A070910 -id:A070910 - cp's OEIS frontend

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

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

Offset: 1

Jean-François Alcover, Sep 25 2014

Equals 1+A052119.

			1.697774657964007982006790592551752599486658262998...

MathOverflow, Is any particular algebraic number known to have unbounded continued fraction coefficients?
Eric Weisstein's MathWorld, Continued Fraction Constant
Eric Weisstein's MathWorld, Continued Fraction

Cf. A001040, A001053, A052119, A060997, A070910, A096789.

FromContinuedFraction[Join[{1}, Range[50]]] // RealDigits[#, 10, 105]& // First
(* or *) 1+BesselI[1, 2]/BesselI[0, 2] // RealDigits[#, 10, 105]& // First

1+besseli(1,2)/besseli(0,2) \\ Charles R Greathouse IV, Oct 23 2023

1 + I_1(2) / I_0(2), where I_n(x) gives the modified Bessel function of the first kind.

A261879 Decimal expansion of BesselI(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 19 2015

			0.212739959239852655272354393375932037291752272915691833255184450497...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's MathWorld, Modified Bessel Function of the First Kind

Cf. A070910 (BesselI(0,2)), A096789 (BesselI(1,2)), A229020 (BesselI(2,2)).

Mathematica
```
RealDigits[BesselI[3, 2], 10, 105][[1]]
```

besseli(3,2) \\ Altug Alkan, Nov 19 2015

Sum_{k>=1} 1/((k - 2)!*(k + 1)!).

Also S(2) - S(1), using Peter Bala's notation in A229020.

A334379 Decimal expansion of Sum_{k>=0} 1/((2*k)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/0!^2 + 1/2!^2 + 1/4!^2 + 1/6!^2 + ... = 1.25173804073865146774451594773...

Index entries for sequences related to Bessel functions or polynomials

Cf. A010050, A049470, A070910, A073743, A091681, A197036, A334378.

RealDigits[(BesselI[0, 2] + BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) + besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) + BesselJ(0,2))/2.

Continued fraction: 1 + 1/(4 - 4/(145 - 144/(901 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n - 1))^2. - Peter Bala, Feb 22 2024

A348607 Decimal expansion of BesselJ(1,2).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.

Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

A363679 Decimal expansion of the sum of the reciprocals of triangular polygorials A006472.

Original entry on oeis.org

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

Offset: 1

Kelvin Voskuijl, Aug 17 2023

			2.3948330992734047165226326364363731519686370070913624447267975638...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A001113 (of factorials), A070910 (of factorials squared).
Cf. A365077 (continued fraction).
Cf. A006472.

RealDigits[BesselI[1, 2 Sqrt[2]]/Sqrt[2] 10, 120][[1]]

Equals BesselI(1, 2*sqrt(2))/sqrt(2).

A141827 a(n) = (n^3*a(n-1) - 1)/(n - 1) for n >= 2, with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 31, 418, 8917, 278656, 12037939, 688168846, 50334635593, 4586743668412, 509638185379111, 67832842473959674, 10655922890454756061, 1950921882527424922168, 411794588127327229725307, 99271909637837814308779366, 27107849458438912493917352209

Offset: 0

Peter Bala, Jul 09 2008, Oct 06 2008

For related recurrences of the form a(n) = (n^k*a(n-1)-1)/(n-1) see A001339, A007808 (both k = 2) and A141828 (k = 4). a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.
From Peter Bala, Jul 02 2016: (Start)
For k = 1,2,3,... and x in Z, the recurrence equation a(n) = (n^k*a(n-1) - 1)/(n - 1) with starting value a(1) = x produces an integer sequence. This is because the sequence also satisfies the second-order recurrence a(n) = (1 + (n^k - 1)/(n - 1))*a(n-1) - (n - 1)^(k-1)*a(n-2) with integer starting values a(1) = x, a(2) = x*2^k - 1. Here we take k = 3 and x = 4.
The solution to the recurrence is a(n) = n*n!^(k-1)*( x - Sum_{i = 2..n} 1/(i*(i-1)*i!^(k-1)) ). Hence limit (n -> inf) a(n)/(n*n!^(k-1)) equals the constant x - Sum_{i = 2..inf} 1/(i*(i-1)*i!^(k-1)). Note that the sequence b(n) := n*n!^(k-1) satisfies the same second-order recurrence but with starting values b(1) = 1, b(2) = 2^k. From this observation one can get a generalized continued fraction expansion for a(n)/b(n) and hence, by going to the limit, for the constant x - Sum_{i = 2..n} 1/(i*(i-1)*i!^(k-1)). See, for example, A130820.  (End)

