A249403 -id:A249403 - cp's OEIS frontend

A174968 Decimal expansion of (1 + sqrt(2))/2.

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

Offset: 1

Klaus Brockhaus, Apr 02 2010

a(n) is the diameter of the circle around the Vitruvian Man when the square has sides of unit length. See illustration in links. - Kival Ngaokrajang, Jan 29 2015
The iterated function z^2 - 1/4, starting from z = 0, gives a pretty good rational approximation of (-1)((1 + sqrt(2))/2 - 1) to more than eight decimal digits after just twenty steps. - Alonso del Arte, Apr 09 2016
This sequence describes the minimum Euclidean length of the optimal solution of the well-known Nine dots puzzle, published in Sam Loyd’s Cyclopedia of puzzles (1914), p. 301 since a valid polygonal chain satisfying the conditions of the above-mentioned problem is (0, 1)-(0, 3)-(3, 0)-(0, 0)-(2, 2), and its total length is equal to 5*(1 + sqrt(2)) = 12.071... (i.e., 10*(1 + sqrt(2))/2). - Marco Ripà, Jul 22 2024

			1.20710678118654752440084436210484903928483593768847...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Kival Ngaokrajang, Illustration of Vitruvian Man and construction rule.
Marco Ripà, Shortest polygonal chains covering each planar square grid, arXiv:2207.08708v3 [math.CO], 2024.
Wikipedia, Vitruvian Man.
Index entries for algebraic numbers, degree 2.

Cf. A002193 (decimal expansion of sqrt(2)), A010685 (continued fraction expansion of (1 + sqrt(2))/2), A079291, A249403.

Apart from initial digits the same as A157214 and A010503.

First@RealDigits[(1 + Sqrt[2])/2, 10, 105] (* Michael De Vlieger, Jan 29 2015 *)

(1+sqrt(2))/2 \\ Altug Alkan, Apr 16 2016

Equals Product_{k>=2} (1 + (-1)^k/A079291(k)). - Amiram Eldar, Dec 03 2024

A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 16 2012

We note that the maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is equal to (1+sqrt(2))/2 for n=2 and 4/3 for n=3. Moreover it can be proved that M(n) < (1 + 1/n)^(n-1) - it is a finite version of Carleman's inequality (see the paper of Witula et al. for the proof). The sequence M(n), n=2,3,..., is increasing.

The decimal expansions of M(5) and M(6) are A219246 and A219336, respectively.

			M(4) = 1.42084438540961...

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008.

Cf. A219246, A219336, A249403.

RealDigits[N[Root[387420489 + 22039921152*#1 + 373658292864*#1^2 + 12841816536576*#1^3 + 274965186525696*#1^4 - 201976270848000*#1^5 + 42624005978423296*#1^6 + 342213608420278272*#1^7 + 660475521813381120*#1^8 - 2629784260986273792*#1^9 + 41447678188009291776*#1^10 + 427447433656163893248*#1^11 - 198705178996352483328*#1^12 - 2098418839125516877824*#1^13 + 16905530303693690241024*#1^14 + 14417509185682352898048*#1^15 - 20033038006659651207168*#1^16 - 149735761790067869220864*#1^17 + 18738444188050884919296*#1^18 + 361130725214496730644480*#1^19 + 220843507713085418766336*#1^20 - 1387347813563214701002752*#1^21 + 1472163837099830446915584*#1^22 - 654295038711035754184704*#1^23 + 109049173118505959030784*#1^24 & , 4], 105]][[1]] (* Vaclav Kotesovec, Oct 26 2014 *)

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 16 2012

The maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is discussed in A219245 - see also the paper of Witula et al. for the proofs.

The decimal expansions of M(4) and M(6) are A219245 and A219336, respectively.

			1.486353228963....

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008

Cf. A219245, A219336, A249403.

RealDigits[c5/.FindRoot[{1+x2/2+x3/3+x4/4+x5/5==c5, x2/2+x3/3+x4/4+x5/5==c5*x2^2, x3/3+x4/4+x5/5==c5*x3^3/x2^2, x4/4+x5/5==c5*x4^4/x3^3, x5/5==c5*x5^5/x4^4},{{c5,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2}},WorkingPrecision->120],10,105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 18 2012

The decimal expansions of M(4) and M(5) are A219245 and A219246, respectively.

			1.537937556520034931368158716...

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008.

Cf. A219245, A219246, A249403.

RealDigits[c6/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6*x2^2, x3/3 + x4/4 + x5/5 + x6/6 == c6*x3^3/x2^2, x4/4 + x5/5 + x6/6 == c6*x4^4/x3^3, x5/5 + x6/6 == c6*x5^5/x4^4, x6/6 == c6*x6^6/x5^5},{{c6,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2},{x6,1/2}},WorkingPrecision->120],10,105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

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A174968 Decimal expansion of (1 + sqrt(2))/2.

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A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

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A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

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A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.

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A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.