A099879 -id:A099879 - cp's OEIS frontend

A257574 Continued square root map applied to the sequence of positive even numbers, (2, 4, 6, 8, ...).

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

Offset: 1

N. J. A. Sloane, May 02 2015

The continued square root or CSR map applied to a sequence b = (b(1), b(2), b(3), ...) is the number CSR(b) := sqrt(b(1)+sqrt(b(2)+sqrt(b(3)+sqrt(b(4)+...)))).

Taking out a factor sqrt(2), one gets CSR(2, 4, 6, 8, ...) = sqrt(2) CSR(1, 1, 3/8, 1/32, ...) < A002193*A001622 = (sqrt(5)+1)/sqrt(2). - M. F. Hasler, May 01 2018

			sqrt(2 + sqrt(4 + sqrt(6 + sqrt(8 + ...)))) = 2.1584768723110397656558534...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..400
A. Herschfeld, On Infinite Radicals, Amer. Math. Monthly, 42 (1935), 419-429.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Cf. A072449, A257575..A257581, A105817, A099879, A001622, A105546.

(CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); t=0;for(N=5,oo,(t==t=Str(CSR([1..2*N]*2)))&&break;print(2*N": "t)) \\ Allows to see the convergence, which is reached when length of vector ~ precision [given as number of digits]. Using Str() to avoid infinite loop when internal representation is "fluctuating". - M. F. Hasler, May 04 2018

a(27)-a(87) from Hiroaki Yamanouchi, May 03 2015

Edited by M. F. Hasler, May 01 2018

A105817 Decimal expansion of the Fibonacci nested radical.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 21 2005

The continued fraction expression of this is A105818. "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane]. We know the asymptotic limit of Fibonacci numbers is Phi^n (Binet expansion) and that Phi^n < 2^n and hence that the Fibonacci Nested Radical converges.

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

			1.66198246232781155796760608181513129505616756246503500829906806743...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
Wikipedia, Tirukkannapuram Vijayaraghavan

Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105818, A239349 for other nested radicals.

Cf. A151558.

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Fibonacci[ Range[50]]]], 10, 111][[1]] (* Robert G. Wilson v, Apr 21 2005 *)

Sqrt(1 + sqrt(1 + sqrt(2 + sqrt(3 + sqrt(5 + ... + sqrt(Fibonacci(n)=A000045)))).

A105815 Decimal expansion of the semiprime nested radical.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 21 2005

The semiprime nested radical is defined by the infinite recursion: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))). This converges by the criterion of T. Vijayaraghavan that "the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane A072449]. The continued fraction representation of this constant is A105816.

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

			2.66352563480685654498944673272195514599922982689272932914833705868...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
Steven R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, 2003, p. 8.

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
Wikipedia, Tirukkannapuram Vijayaraghavan.

For other nested radicals, see A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349.

Cf. A001358.

fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Select[ Range[260], fQ[ # ] &]]], 10, 111][[1]] (* Robert G. Wilson v, May 31 2005 *)

Limit_{n -> infinity} sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))), where semiprime(n) = A001358(n).

A105816 Continued fraction expansion of the semiprime nested radical (A105815).

Original entry on oeis.org

2, 1, 1, 1, 34, 1, 2, 2, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 9, 7, 1, 9, 1, 5, 1, 5, 1, 2, 7, 2, 2, 3, 5, 2, 1, 10, 8, 2, 3, 1, 1, 1, 12, 1, 1, 5, 4, 4, 2, 1, 1, 2, 2, 4, 13, 2, 2, 12, 3, 11, 15, 2, 2, 2, 23, 8, 1, 1, 3, 1, 2, 8, 19, 1, 5, 2, 7, 4, 1, 82, 22, 1, 1, 1, 2, 1, 1, 9, 1, 1, 1, 15, 8, 12, 2, 11, 1, 15

Offset: 0

Jonathan Vos Post, Apr 21 2005

The semiprime nested radical is defined by the infinite recursion: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))). This converges by the criterion of T. Vijayaraghavan that "the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane A072449].

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

			2.66352563480685654498944673272195514599922982689272932914833705868...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.
S. R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
Wikipedia, Tirukkannapuram Vijayaraghavan

Cf. A001358; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349 for other nested radicals.
From Robert G. Wilson v: (Start)
Cf. A072449, Decimal expansion of limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...
Cf. A083869, a(1)=1 then a(n) is the least k>=1 such that the nested radical sqrt(a(1)^2+sqrt(a(2)^2+sqrt(a(3)^2+(....+sqrt(a(n)^2)))...) is an integer.
Cf. A099874, Decimal expansion of a nested radical: cubeRoot(1 + cubeRoot(2 + cubeRoot(3 + cubeRoot(4 + ...
Cf. A099876, Decimal expansion of a nested radical: sqrt(1! + sqrt(2! + sqrt(3! + ...
Cf. A099877, Decimal expansion of a nested radical: sqrt(1^2 + cubeRoot(2^3 + 4thRoot(3^4 + 5thRoot(4^5 + ...
Cf. A099878, Decimal expansion of a nested radical: sqrt(1 + cubeRoot(2 + 4thRoot(3 + 5thRoot(4 + ...
Cf. A099879, Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...
(End)

fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; t = Select[ Range[ 300], fQ[ # ] &]; f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + t[[k]]]; k-- ]; s]; ContinuedFraction[ f[90], 99] (* Robert G. Wilson v, Apr 21 2005 *)

continued fraction representation of: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n)=A001358(n))))).