Cf. A001339, A007808, A070910, A082425, A096789, A130820, A141828.

a(n) := n -> n!^2*sum((n-k+1)*(k+1)/k!^2, k = 0..n): seq(a(n), n = 0..16);

a[0] = 1; a[1] = 4; a[n_] := a[n] = (n^3 a[n - 1] - 1)/(n - 1); Table[a@ n, {n, 0, 14}] (* or *)
Table[n!^2 Sum[(n - k + 1) (k + 1)/k!^2, {k, 0, n}], {n, 0, 14}] (* or *)
Table[n n!^2 (4 - Sum[ 1/(k!^2*k*(k - 1)), {k, 2, n}]), {n, 0, 14}] /. 0 -> 1 (* Michael De Vlieger, Jul 03 2016 *)

Sum_{n = 0..inf} a(n)*x^n/n!^2 = 1/(1 - x)^2*Sum_{n = 0..inf} (n + 1)*x^n/n!^2.
a(n) = n!^2*Sum_{k = 0..n} (n - k + 1)(k + 1)/k!^2.
a(n) = n*n!^2*(4 - Sum_{k = 2..n} 1/(k!^2*k*(k - 1))).
Congruence property: a(n) == (1 + n + n^2) (mod n^3).
The recurrence a(n) = (n^2 + n + 2)*a(n-1) - (n - 1)^2*a(n-2), n >= 2, shows that a(n) is always a positive integer. The sequence b(n) := n*n!^2 also satisfies the same recurrence with b(0) = 0, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/(n*n!^2) = 4 - 1^2/(8 - 2^2/(14 - 3^2/(22 -...-(n - 1)^2/(n^2 + n + 2)))), for n > 1. a(n)*b(n+1) - b(n)*a(n+1) = n!^2.
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 0..inf} (n + 1)/n!^2 = BesselI(0,2) + BesselI(1,2) = 3.87022 21569 ..., using the values of the modified Bessel function, BesselI(0, 2) = 2.27958 53023 ... and BesselI(1, 2) = 1.59063 68546 ... (see A070910 and A096789; Cf. A130820). This yields the continued fraction expansion BesselI(0,2) + BesselI(1,2) = 4 - 1^2/(8 - 2^2/(14 - 3^2/(22 -...-(n - 1)^2/(n^2 + n + 2 - ... )))).
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 1..inf} (n + n^2)/n!^2 = Sum_{n = 1..inf} n^3/n!^2 = 1/2 * Sum_{n = 1..inf} n^4/n!^2.
Limit_{n -> infinity} a(n)/(n*n!^2) = Sum_{n = 0..inf} A001405 (n)/n!.
Limit_{n -> infinity} a(n)/(n*n!^2) = 1 + Sum_{n = 0..inf} 1/(Product_{k = 0..n} A008619(k)).

a(15)-a(16) from Jason Yuen, Jan 31 2025

A328378 Number of permutations of length n that possess the maximal sum of distances between contiguous elements.

Original entry on oeis.org

1, 1, 2, 4, 2, 8, 8, 48, 72, 576, 1152, 11520, 28800, 345600, 1036800, 14515200, 50803200, 812851200, 3251404800, 58525286400, 263363788800, 5267275776000, 26336378880000, 579400335360000, 3186701844480000, 76480844267520000, 458885065605120000, 11931011705733120000

