 14 2003
Smallest number greater than n, not a multiple of n, but containing it in binary representation. - Reinhard Zumkeller, Oct 06 2003
Numbers n such that phi(2n) = phi(n), where phi is Euler's totient (A000010). - Lekraj Beedassy, Aug 27 2004
Pi*sqrt(2)/4 = Sum_{n>=0} (-1)^floor(n/2)/(2n+1) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 ... [since periodic f(x)=x over -Pi < x < Pi = 2(sin(x)/1 - sin(2x)/2 + sin(3x)/3 - ...) using x = Pi/4 (Maor)]. - Gerald McGarvey, Feb 04 2005
For n > 1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
a(n) = shortest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1.
First differences of squares (A000290). - Lekraj Beedassy, Jul 15 2006
The odd numbers are the solution to the simplest recursion arising when assuming that the algorithm "merge sort" could merge in constant unit time, i.e., T(1):= 1, T(n):= T(floor(n/2)) + T(ceiling(n/2)) + 1. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 14 2006
2n-5 counts the permutations in S_n which have zero occurrences of the pattern 312 and one occurrence of the pattern 123. - David Hoek (david.hok(AT)telia.com), Feb 28 2007
For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A001248(k)^(n-1)); a(n) = A000005(A000302(n-1)) = A000005(A001019(n-1)) = A000005(A009969(n-1)) = A000005(A087752(n-1)). - Reinhard Zumkeller, Mar 04 2007
For n > 2, a(n-1) is the least integer not the sum of < n n-gonal numbers (0 allowed). - Jonathan Sondow, Jul 01 2007
A134451(a(n)) = abs(A134452(a(n))) = 1; union of A134453 and A134454. - Reinhard Zumkeller, Oct 27 2007
Numbers n such that sigma(2n) = 3*sigma(n). - Farideh Firoozbakht, Feb 26 2008
a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller, Apr 17 2008
Number of divisors of 4^(n-1) for n > 0. - J. Lowell, Aug 30 2008
Equals INVERT transform of A078050 (signed - cf. comments); and row sums of triangle A144106. - Gary W. Adamson, Sep 11 2008
Odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1). - Pierre CAMI, Sep 27 2008
A000035(a(n))=1, A059841(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
Multiplicative closure of A065091. - Reinhard Zumkeller, Oct 14 2008
a(n) is also the maximum number of triangles that n+2 points in the same plane can determine. 3 points determine max 1 triangle; 4 points can give 3 triangles; 5 points can give 5; 6 points can give 7 etc. - Carmine Suriano, Jun 08 2009
Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). - Philippe Deléham, Sep 17 2009
Also the 3-rough numbers: positive integers that have no prime factors less than 3. - Michael B. Porter, Oct 08 2009
Or n without 2 as prime factor. - Juri-Stepan Gerasimov, Nov 19 2009
Given an L(2,1) labeling l of a graph G, let k be the maximum label assigned by l. The minimum k possible over all L(2,1) labelings of G is denoted by lambda(G). For n > 0, this sequence gives lambda(K_{n+1}) where K_{n+1} is the complete graph on n+1 vertices. - K.V.Iyer, Dec 19 2009
A176271 = odd numbers seen as a triangle read by rows: a(n) = A176271(A002024(n+1), A002260(n+1)). - Reinhard Zumkeller, Apr 13 2010
For n >= 1, a(n-1) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n-1)) = 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179084 and A179085. - Reinhard Zumkeller, Jun 28 2010
For n>0, continued fraction [1,1,n] = (n+1)/a(n); e.g., [1,1,7] = 8/15. - Gary W. Adamson, Jul 15 2010
Numbers that are the sum of two sequential integers. - Dominick Cancilla, Aug 09 2010
Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h and n in A000027), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 4). Also a(n)^2 - 1 == 0 (mod 8). - Bruno Berselli, Nov 17 2010
A004767 = a(a(n)). - Reinhard Zumkeller, Jun 27 2011
A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012
a(n) is the minimum number of tosses of a fair coin needed so that the probability of more than n heads is at least 1/2. In fact, Sum_{k=n+1..2n+1} Pr(k heads|2n+1 tosses) = 1/2. - Dennis P. Walsh, Apr 04 2012
A007814(a(n)) = 0; A037227(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2012
1/N (i.e., 1/1, 1/2, 1/3, ...) = Sum_{j=1,3,5,...,infinity} k^j, where k is the infinite set of constants 1/exp.ArcSinh(N/2) = convergents to barover(N). The convergent to barover(1) or [1,1,1,...] = 1/phi = 0.6180339..., whereas c.f. barover(2) converges to 0.414213..., and so on. Thus, with k = 1/phi we obtain 1 = k^1 + k^3 + k^5 + ..., and with k = 0.414213... = (sqrt(2) - 1) we get 1/2 = k^1 + k^3 + k^5 + .... Likewise, with the convergent to barover(3) = 0.302775... = k, we get 1/3 = k^1 + k^3 + k^5 + ..., etc. - Gary W. Adamson, Jul 01 2012
Conjecture on primes with one coach (A216371) relating to the odd integers: iff an integer is in A216371 (primes with one coach either of the form 4q-1 or 4q+1, (q > 0)); the top row of its coach is composed of a permutation of the first q odd integers. Example: prime 19 (q = 5), has 5 terms in each row of its coach: 19: [1, 9, 5, 7, 3] ... [1, 1, 1, 2, 4]. This is interpreted: (19 - 1) = (2^1 * 9), (19 - 9) = (2^1 * 5), (19 - 5) = (2^1 - 7), (19 - 7) = (2^2 * 3), (19 - 3) = (2^4 * 1). - Gary W. Adamson, Sep 09 2012
A005408 is the numerator 2n-1 of the term (1/m^2 - 1/n^2) = (2n-1)/(mn)^2, n = m+1, m > 0 in the Rydberg formula, while A035287 is the denominator (mn)^2. So the quotient a(A005408)/a(A035287) simulates the Hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013
This sequence has unique factorization. The primitive elements are the odd primes (A065091). (Each term of the sequence can be expressed as a product of terms of the sequence. Primitive elements have only the trivial factorization. If the products of terms of the sequence are always in the sequence, and there is a unique factorization of each element into primitive elements, we say that the sequence has unique factorization. So, e.g., the composite numbers do not have unique factorization, because for example 36 = 4*9 = 6*6 has two distinct factorizations.) - Franklin T. Adams-Watters, Sep 28 2013
These are also numbers k such that (k^k+1)/(k+1) is an integer. - Derek Orr, May 22 2014
a(n-1) gives the number of distinct sums in the direct sum {1,2,3,..,n} + {1,2,3,..,n}. For example, {1} + {1} has only one possible sum so a(0) = 1. {1,2} + {1,2} has three distinct possible sums {2,3,4} so a(1) = 3. {1,2,3} + {1,2,3} has 5 distinct possible sums {2,3,4,5,6} so a(2) = 5. - Derek Orr, Nov 22 2014
The number of partitions of 4*n into at most 2 parts. - Colin Barker, Mar 31 2015
a(n) is representable as a sum of two but no fewer consecutive nonnegative integers, e.g., 1 = 0 + 1, 3 = 1 + 2, 5 = 2 + 3, etc. (see A138591). - Martin Renner, Mar 14 2016
Unique solution a( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the number of maximal and maximum cliques in the n-centipede graph. - Eric W. Weisstein, Dec 01 2017
Lexicographically earliest sequence of distinct positive integers such that the average of any number of consecutive terms is always an integer. (For opposite property see A042963.) - Ivan Neretin, Dec 21 2017
Maximum number of non-intersecting line segments between vertices of a convex (n+2)-gon. - Christoph B. Kassir, Oct 21 2022
a(n) is the number of parking functions of size n+1 avoiding the patterns 123, 132, and 231. - Lara Pudwell, Apr 10 2023

