A010466 -id:A010466 - cp's OEIS frontend

A194382 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(8) and < > denotes fractional part.

4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 40, 46, 52, 58, 64, 104, 110, 116, 122, 128, 134, 138, 140, 144, 146, 150, 152, 156, 158, 162, 164, 168, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 204, 208, 210, 214, 216, 220, 222, 226, 228

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010466, A194368, A194383.

r = Sqrt[8]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]   (* A194381 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]   (* A194382 *)
%/2  (* A194383 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 600}];
Flatten[Position[t3, 1]]   (* A194384 *)

A194384 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(8) and < > denotes fractional part.

Original entry on oeis.org

5, 11, 17, 23, 29, 139, 145, 151, 157, 163, 169, 173, 174, 175, 179, 180, 181, 185, 186, 187, 191, 192, 193, 197, 198, 199, 203, 209, 215, 221, 227, 233, 343, 349, 355, 361, 367, 373, 377, 378, 379, 383, 384, 385, 389, 390, 391, 395, 396, 397, 401

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010466, A194368, A194381, A194382, A194383.

r = Sqrt[8]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]   (* A194381 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]   (* A194382 *)
%/2  (* A194383 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 600}];
Flatten[Position[t3, 1]]   (* A194384 *)

A274540 Decimal expansion of exp(sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).
From Peter Bala, Oct 23 2019: (Start)
The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.
Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

			c = 4.113250378782927517173581815140304502401663943151...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Todd Trimble and Vishal Lama, Continued fraction for e, Todd and Vishal’s blog 2008/08/04
Index entries for transcendental numbers

Cf. A274541, A274542, A036039, A135002, A135004, A234473.

Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.

Digits := 80: evalf(exp(sqrt(2))); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=1 then (1 + sqrt(2)) else 1 fi: end: Digits := 49; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2))); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)).

c = lim_{n -> infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(1) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More terms from Jon E. Schoenfield, Mar 15 2018

A280725 Decimal expansion of 22*sin(Pi/22).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 07 2017

The ratio of the perimeter of a regular 11-gon (hendecagon) to its diameter (largest diagonal).

Also least positive root of x^5 - 11x^4 - 484x^3 + 3993x^2 + 43923x - 161051.

			3.130926442012273089763438709560132713403129944771655225197821304298120771...

Index entries for algebraic numbers, degree 5.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280819 (n=12).

