A010503 -id:A010503 - cp's OEIS frontend

A270394 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Fibonacci(k+1).

2, 3, 9, 59, 9437, 62059971, 2813586350787717, 8534689167911295735140758101600, 54171527001975050997893888972139886506909953999125751170768531

Offset: 1

Clark Kimberling, Mar 22 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(1/2) = 1/2 + 1/(2*3) + 1/(3*9) + 1/(5*59) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000045, A010503.

r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

A270517 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/k!.

Original entry on oeis.org

2, 3, 5, 6, 52, 668, 171510, 4590170768, 17103459833953822083, 31906466290986600582512428032058109695, 237271596693541800921324673318278335675822001026279366213434934428597656224

Offset: 1

Clark Kimberling, Mar 30 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(1/2) = 1/(1*2) + 1/(2*3) + 1/(6*5) + 1/(24*6) + ...

Clark Kimberling, Table of n, a(n) for n = 1..13
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000142, A010503.

r[k_] := 1/k!; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

r(k) = 1/k!;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

A270546 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(2k-1).

Original entry on oeis.org

2, 2, 5, 325, 200533, 65627675599, 22975481891957121466348, 1958997403653886589078102754522745217186637162, 141280756113351994103874857935521871912536028357392961997286697261498102983722388787617517574

Offset: 1

Clark Kimberling, Apr 02 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(1/2) = 1/(1*2) + 1/(3*2) + 1/(5*5) + 1/(7*325) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A005408, A010503.

r[k_] := 1/(2k-1); f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt(1/2); Table[n[x, k], {k, 1, z}]

r(k) = 1/(2*k-1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Jan 14 2017

The corresponding values for two to nine points have simple expressions:
  N ... d_min
  2 ... sqrt(2)            (A002193)
  3 ... sqrt(6) - sqrt(2)  (A120683)
  4 ... 1                  (A000007)
  5 ... sqrt(2) / 2        (A010503)
  6 ... sqrt(13) / 6       (A381485)
  7 ... 4 - 2*sqrt(3)      (A379338)
  8 ... sqrt(2 - sqrt(3))  (A101263)
  9 ... 1 / 2              (A020761)
In contrast, the value for ten points has a minimal polynomial of degree 18.
The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...

			0.421279543983903432768821760650298...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

C. de Groot, R. Peikert, and D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
Eckard Specht, The best known packings of equal circles in a square.
Jeremy Tan, Sympy (Python) program.
Index entries for algebraic numbers, degree 18.

Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.

RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 24 2025 *)

my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]

d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).
One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025
Equals A232738/(2*A343056). - R. J. Mathar, Sep 05 2025

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Cos[Pi/32], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

A380703 Decimal expansion of the obtuse vertex angle, in radians, in a (small) triakis octahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 30 2025

			2.045535838349167749400916817638245527237744672...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triakis octahedron.

Cf. A380702 (face obtuse angles).

Cf. A010503, A378351, A378352, A378353, A378354.

First[RealDigits[ArcCos[1/4 - Sqrt[2]/2], 10, 100]]

Equals arccos(1/4 - sqrt(2)/2) = arccos(1/2 + A010503).

Equals Pi - 2*A380702.

A020771 Decimal expansion of 1/sqrt(14).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(14) = 0.267261241912424384684553480879753521554001414841337639021696104808573939736... [Vladimir Joseph Stephan Orlovsky, May 30 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[N[1/Sqrt[14],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals 1/A010471 = A010503 * A020764. - R. J. Mathar, Nov 19 2024

A270916 (r,1)-greedy sequence, where r(k) = 1/(k*sqrt(2)).

Original entry on oeis.org

1, 2, 3, 5, 65, 6529, 136091233, 41625259047416909, 2189507051227161558033650829868135, 75931290362065676573711484986356332365619562746656079489987281066955

Offset: 1

Clark Kimberling, Apr 09 2016

Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers. Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x. The sequence (a(n)) is the (r,x)-greedy sequence. We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x. See A270744 for a guide to related sequences.

			a(1) = ceiling(r(1)) = ceiling(1/sqrt(2)) = ceiling(0.707...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1)) = 2;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2)) = 3.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.707..., 0.883..., 0.962..., 0.997..., 0.99998..., 0.9999999992...

Cf. A001620, A270744, A010503.

$MaxExtraPrecision = Infinity; z = 16;
r[k_] := N[1/(k*Sqrt[2]), 1000]; f[x_, 0] = x;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = 1; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]

a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).
r(1)/a(1) + ... + r(n)/a(n) + ... = 1.
Conjecture: a(n) = A270582(n-1). - R. J. Mathar, Jun 02 2016

A371466 Decimal expansion of Product_{k>=1} (1 - 1/(3*k+1)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.8833193751427249786568447498242193512859342691...

Cf. A003881, A010503, A086089, A096427, A224268, A290570, A371467.

RealDigits[Gamma[1/3]^3/(4 Sqrt[3] Pi), 10, 100][[1]]

Equals Gamma(1/3)^3 / (4 * sqrt(3) * Pi).
Equals A290570/2. - Hugo Pfoertner, Mar 31 2024
Equals Integral_{x=0..1} (1-x^3)^(1/3) dx. - Mikhail Kurkov, Jun 29 2025

cp's OEIS Frontend

A270394 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Fibonacci(k+1).

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A270517 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/k!.

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A270546 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/(2k-1).

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A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

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A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

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A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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A380703 Decimal expansion of the obtuse vertex angle, in radians, in a (small) triakis octahedron face.

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