			G.f. = q + 3*q^3 + 5*q^5 + 7*q^7 + 9*q^9 + 11*q^11 + 13*q^13 + 15*q^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
T. Dantzig, The Language of Science, 4th Edition (1954) page 276.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology, p. 264.
D. Hök, Parvisa mönster i permutationer [Swedish], (2007).
E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 203-205.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
K. Ciesielski and Z. Pogoda, On ordering the natural numbers, or the Sharkovski theorem, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See p. 35.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, theorem 2.5, k=4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 935
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Bridges.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
Michael Somos, Rational Function Multiplicative Coefficients
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Leo Tavares, Illustration: Triangular Sides
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Davenport-Schinzel Sequence
Eric Weisstein's World of Mathematics, Gnomonic Number
Eric Weisstein's World of Mathematics, Inverse Cotangent,
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cotangent
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Inverse Tangent
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Nexus Number
Eric Weisstein's World of Mathematics, Odd Number
Eric Weisstein's World of Mathematics, Pythagorean Triple
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000027, A005843, A065091.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A128200, A000290, A078050, A144106, A109613, A167875.
Cf. A001651 (n=1 or 2 mod 3), A047209 (n=1 or 4 mod 5).
Cf. A003558, A216371, A179480 (relating to the Coach theorem).
Cf. A000754 (boustrophedon transform).

