A224269 -id:A224269 - cp's OEIS frontend

A227626 Consider the spiral of Theodorus (A072895). This sequence is closely related to A224269 and gives the number of k successive revolutions such that the triangles are closer to 360 degrees than any previous triangles.

1, 2, 4, 6, 22, 30, 45, 53, 211, 242, 429, 554, 917, 1239, 1738, 2161, 2986, 3005, 3101, 3307, 4800, 6385, 7308, 15148, 16668, 19287, 28103, 72754, 143406, 457425, 955117, 1129313, 2290339, 7362039, 11374333, 11711400, 11778444, 11896240, 14221855, 31972242

Offset: 1

Herbert Kociemba, Jul 18 2013

nonn

Herbert Kociemba, The Spiral of Theodorus

Cf. A072895, A224269.

k=minDist=1; lst={}; K=-2.1577829966594462209291427868295777235; num[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]] While[k<40000000, n=num[k]; If[!NumberQ[n], Print[k," Stop"]; Break[]]; a=2Pi-Mod[K+2 Sqrt[n]+1/(6 Sqrt[n]),2Pi]; b=Mod[K+2 Sqrt[n+1]+1/(6 Sqrt[n+1]),2Pi]; If[a

A088306 Integers n with tan n > |n|, ordered by |n|.

Original entry on oeis.org

1, -2, -11, -33, -52174, 260515, -573204, 37362253, -42781604, 122925461, 534483448, 3083975227, 902209779836, -2685575996367, -65398140378926, 74357078147863, 214112296674652, 642336890023956, -5920787228742393, -12055686754159438, 18190586279576483, -48436859313312404

Offset: 1

Paul Boddington, Nov 05 2003

sign

Name was "Positive integers n with |tan n| > n." before signs were added. The sign here shows whether tan(|n|) is positive or negative.
That this sequence is infinite was proved by Bellamy, Lagarias and Lazebnik. It seems not to be known whether there are infinitely many n with tan n > n.
At approximately 2.37e154, there is a value of n which has tan(n)/n > 556. - Phil Carmody, Mar 04 2007 [This is index 214 in the b-file.]
As n increases, log(|a(n)|)/n seems to approach Pi/2; this is similar to what would be expected if an integer sequence were created by drawing many random numbers independently from a uniform distribution on the interval [-Pi/2,+Pi/2] and including in the sequence only those integers j for which the j-th random number x_j happened to satisfy |x_j| < 1/j (and applying to j the sign of x_j). - Jon E. Schoenfield, Aug 19 2014; updated Nov 07 2014 to reflect the change in the sequence's Name)

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
D. Bellamy, J. C. Lagarias and F. Lazebnik, Proposed problem: large values of Tan n
Jon E. Schoenfield, Magma program

Cf. A000503, A224269, A079330, A088989.

Cf. A249836 (subsequence of positive terms).

a:=proc(n) if abs(evalf(tan(n)))>n then n else fi end: seq(a(n),n=1..100000); # Emeric Deutsch, Dec 18 2004

Select[Range[600000],Abs[Tan[#]]>#&] (* Harvey P. Dale, Nov 30 2012 *)

is(n)=tan(n)>abs(n) \\ Charles R Greathouse IV, Nov 07 2014

More terms from Jon E. Schoenfield, Aug 17 2014

Signs added and other edits by Franklin T. Adams-Watters, Sep 09 2014

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

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.

cp's OEIS Frontend

A227626 Consider the spiral of Theodorus (A072895). This sequence is closely related to A224269 and gives the number of k successive revolutions such that the triangles are closer to 360 degrees than any previous triangles.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A088306 Integers n with tan n > |n|, ordered by |n|.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A226317 Decimal expansion of the constant of Theodorus.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula