A105459 -id:A105459 - cp's OEIS frontend

A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

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

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			1.8265078108584785588157654061303219730994914849434906683229013637764992...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), page 785.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663 (constant K').

Cf. A105459, A226317, A234014.

RealDigits[ Sum[ (Zeta[k + 1/2] - 1)/(2 k + 1), {k, 0, 370}], 10,  111][[1]]

PARI
```
sum(k=0,340,(zeta(k+1/2)-1)/(2*k+1))
```

A072895 Least k for the Theodorus spiral to complete n revolutions.

Original entry on oeis.org

17, 54, 110, 186, 281, 396, 532, 686, 861, 1055, 1269, 1503, 1757, 2030, 2323, 2636, 2968, 3320, 3692, 4084, 4495, 4927, 5377, 5848, 6338, 6849, 7378, 7928, 8497, 9087, 9695, 10324, 10972, 11640, 12328, 13036, 13763, 14510, 15277, 16063, 16869

Offset: 1

Robert G. Wilson v, Jul 29 2002

nonn

"For n = 16 (which gives sqrt(17)) this sum is 351.15 degree, while for n = 17 the sum is 364.78 degree. That is, perhaps Theodorus stopped at sqrt(17) simply because for n > 16 his spiral started to overlap itself and the drawing became 'messy.'" Nahin p. 34.
There exists a constant c = 1.07889149832972311... such that b(n) = IntegerPart[(Pi*n + c)^2 - 1/6] differs at most by 1 from a(n) for all n>=1. At least for n<=4000 we indeed have a(n)=b(n). - Herbert Kociemba, Sep 12 2005
The preceding constant and function b(n) = a(n) for all n < 21001. - Robert G. Wilson v, Mar 07 2013; Update: b(n) = a(n) for all n < 10^9. - Herbert Kociemba, Jul 15 2013
The preceding constant, c, is actually is -K/2, where K is the Hlawka's Schneckenkonstante (A105459). - Robert G. Wilson v, Jul 10 2013

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, page 35.
Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton University Press, Princeton, NJ. 1998, pgs 33-34.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 76.

Robert G. Wilson v, Table of n, a(n) for n = 1..21000
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
Herbert Kociemba, The Spiral of Theodorus

Cf. A002194, A105459.

s = 0; k = 1; lst = {}; Do[ While[s < (2Pi)n, (* change the value in the parentheses to change the angle *) s = N[s + ArcTan[1/Sqrt@k], 32]; k++]; AppendTo[lst, k - 1], {n, 50}]; lst (* Robert G. Wilson v, Oct 14 2012 *)
K = -2.15778299665944622; f[n_] := Floor[(n*Pi - K/2)^2 - 1/6]; Array[f, 41] (* Robert G. Wilson v, Jul 10 2013 *)
K = -2.1577829966594462209291427868295777235; a[n_] := Module[{a = -(K/2) + n Pi, b}, b = a^2 - 1/6; If[Floor[b] == Floor[b + 1/(144 a^2)], Floor[b], Undefined]] (* defined at least for all n < 10^9, Herbert Kociemba, Jul 15 2013 *)

a(n) = k means k is the least integer such that Sum_{i=1..k} arctan(1/sqrt(i)) > 2n*Pi.

a(n) = A137515(n) + 1. - Robert G. Wilson v, Feb 27 2013

A226317 Decimal expansion of the constant of Theodorus.

Original entry on oeis.org

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

Offset: 1

Walter Gautschi (wxg(AT)cs.purdue.edu), Robert G. Wilson v, and Jean-François Alcover, Apr 15 2013

The decimal expansion of the Sum {k>=1} 1/(k^(3/2) + k^(1/2)).
This constant was first identified by Professor Philip J. Davis.
This constant is not in Steven R. Finch, Mathematical Constants, Cambridge, 2003, nor is it in the Inverse Symbolic Calculator (originally by Simon Plouffe & the Borwein brothers).

			1.86002507922119030718069591571714332466652412152345149304919950359788...

Philip J. Davis, Spirals: From Theodorus to Chaos, AK Peters, 1993.
Julian R. Havil, The Irrationals: A Story of the Numbers You Can't Count On, Princeton University Press, Princeton NJ, 2012, page 277.

Robert G. Wilson v, Table of n, a(n) for n = 1..1024
David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
Ewan Brinkman, Robert Corless, and Veselin Jungic, The Theodorus Variation, Maple Transactions, Vol. 1, No. 2 (2021), Article 14500.
Steven Finch, Constant of Theodorus [broken link. Probably removed from CiteSeerX]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663.
Walter Gautschi, Purdue University, The Spiral of Theodorus, Numerical Analysis, and Special Functions.
Kevin Ryde, Math-PlanePath, Theodorus Spiral.
Jörg Waldvogel, Analytic Continuation of the Theodorus Spiral, Seminar für Angewandte Mathematik, ETH Zürich, 2008.
Eric Weisstein's World of Mathematics, Theodorus's Constant.