List([0..100],n->2*n+1); # Muniru A Asiru, Oct 16 2018

a005408 n = (+ 1) . (* 2)
a005408_list = [1, 3 ..]  -- Reinhard Zumkeller, Feb 11 2012, Jun 28 2011

Magma
```
[ 2*n+1 : n in [0..100]];
```

A005408 := n->2*n+1;
A005408:=(1+z)/(z-1)^2; # Simon Plouffe in his 1992 dissertation

Table[2 n - 1, {n, 1, 50}] (* Stefan Steinerberger, Apr 01 2006 *)
Range[1, 131, 2] (* Harvey P. Dale, Apr 26 2011 *)
2 Range[0, 20] + 1 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{2, -1}, {1, 3}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

makelist(2*n+1, n, 0, 30); /* Martin Ettl, Dec 11 2012 */

PARI
```
{a(n) = 2*n + 1}
```

first(n) = Vec((1 + x)/(1 - x)^2 + O(x^n)) \\ Iain Fox, Dec 29 2017

a=lambda n: 2*n+1 # Indranil Ghosh, Jan 04 2017

[2*n+1 for n in range(100)] # G. C. Greubel, Nov 28 2018

a(n) = 2*n + 1. a(-1 - n) = -a(n). a(n+1) = a(n) + 2.
G.f.: (1 + x) / (1 - x)^2.
E.g.f.: (1 + 2*x) * exp(x).
G.f. with interpolated zeros: (x^3+x)/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*(exp(x)+exp(-x))/2. - Geoffrey Critzer, Aug 25 2012
a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Mar 30 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 2*u) * (1 - 2*u + 16*v) - (u - 4*v)^2 * (1 + 2*u + 2*u^2). - Michael Somos, Mar 30 2007
a(n) = b(2*n + 1) where b(n) = n if n is odd is multiplicative. [This seems to say that A000027 is multiplicative? - R. J. Mathar, Sep 23 2011]
From Hieronymus Fischer, May 25 2007: (Start)
a(n) = (n+1)^2 - n^2.
G.f. g(x) = Sum_{k>=0} x^floor(sqrt(k)) = Sum_{k>=0} x^A000196(k). (End)
a(0) = 1, a(1) = 3, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = A000330(A016777(n))/A000217(A016777(n)). - Pierre CAMI, Sep 27 2008
a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. - Jaroslav Krizek, Sep 05 2009
a(n) = (n - 1) + n (sum of two sequential integers). - Dominick Cancilla, Aug 09 2010
a(n) = 4*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. - Bruno Berselli, Nov 17 2010
n*a(2n+1)^2+1 = (n+1)*a(2n)^2; e.g., 3*15^2+1 = 4*13^2. - Charlie Marion, Dec 31 2010
arctanh(x) = Sum_{n>=0} x^(2n+1)/a(n). - R. J. Mathar, Sep 23 2011
a(n) = det(f(i-j+1))A113311(n);%20for%20n%20%3C%200%20we%20have%20f(n)=0.%20-%20_Mircea%20Merca">{1<=i,j<=n}, where f(n) = A113311(n); for n < 0 we have f(n)=0. - _Mircea Merca, Jun 23 2012
G.f.: Q(0), where Q(k) = 1 + 2*(k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
a(n) = floor(sqrt(2*A000384(n+1))). - Ivan N. Ianakiev, Jun 17 2013
a(n) = 3*A000330(n)/A000217(n), n > 0. - Ivan N. Ianakiev, Jul 12 2013
a(n) = Product_{k=1..2*n} 2*sin(Pi*k/(2*n+1)) = Product_{k=1..n} (2*sin(Pi*k/(2*n+1)))^2, n >= 0 (undefined product = 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
Noting that as n -> infinity, sqrt(n^2 + n) -> n + 1/2, let f(n) = n + 1/2 - sqrt(n^2 + n). Then for n > 0, a(n) = round(1/f(n))/4. - Richard R. Forberg, Feb 16 2014
a(n) = Sum_{k=0..n+1} binomial(2*n+1,2*k)*4^(k)*bernoulli(2*k). - Vladimir Kruchinin, Feb 24 2015
a(n) = Sum_{k=0..n} binomial(6*n+3, 6*k)*Bernoulli(6*k). - Michel Marcus, Jan 11 2016
a(n) = A000225(n+1) - A005803(n+1). - Miquel Cerda, Nov 25 2016
O.g.f.: Sum_{n >= 1} phi(2*n-1)*x^(n-1)/(1 - x^(2*n-1)), where phi(n) is the Euler totient function A000010. - Peter Bala, Mar 22 2019
Sum_{n>=0} 1/a(n)^2 = Pi^2/8 = A111003. - Bernard Schott, Dec 10 2020
Sum_{n >= 1} (-1)^n/(a(n)*a(n+1)) = Pi/4 - 1/2 = 1/(3 + (1*3)/(4 + (3*5)/(4 + ... + (4*n^2 - 1)/(4 + ... )))). Cf. A016754. - Peter Bala, Mar 28 2024
a(n) = A055112(n)/oblong(n) = A193218(n+1)/Hex number(n). Compare to the Sep 27 2008 comment by Pierre CAMI. - Klaus Purath, Apr 23 2024
a(k*m) = k*a(m) - (k-1). - Ya-Ping Lu, Jun 25 2024
a(n) = A000217(a(n))/n for n > 0. - Stefano Spezia, Feb 15 2025