Offset changed by Andrew Howroyd, Aug 03 2024

A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).

Original entry on oeis.org

1, 1, 1, 1, 23, 18, 1, 1, 1, 1, 1, 1, 2, 1, 22, 2, 1, 53, 1, 1, 10, 1, 1, 17, 2, 4, 1, 27, 1, 2, 422, 3, 3, 13, 12, 5, 28, 1, 3, 1, 2, 1, 3, 2, 4, 6, 6, 3, 5, 50, 1, 1, 6, 3, 2, 1, 118, 2, 1, 1, 2, 6, 1, 4, 1, 1, 5, 2, 3, 3, 16, 1, 4, 6, 2, 2, 22, 4, 3, 10, 1, 1, 49, 5, 1, 1, 12, 1, 1, 3, 13, 3, 10, 1, 2

Offset: 0

Jonathan Vos Post, Apr 21 2005

The decimal expansion of this is A105817. "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane]. We know the asymptotic limit of Fibonacci numbers is Phi^n (Binet expansion) and that Phi^n < 2^n and hence that the Fibonacci Nested Radical converges.

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

			1.66198246232781155796760608181513129505616756246503500829906806743...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
Wikipedia, Tirukkannapuram Vijayaraghavan

Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105817, A239349 for other nested radicals.

f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + Fibonacci[k]]; k-- ]; s]; ContinuedFraction[ f[46], 95] (* Robert G. Wilson v, Apr 21 2005 *)

Sqrt(1 + Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(5 + ... + Sqrt(Fibonacci(n) = A000045)))).

Offset changed by Andrew Howroyd, Aug 03 2024

A296041 Decimal expansion of sqrt(1 + sqrt(3 + sqrt(6 + sqrt(10 + sqrt(15 + ... + sqrt(k*(k + 1)/2 + ...)))))).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 03 2017

			1.86445895816348813235200371527394378415642206982664...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A000217, A072449, A099876, A099879, A105546, A105817, A239349, A257574, A257575, A257576, A257577, A257578.

A296040 Decimal expansion of sqrt(1^1 + sqrt(2^2 + sqrt(3^3 + sqrt(4^4 + sqrt(5^5 + ...))))).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 03 2017

			2.0661766864240510842686019157724311005106721685821046...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A000312, A072449, A099876, A099879, A105546, A105817, A239349, A257574, A257575, A257576, A257577, A257578.

A296042 Decimal expansion of sqrt(1 + 2sqrt(2 + 3sqrt(3 + 4sqrt(4 + 5sqrt(5 + 6*sqrt(6 + ...)))))).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 03 2017

Decimal expansion of sqrt(1 + sqrt(8 + sqrt(432 + sqrt(1327104 + ... + sqrt(k*A030450(k) + ...))))).

			3.0833551418306944580511425800881719306014784933...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A030450, A072449, A099876, A099879, A105546, A105817, A239349, A254689, A257574, A257575, A257576, A257577, A257578.

A347073 Decimal expansion of sqrt(12 + sqrt(23 + sqrt(34 + sqrt(45 + sqrt(5*6 + ...))))).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 15 2021

			2.276781083523092607713497171325417324079...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A002378, A072449, A099876, A099879, A296041.

cp's OEIS Frontend

A257574 Continued square root map applied to the sequence of positive even numbers, (2, 4, 6, 8, ...).

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A105817 Decimal expansion of the Fibonacci nested radical.

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A105815 Decimal expansion of the semiprime nested radical.

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A105816 Continued fraction expansion of the semiprime nested radical (A105815).

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A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).

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A296041 Decimal expansion of sqrt(1 + sqrt(3 + sqrt(6 + sqrt(10 + sqrt(15 + ... + sqrt(k*(k + 1)/2 + ...)))))).

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A296040 Decimal expansion of sqrt(1^1 + sqrt(2^2 + sqrt(3^3 + sqrt(4^4 + sqrt(5^5 + ...))))).

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A296042 Decimal expansion of sqrt(1 + 2*sqrt(2 + 3*sqrt(3 + 4*sqrt(4 + 5*sqrt(5 + 6*sqrt(6 + ...)))))).

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A296042 Decimal expansion of sqrt(1 + 2sqrt(2 + 3sqrt(3 + 4sqrt(4 + 5sqrt(5 + 6*sqrt(6 + ...)))))).

A347073 Decimal expansion of sqrt(12 + sqrt(23 + sqrt(34 + sqrt(45 + sqrt(5*6 + ...))))).