Cf. A072895, A105459, A224269.

Digits := 102: evalf(sum((k^(3/2) + k^(1/2))^(-1), k=1..infinity));
# Peter Luschny, Feb 28 2022

digits = 100; 2/Sqrt[Pi]*NIntegrate[(-Exp[t^2])*Log[1 - Exp[-t^2]] - 1, {t, 0, Infinity}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First
(* or *)
a = NSum[1/(k^(3/2) + k^(1/2)), {k, 1, Infinity}, AccuracyGoal -> 2^8, PrecisionGoal -> 2^8, WorkingPrecision -> 2^8, NSumTerms -> 2^15]; RealDigits[a, 10, 105][[1]]

sumpos(k=1,1/sqrt(k)/(1+k)) \\ Charles R Greathouse IV, Aug 29 2013

sumalt(k=0,zeta(k+3/2)*(-1)^k) \\ Charles R Greathouse IV, Aug 29 2013

Sum_{k>=1} 1/(k^(3/2) + k^(1/2)).

Equals -(2/sqrt(Pi)) * Integral_{x>=0} (exp(x^2)*log(1-exp(-x^2))+1) dx (Waldvogel, 2008). - Amiram Eldar, Jul 19 2022

A185051 Continued fraction expansion of Hlawka's Schneckenkonstante K = -2.157782...

Original entry on oeis.org

-3, 1, 5, 2, 1, 24, 9, 2, 2, 1, 1, 6, 8, 11, 2, 44, 1, 5, 3, 424, 1, 5, 39, 2, 1, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 4, 2, 1, 1, 15, 2, 3, 3, 3, 2, 1, 45, 15, 10, 16, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 1, 2, 3, 2, 14, 3, 5, 1, 2, 1, 19, 1, 4, 16, 5, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 16, 65, 2, 3, 3, 1, 5, 3, 1, 11, 2, 1, 3, 1, 1, 2, 5, 2, 11, 1, 2, 2, 1, 10, 1, 1, 2

Offset: 0

David Brink, Jan 22 2012

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]

Cf. A105459 for decimal expansion.

A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 7, 18, 33, 54, 80, 110, 146, 186, 231, 282, 337, 397, 462, 532, 607, 687, 772, 861, 956, 1056, 1160, 1270, 1384, 1504, 1628, 1757, 1891, 2030, 2174, 2323, 2477, 2636, 2800, 2969, 3142, 3321, 3504, 3693, 3886, 4084, 4288, 4496, 4709, 4927, 5150, 5378, 5611, 5849, 6091, 6339

Offset: 0

M. F. Hasler, Mar 31 2022

nonn

The Spiral of Theodorus (a.k.a. Pythagorean spiral or Pythagoras's snail) results from constructing the square roots sqrt(2), sqrt(3), sqrt(4), ... as hypotenuse of the right triangle having the previous one as longer leg, and the shorter leg equal to 1, starting with the segment [0, 1] for sqrt(1). In the complex plane, this construction corresponds to a sequence z(n+1) = z(n) + i*z(n)/|z(n)|, starting with z(1) = 1 (and z(0) = 0 by convention).
This sequence lists those n for which the endpoint is closer to the x-axis than the preceding and next one (plus the two initial points 0 and 1 which are the only ones lying directly on the x-axis). In terms of the complex sequence z(n), this means the indices n such that abs(Im(z(n)) <=  abs(Im(z(n +- 1))).
The two initial terms correspond to the points (0, 0) and (1, 0), which can be regarded as the start of the Pythagorean spiral, and are the only points exactly on the x-axis.
There is a smooth complex-valued function z: [0,oo) -> C, t |-> z(t), which interpolates the spiral in non-integer values. In terms of this function, a(n) = round(t(n)) where t(n) is the n-th zero of the imaginary part of z (if indexing starts with 0 for t(0) = 0, then t(1) = 1, and t(2) ~ 6.8 where z(6.8) = -1). (This function has a natural extension to the whole of R including also the negative real line, but we don't consider negative arguments here.)

Christian Mercat, Escargot de Pythagore: Spirale des racines carrées d'entiers, GeoGebra.org, Dec. 12, 2015
Wikipedia, Spiral of Theodorus, as of March 14, 2022

Cf. A072895, A137515, A105459 (Hlawka's "snail constant").

z=!t=y=0; [n | n<-[0..10^4], ![t < t=y, y > y=abs(imag(z))]*z*=1+I/abs(z)]

a(2n+1) = A072895(n) or A072895(n)+1, for all n > 0.

A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

Original entry on oeis.org

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

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			2.184009470267851952894734157852949070443908406263229420200251207928354...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol 119, No. 9 (November 2012), page 785.

Cf. A105459, A226317, A234015.

RealDigits[ Sum[ Zeta[k + 1/2] - 1, {k, 1, 355}], 10, 111][[1]]

PARI
```
sum(k=1,340,zeta(k+1/2)-1)
```

A351861 Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.

Original entry on oeis.org

2, 1, -1, -1, 5, 1, -521, -29, 1067, 13221, -538019, -692393, 2088537, 3155999, -27611845, -33200670659, 1202005038007, 40366435189, -29289910899229, -14754517273097, 1825124640773023, 18449097055233961, -250479143430425927, -1976767636081931863, 1419438523008706978221

Offset: 0

Robert B Fowler, Feb 22 2022

S(i) is the sum of the angles of the first i-1 triangles in the Spiral of Theodorus (in radians). [Corrected by Robert B Fowler, Oct 23 2022]
S(i) = K + sqrt(i) * (2 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...) where K is Hlawka's Schneckenkonstante, K = A105459 * (-1) = -2.1577829966... .
The coefficients in the polynomial series are a(n)/A351862(n). The series is asymptotic, but is very accurate even for low values of i.

			2/1 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

David Brink, The Spiral of Theodorus and sums of zeta values at half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pages 779-786.
David Brink, The Spiral of Theodorus and sums of zeta values at half-integers, July 2012. There are errors in equation (9): the denominator factors n, n^2, n^3, n^4, n^5 should actually be n, n^3, n^5, n^7, n^9, respectively.
Detlef Gronau, The Spiral of Theodorus, The American Mathematical Monthly, Vol. 111, No. 3 (March 2004), pages 230-237.
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math. 89 (1980) pages 19-44. [For a summary in English, see the Davis reference, pages 157-167.]
Herbert Kociemba, The Spiral of Theodorus, 2018.
Wikipedia, Spiral of Theodorus

Cf. A351862 (denominators).
Cf. A105459, A185051 (Hlawka's constant).
Cf. A027641, A027642 (Bernoulli numbers).
Cf. A072895, A224269 (spiral revolutions).

c[0] = 2; c[n_] := ((2*n - 2)!/(n - 1)!) * Sum[(-1)^(n + 1) * BernoulliB[n - k] * k!/(4^(n - k - 1) * (2*k + 1)! * (n - k)!), {k, 0, n}]; Numerator @ Array[c, 30, 0] (* Amiram Eldar, Feb 22 2022 *)

a(n) = {numerator(if(n==0, 2, ((2*n-2)!/(n-1)!) * sum(k=0, n, (-1)^(n+1) * bernfrac(n-k) * k! / (4^(n-k-1) * (2*k+1)! * (n-k)!))))} \\ Andrew Howroyd, Feb 22 2022

Let r(n) = ((2*n-2)! / (n-1)!) * Sum_{k=0..n} ((-1)^(n+1)*B(n-k)*k!) / ((4^(n-k-1) * (2*k+1)! * (n-k)!) ) for n > 0, where B(n-k) are Bernoulli numbers. Then:

a(n) = numerator(r(n)) for n >= 1 and additionally a(0) = 2.

A351862 Denominators of the coefficients in a series for the angles in the Spiral of Theodorus.

Original entry on oeis.org

1, 6, 120, 840, 8064, 4224, 2196480, 199680, 5013504, 74088448, 1568931840, 1899233280, 2411724800, 2831155200, 8757706752, 6968215339008, 76890652016640, 1488206168064, 289223097712640, 74371653697536, 2197648866017280, 10176804748787712, 29785769996451840

Offset: 0

Robert B Fowler, Feb 22 2022

S(i) is the sum of the angles in the first i-1 triangles of the Spiral of Theodorus (in radians). [corrected by Robert B Fowler, Oct 23 2022]
S(i) = K + sqrt(i) * (2 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...) where K is Hlawka's Schneckenkonstante = A105459 * (-1) = -2.1577829966... .
The coefficients in the polynomial series are A351861(n)/a(n). The series is asymptotic, but is accurate for even very low values of i.
See A351861 for the numerators, as well as references, links, and crossrefs.

			2/1 + 1/(6*i) - 1/(120*i^2) - 1/(840*i^3) + ...

Cf. A351861 (numerators).

c[0] = 2; c[n_] := ((2*n - 2)!/(n - 1)!) * Sum[(-1)^(n + 1) * BernoulliB[n - k] * k!/(4^(n - k - 1) * (2*k + 1)! * (n - k)!), {k, 0, n}]; Denominator @ Array[c, 30, 0] (* Amiram Eldar, Feb 22 2022 *)

cp's OEIS Frontend

A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

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A072895 Least k for the Theodorus spiral to complete n revolutions.

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A226317 Decimal expansion of the constant of Theodorus.

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A185051 Continued fraction expansion of Hlawka's Schneckenkonstante K = -2.157782...

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A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.

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A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

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A351861 Numerators of the coefficients in a series for the angles in the Spiral of Theodorus.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351862 Denominators of the coefficients in a series for the angles in the Spiral of Theodorus.

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