Incorrect comment and example removed by Joerg Arndt, Mar 11 2010

Peripheral comments deleted by N. J. A. Sloane, May 09 2022

A073743 Decimal expansion of cosh(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of cos(i). - N. J. A. Sloane, Feb 12 2010
cosh(x) = (e^x + e^(-x))/2.
Equals Sum_{n>=0} 1/A010050(n). See Gradsteyn-Ryzhik (0.245.5). - R. J. Mathar, Oct 27 2012
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.54308063481524377847790562075...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:6 at page 20.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068118 (continued fraction), A073742, A073744, A073745, A073746, A073747, A049470, A137204.

Digits:=100: evalf(cosh(1)); # Wesley Ivan Hurt, Nov 18 2014

RealDigits[Cosh[1],10,120][[1]] (* Harvey P. Dale, Aug 03 2014 *)

PARI
```
cosh(1)
```

Continued fraction representation: cosh(1) = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). See A051396 for proof. Cf. A049470 (cos(1)) and A073742 (sinh(1)). - Peter Bala, Sep 05 2016
Equals Product_{k>=0} 1 + 4/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073746 = A137204/2. - Hugo Pfoertner, Dec 27 2024

A073742 Decimal expansion of sinh(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Decimal expansion of u > 0 such that 1 = arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (u,y(u)). - Clark Kimberling, Jul 04 2020

			1.17520119364380145688238185059...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:7 at page 20.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068139 (continued fraction), A073743, A073744, A073745, A073746, A073747, A049469, A049470, A174548.

First@ RealDigits@ N[Sinh@ 1, 120] (* Michael De Vlieger, Sep 04 2016 *)

PARI
```
sinh(1)
```

Equals (e - e^(-1))/2.
Equals sin(i)/i. - N. J. A. Sloane, Feb 12 2010
Equals Sum_{n>=0} 1/A009445(n). See Gradsteyn-Ryzhik (0.245.6.) - R. J. Mathar, Oct 27 2012
Continued fraction representation: sinh(1) = 1 + 1/(6 - 6/(21 - 20/(43 - 42/(73 - ... - (2*n - 1)*(2*n - 2)/((2*n*(2*n + 1) + 1) - ... ))))). See A051397 for proof. Cf. A049469. - Peter Bala, Sep 02 2016
Equals Product_{k>=1} 1 + 1/(k * Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073745 = A174548/2. - Hugo Pfoertner, Dec 27 2024

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

A073744 Decimal expansion of tanh(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of tan(i)/i. - N. J. A. Sloane, Feb 12 2010
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)).
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.76159415595576488811945828260...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Hyperbolic Functions
Index entries for transcendental numbers

Cf. A004273 (continued fraction), A073747 (coth(1)=1/A073744), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)).