evalf(22*sin(Pi/22),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[22*Sin[Pi/22], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
22*sin(Pi/22)
```

A305310 Numbers k(n) used for Cassels's Markoff forms MF(n) corresponding to the conjectured unique Markoff triples MT(n) with maximal entry m(n) = A002559(n), for n >= 1.

Original entry on oeis.org

0, 1, 2, 5, 12, 13, 34, 70, 75, 89, 179, 233, 408, 507, 610, 1120, 1597, 2378, 2673, 2923, 3468, 4181, 6089, 10946, 13860, 15571, 16725, 19760, 23763, 28657, 39916, 51709, 80782, 75025, 113922, 162867, 206855, 196418, 249755, 353702

Offset: 1

Wolfdieter Lang, Jun 26 2018

nonn

For these Markoff forms see Cassels, p. 31. A link to the two original Markoff references is given in A305308.
MF(n) = f_{m(n)}(x, y) = m(n)*F_{m(n)}(x, y) = m(n)*x^2 + (3*m(n) - 2*k(n))*x*y + (l(n) - 3*k(n))*y^2, with the Markoff number m = m(n) = A002559(n) and l(n) = (k(n)^2 + 1)/m(n), for n >= 1.
Every m(n) is proved to appear as largest member of a Markoff triple MT(n) = (m_1(n), m_2(n), m(n)), with positive integers m_1(n) < m_2(n) < m(n) for n >= 3 (MT(1) = (1, 1, 1) and MT(2) = (1, 1, 2)) satisfying the Markoff equation m_1(n)^2 + m_2(n)^2 + m(n)^2 = 3*m_1(n)*m_2(n)*m(n). The famous Markoff uniqueness conjecture is that m(n) as largest member determines exactly one ordered triple MT(n). See, e.g., the Aigner reference, pp. 38-39, and Corollary 3.5, p. 48. [In numerating the sequence with n related to A002559(n) this conjecture is assumed to be true. - Wolfdieter Lang, Jul 29 2018]
The nonnegative integers k(n) are defined for the Markoff forms given by Cassels by k(n) = min{k1(n), k2(n)}, where  m_1(n)*k1(n) - m_2(n) == 0 (mod m(n)), with 0 <= k1(n) < m(n), and  m_2(n)*k2(n) - m_1(n) == 0 (mod m(n)), with 0 <= k2(n) < m(n). The k1 and k2 sequences are k1 = [0, 1, 2, 5, 17, 13, 34, 99, 119, 89, 179, 233, 577, 818, 610, 1777, 1597, 3363, 2673, 2923, 5609, 4181, 6089, 10946, 19601, 22095, 26536, 31881, 38447, 28657, 39916, 51709, 114243, 75025, 113922, 263522, 206855, 196418, 396263, 572063, ...], and k2 = [0, 1, 3, 8, 12, 21, 55, 70, 75, 144, 254, 377, 408, 507, 987, 1120, 2584, 2378, 3793, 4638, 3468, 6765, 8612, 17711, 13860, 15571, 16725, 19760, 23763, 46368, 56641, 83428, 80782, 121393, 180763, 162867, 292538, 317811, 249755, 353702, ...].
The discriminant of the form MF(n) = f_{m(n)}(x, y) is D(n) = 9*m(n)^2 - 4. D(n) = A305312(n), for n >= 1. Because D(n) > 0 (not a square) this is an indefinite binary quadratic form, for n >= 1. See Cassels Fig. 2 on p. 32 for the Markoff tree with these forms.
The quadratic irrational xi, determined by the solution with positive square root of f_{m(n)}(x, 1) = 0, is xi(n) = ((2*k - 3*m) + sqrt(D))/(2*m) (the argument n has been dropped). The regular continued fraction is eventually periodic, but not purely periodic. One can find equivalent Markoff forms determining purely periodic quadratic irrationals. The corresponding k sequence is given in A305311.
For the approximation of xi(n) with infinitely many rationals (in lowest terms) Perron's unimodular invariant M(xi) enters. For quadratic irrationals M(xi) < 3, and the values coincide with the discrete Lagrange spectrum < 3: M(xi(n)) = Lagrange(n) = sqrt{D(n)}/m(n), n >= 1. For n=1..4 see A002163, A010466, A200991 and A305308.

			n = 5: a(5) = k(5) = 12 because m(5) = A002559(5) = 29 with the triple MT(5) = (2, 5, 29). Whence 2*k1(5) - 5 == 0 (mod 29) for k1(5) = 17 < 29, and 5*k2(5) - 2 == 0 (mod 29) leads to k2(5) = 12. The smaller value is k2(5) = k(5) = 12. This leads to the form coefficients MF(5) = [29, 63, -31].
The forms MF(n) = [m(n), 3*m(n) - k(n), l(n) - 3*k(n)] with l(n) := (k(n)^2 + 1)/m(n) begin: [1, 3, 1], [2, 4, -2], [5, 11, -5], [13, 29, -13], [29, 63, -31], [34, 76, -34], [89, 199, -89], [169, 367, -181], [194, 432, -196], [233, 521, -233], [433, 941, -463], [610, 1364, -610], [985, 2139, -1055], [1325, 2961, -1327], [1597, 3571, -1597], [2897, 6451, -2927], [4181, 9349, -4181], [5741, 12467, -6149], [6466, 14052, -6914], [7561, 16837, -7639] ... .
The quadratic irrationals xi(n) = ((2*k(n) - 3*m(n)) + sqrt(D(n)))/(2*m(n)) begin: (-3 + sqrt(5))/2, -1 + sqrt(2), (-11 + sqrt(221))/10, (-29 + sqrt(1517))/26, (-63 + sqrt(7565))/58, (-19 + 5*sqrt(26))/17, (-199 + sqrt(71285))/178, (-367 + sqrt(257045))/338, (-108 + sqrt(21170))/97, (-521 + sqrt(488597))/466, (-941 + sqrt(1687397))/866, (-341 + sqrt(209306))/305, (-2139 + sqrt(8732021))/1970, (-2961 + sqrt(15800621))/2650, (-3571 + sqrt(22953677))/3194, (-6451 + sqrt(75533477))/5794, (-9349 + sqrt(157326845))/8362, (-12467 + 5*sqrt(11865269))/11482, (-3513 + 5*sqrt(940706))/3233, (-16837 + sqrt(514518485))/15122, ... .
The invariant M(xi(n)) = Lagrange(n) numbers begin with n >=1: sqrt(5), 2*sqrt(2), (1/5)*sqrt(221), (1/13)*sqrt(1517), (1/29)*sqrt(7565), (10/17)*sqrt(26), (1/89)*sqrt(71285), (1/169)*sqrt(257045), (2/97)*sqrt(21170), (1/233)*sqrt(488597), (1/433)*sqrt(1687397), (2/305)*sqrt(209306), (1/985)*sqrt(8732021), (1/1325)*sqrt(15800621), (1/1597)*sqrt(22953677), (1/2897)*sqrt(75533477), (1/4181)*sqrt(157326845), (5/5741)*sqrt(11865269), (10/3233)*sqrt(940706), (1/7561)*sqrt(514518485), ... .

Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.
J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 172-180 and 222-224.
Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, Sitzungsber. Heidelberger Akademie der Wiss., 1921, 4. Abhandlung, pp. 1-17 , and part II., 8. Abhandlung, pp.1-12, Carl Winters Universitätsbuchhandlung.

Wolfdieter Lang, A Note on Markoff Forms Determining Quadratic Irrationals with Purely Periodic Continued Fractions

Cf. A002559, A305308, A305311, A305312, A305313, A305314.

a(n) = k(n) has been defined in terms of the (conjectured unique) ordered Markoff triple MT(n) = (m_1(n), m_2(n), m(n)) with m(n) = A002559(n) in the comment above as k(n) = min{k1(n), k2(n)}, where m_1(n)*k1(n) - m_2(n) == 0 (mod m(n)), with 0 <= k1(n) < m(n), and m_2(n)*k2(n) - m_1(n) == 0 (mod m(n)), with 0 <= k2(n) < m(n).

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			11.3137084989847603904135097936775846285573750030...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377341 (surface area), A020797 (circumradius/10), A152623 (midradius).
Cf. A131594 (analogous for a regular octahedron).
Cf. A002193, A010466, A010487.

First[RealDigits[8*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "Volume"], 10, 100]]

Equals 8*sqrt(2) = 8*A002193 = 4*A010466 = 2*A010487.

A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 01 2025

The triaugmented triangular prism is Johnson solid J_51.

			1.140119483078766847782705947481317131020537251141...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triaugmented triangular prism.

Cf. A097715 (surface area).

Cf. A002194, A010466, A010503, A120011, A385505.

First[RealDigits[(Sqrt[8] + Sqrt[3])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J51", "Volume"], 10, 100]]

Equals 1/sqrt(2) + sqrt(3)/4 = A010503 + A120011 = (A010466 + A002194)/4.

Equals the largest root of 256*x^4 - 352*x^2 + 25.

A200991 Decimal expansion of square root of 221/25.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Dec 06 2011

This is the third Lagrange number, corresponding to the third Markov number (5). With multiples of the golden ration and sqrt(2) excluded from consideration, the Hurwitz irrational number theorem uses this Lagrange number to obtain very good rational approximations for irrational numbers.

Continued fraction is 2 followed by 1, 36, 3, 148, 3, 36, 1, 4 repeated.

			2.9732137494637011045224016...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 187

Eric Weisstein's World of Mathematics, Lagrange Number.

Cf. A002163 (the first Lagrange number), A010466 (the second Lagrange number).

Mathematica
```
RealDigits[Sqrt[221/25], 10, 100][[1]]
```

sqrt(221)/5 \\ Charles R Greathouse IV, Dec 06 2011

With m = 5 being a Markov number (A002559), L = sqrt(9 - 4/m^2).

A265303 Decimal expansion of Sum_{k>=1} (x-c(2k-1)), where c = convergents to (x = sqrt(8)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 13 2015

			sum = 0.8577206761361919660042227352265446144...

Cf. A010466, A265304, A265305, A265288 (guide).

x = Sqrt[8]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265303 *)
RealDigits[s2, 10, 120][[1]]  (* A265304 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265305 *)

A265304 Decimal expansion of Sum_{k>=1} c(2k), where c = convergents to (x = sqrt(8)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 13 2015

			sum = 0.176627764305151806435705969450793857731839184448725557757777...

Cf. A010466, A265303, A265305, A265288 (guide).

x = Sqrt[8]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265303 *)
RealDigits[s2, 10, 120][[1]]  (* A265304 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265305 *)

cp's OEIS Frontend

A194382 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(8) and < > denotes fractional part.

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A194384 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(8) and < > denotes fractional part.

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A274540 Decimal expansion of exp(sqrt(2)).

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A280725 Decimal expansion of 22*sin(Pi/22).

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Maple

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PARI

A305310 Numbers k(n) used for Cassels's Markoff forms MF(n) corresponding to the conjectured unique Markoff triples MT(n) with maximal entry m(n) = A002559(n), for n >= 1.

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A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

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Mathematica

Formula

A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

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Mathematica

Formula

A200991 Decimal expansion of square root of 221/25.

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