Offset: 0

Tomás Roca Sánchez, Oct 14 2019

nonn

From Andrew Howroyd, Oct 16 2019: (Start)
No permutation with maximal sum of distances between contiguous elements can contain three contiguous elements a, b, c such that a < b < c or a > b > c. Otherwise removing b will not alter the sum and then appending b to the end of the permutation will increase it so that the original permutation could not have been maximal. In this sense all solution permutations are alternating.
For odd n consider an alternating permutation of the form p_1 p_2 ... p_n with p_1 > p2, p_2 < p_3, etc. The sum of distances is given by (p_1 + 2*p_3 + 2*p_5 + ... 2*p_{n-2} + p_n) - 2*(p_2 + p_4 + ... p_{n-1}). This is maximized by choosing the central odd p_i to be as highest possible and the even p_i to be least possible but other than that the order does not alter the sum. Similar arguments can be made for p_1 < p_2 and for the case when n is even.
The above considerations lead to a formula for this sequence with the maximum sum being given by A047838(n). (End)

			(1,3,2) is a permutation of length 3 with distance sum |1-3| + |3-2| = 2 + 1 = 3. For n = 3, the 4 permutations with maximum sum of distances are (1,3,2), (2,1,3), (2,3,1) and (3,1,2).

Alois P. Heinz, Table of n, a(n) for n = 0..508
Tomás Roca Sánchez, Github Python program along with explanations.

Cf. A047838 is the maximum distance for every length n, except for n = 0 and n = 1.

Cf. A070910, A096789.

A328378[n_]:=If[n<2,1,2(Floor[n/2]-1)!^2If[Divisible[n,2],1,n-1]];Array[A328378,30,0] (* Paolo Xausa, Aug 13 2023 *)

a(n)={if(n<2, n>=0, 2*(n\2-1)!^2*if(n%2, n-1, 1))} \\ Andrew Howroyd, Oct 16 2019

Python
```
# See Github link
```

a(2*n) = 2*(n-1)!^2 for n > 0; a(2*n+1) = 4*n!*(n-1)! for n > 0. - Andrew Howroyd, Oct 16 2019
D-finite with recurrence: - (12*n-20)*a(n) + 4*a(n-1) + (3*n-2)*(n-3)*(n-2)*a(n-2) = 0. - Georg Fischer, Nov 25 2022
Sum_{n>=0} 1/a(n) = BesselI(0, 2)/2 + BesselI(1, 2)/4 + 2 = A070910/2 + A096789/4 + 2. - Amiram Eldar, Oct 03 2023

Terms a(12) and beyond from Andrew Howroyd, Oct 16 2019

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

RealDigits[(BesselI[0, 2] - BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k+1)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) - besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) - BesselJ(0,2))/2.

A354302 a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 2, 9, 41, 1313, 5471, 1181737, 28952557, 1235309099, 150090055529, 30018011105801, 201787741322329, 523033825507476769, 44196358255381786981, 5774990812036553498851, 1949059399062336805862213, 997918412319916444601453057, 3697415655903280160125896583

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Cf. A001044, A006040, A053557, A061354, A070910, A103816, A120265, A143382, A354303 (denominators), A354304.

Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Numerator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Numerators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

A354303 a(n) is the denominator of Sum_{k=0..n} 1 / (k!)^2.

Original entry on oeis.org

1, 1, 4, 18, 576, 2400, 518400, 12700800, 541900800, 65840947200, 13168189440000, 88519495680000, 229442532802560000, 19387894021816320000, 2533351485517332480000, 855006126362099712000000, 437763136697395052544000000, 1621968544942912438272000000

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...

Cf. A001044, A006040, A053556, A061355, A070910, A143383, A354302 (numerators), A354305.

Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Denominator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator

Denominators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).

cp's OEIS Frontend

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

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Mathematica

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A261879 Decimal expansion of BesselI(3,2).

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A334379 Decimal expansion of Sum_{k>=0} 1/((2*k)!)^2.

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A348607 Decimal expansion of BesselJ(1,2).

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Sage

Formula

A363679 Decimal expansion of the sum of the reciprocals of triangular polygorials A006472.

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A141827 a(n) = (n^3*a(n-1) - 1)/(n - 1) for n >= 2, with a(0) = 1, a(1) = 4.

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A328378 Number of permutations of length n that possess the maximal sum of distances between contiguous elements.

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Python