Cf. A027641, A027642, A349003.

RealDigits[Tanh[1], 10, 100][[1]] (* Amiram Eldar, Aug 19 2020 *)

PARI
```
tanh(1)
```

Equals Sum_{k>=1} bernoulli(2*k)*2^(2*k)*(2^(2*k)-1)/(2*k)!, where bernoulli(k) = A027641(k)/A027642(k) is the k-th Bernoulli number. - Amiram Eldar, Aug 19 2020
Equal to the continued fraction [0;1,3,5,...,2n-1,...]. - Thomas Ordowski, Oct 22 2024
Equals 1-A349003. - Hugo Pfoertner, Oct 22 2024

A073746 Decimal expansion of sech(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

sech(x) = 2/(e^x + e^(-x)).

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.64805427366388539957497735322...

Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Secant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068118 (continued fraction), A073743 (cosh(1)=1/A073746), A073742 (sinh(1)), A073744 (tanh(1)), A073745 (csch(1)), A073747 (coth(1)), A122045.

RealDigits[Sech[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
1/cosh(1)
```

Equals Sum_{k>=0} E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A073745 Decimal expansion of csch(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

csch(x) = 2/(e^x - e^(-x)).

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.85091812823932154513384276328...

Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068139 (continued fraction), A073742 (sinh(1)=1/A073745), A073743 (cosh(1)), A073744 (tanh(1)), A073746 (sech(1)), A073747 (coth(1)).

RealDigits[Csch[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
1/sinh(1)
```

Equals Sum_{k>=0} B(2*k) * (2 - 2^(2*k)) / (2*k)!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 15 2021

A073821 Decimal expansion of number with continued fraction expansion 0, 2, 4, 6, ... (the even numbers).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 12 2002

			0.44638996589653450704768179519...

Lucas A. Brown, Table of n, a(n) for n = 0..1000

Cf. A005843 (even numbers), A052119 (continued fraction exp. is 0, 1, 2, 3, ...), A073747 (coth(1), continued fraction exp. is odd numbers).

RealDigits[FromContinuedFraction[2Range[0,200]],10,120][[1]] (* Harvey P. Dale, Oct 30 2011 *)

dec_exp(v)= {my(w=contfracpnqn(v)); w[1,1]/w[2,1]+0.0; }
dec_exp(vector(2000,i,2*(i-1)))

BesselI(1, 1)/BesselI(0, 1) (courtesy of the Inverse Symbolic Calculator).

A298242 Decimal expansion of BesselI(1,1/2)/BesselI(0,1/2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.2424996125808019453507023535036354074122660448659455966...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008586, A052119, A073744, A073747, A073821, A296168, A298241, A298243.

RealDigits[BesselI[1, 1/2]/BesselI[0, 1/2], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (1/2)^2/4]/(4 Gamma[2] Hypergeometric0F1[1, (1/2)^2/4]), 10, 110][[1]]

besseli(1,1/2)/besseli(0,1/2) \\ Michel Marcus, Jul 03 2018

Equals 1/(4 + 1/(8 + 1/(12 + 1/(16 + 1/(20 + 1/(24 + ...)))))).

A025164 a(n) = a(n-2) + (2n-1)a(n-1); a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 4, 21, 151, 1380, 15331, 200683, 3025576, 51635475, 984099601, 20717727096, 477491822809, 11958013297321, 323343850850476, 9388929687961125, 291380164177645351, 9624934347550257708, 337164082328436665131, 12484695980499706867555, 487240307321817004499776

Offset: 0

Wouter Meeussen

nonn

Numerators of convergents to coth(1) = 1.313035... = A073747.
Numerator of continued fraction given by C(n) = [ 1; 3, 5, 7, ..., (2n-1)]. - Amarnath Murthy, May 02 2001
Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, ...) in the main diagonal, (1, 1, 1, ...) in the sum diagonal, and the rest zeros. - Gary W. Adamson, Apr 17 2009
We can use the defining recurrence to extend the sequence to negative indices to give a(-n) = A036244(n-1). - Peter Bala, Sep 11 2014

			G.f. = 1 + x + 4*x^2 + 21*x^3 + 151*x^4 + 1380*x^5 + 15331*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Cf. A001040, A001053, A036244, A058798, A073747.

[n le 2 select 1 else (2*n-3)*Self(n-1)+Self(n-2): n in [1..20]]; // Vincenzo Librandi, Apr 22 2015

a:= proc(n) option remember;
      `if`(n<2, 1, a(n-2) +(2*n-1)*a(n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 17 2014

a[ n_ ] := a[n] =a[n-2]+(-1+2 n) a[n-1]; a[0] := 1; a[1] := 1;
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-2]+(2n-1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Mar 25 2012 *)
a[ n_] := Round[ (Exp[1] + Exp[-1]) (BesselK[n - 3/2, 1] + (2 n - 1) BesselK[n - 1/2, 1]) / Sqrt[2 Pi]]; (* Michael Somos, Aug 26 2015 (n>=0) *)
a[ n_] := Module[{ y = Sqrt[1 - 2 x]}, n! SeriesCoefficient[ Cosh[y - 1] / y, {x, 0, n}]]; (* Michael Somos, Aug 26 2015 (n>=0) *)
a[ n_] := (BesselK[ 1/2, 1] BesselI[ n + 1/2, -1] - BesselI[ -1/2, -1] BesselK[ n + 1/2, 1]) / I // FunctionExpand // Simplify; (* Michael Somos, Aug 26 2015 *)
Join[{1}, Convergents[Coth[1], 20] // Numerator] (* Jean-François Alcover, Jun 15 2019 *)

a(n)={if(n<2,1,a(n-2)+(2*n-1)*a(n-1))} \\ Edward Jiang, Sep 11 2014

def A025164(n):
    if n == 0: return 1
    return sloane.A001147(n)*hypergeometric([-n/2+1/2, -n/2], [1/2, -n, 1/2-n], 1)
[round(A025164(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 11 2014

E.g.f.: cosh((1-2*x)^(1/2)-1)/(1-2*x)^(1/2). - Vladeta Jovovic, Jan 30 2004
a(n) = round((exp(1)+exp(-1))*(BesselK(n-3/2, 1)+(2*n-1)*BesselK(n-1/2, 1))/sqrt(2*Pi) ). - Mark van Hoeij, Jul 02 2010
a(n) ~ sqrt(2)*cosh(1)*(2*n)^n/exp(n). - Vaclav Kotesovec, Jan 05 2013
a(n) = A001147(n)*hypergeometric([-n/2+1/2, -n/2], [1/2, -n, 1/2-n], 1) for n >= 1. - Peter Luschny, Sep 11 2014
a(n) = Sum_{k = 0..floor(n/2)} binomial(n - k, k)*( Product_{j = 1 .. n - 2*k} (2*k + 2*j - 1) ) = Sum_{k = 0..floor((n+1)/2)} 2^(2*k - n - 1)*(2*n + 2 - 2*k)!/( (n + 1 - 2*k)!*(2*k)! ). - Peter Bala, Sep 11 2014
a(n) = -i*( BesselK(1/2, 1)*BesselI(n+1/2, -1) - BesselI(-1/2, -1)*BesselK(n+1/2, 1)) for n>=0 (a(0)=1, a(1) = 1). - G. C. Greubel, Apr 21 2015
a(n) = A036244(-1-n) for all n in Z.
0 = a(n)*(-a(n+2)) + a(n+1)*(+a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) if n >= 0. - Michael Somos, Jan 10 2017
Given e.g.f. A(x), then 0 = A(x) + 3*A'(x) + (2*x-1)*A''(x). - Michael Somos, Jan 10 2017
Given g.f. A(x), then 0 = 1 + (x^2+x-1)*A(x) + 2*x^2*A'(x). - Michael Somos, Jan 10 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

More terms from Vladeta Jovovic, Jan 30 2004

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A005408 The odd numbers: a(n) = 2*n + 1.

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A073743 Decimal expansion of cosh(1).

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A073742 Decimal expansion of sinh(1).

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

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A073744 Decimal expansion of tanh(1).

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A073746 Decimal expansion of sech(1).

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