A010471 -id:A010471 - cp's OEIS frontend

A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

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

Offset: 1

Wolfdieter Lang, Nov 27 2017

The regular continued fraction of sqrt(14)/2 is [1, repeat(1, 6, 1, 2)].
The convergents are given in A295336/A295337.
sqrt(14)/2 appears in a regular hexagon inscribed in a circle of radius 1 unit in the following way. Draw a straight line through two opposed midpoints of a side (halving the hexagon). The length between one of the midpoints, say M, and one of the two vertices nearest to the opposed midpoint is sqrt(13)/2 = A295330 units. A circle through M with this length ratio sqrt(13)/2 intersects the line below the hexagon at a point, say P. Then the length ratio between P and one of the two vertices nearest to M is sqrt(14)/2 (from a right triangle (1/2, sqrt(13)/2, sqrt(14)/2)).

			1.87082869338697069279187436615827465087800990388936347315187273366001757815...

Cf. A010471, A295330, A295336/A295337.

First[RealDigits[Sqrt[14]/2, 10, 100]] (* Paolo Xausa, May 23 2025 *)

A248243 Egyptian fraction representation of sqrt(14) (A010471) using a greedy function.

Original entry on oeis.org

3, 2, 5, 25, 604, 568947, 524109421430, 456412587974094208278324, 217923503007735559214372603301923745039374715408, 53829867761684622028477476025136774072620218179339699337234480313626745601639126196448075512614

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 14]]

A010123 Continued fraction for sqrt(14).

Original entry on oeis.org

3, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6

Offset: 0

N. J. A. Sloane

			3.741657386773941385583748732... = 3 + 1/(1 + 1/(2 + 1/(1 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 2009

Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 56. [From Olivier Gérard, May 22 2009]
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010471 (decimal expansion).

ContinuedFraction[Sqrt[14],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{3},120,{6,1,2,1}] (* Harvey P. Dale, Jan 16 2017 *)

{ allocatemem(932245000); default(realprecision, 15000); x=contfrac(sqrt(14)); for (n=0, 20000, write("b010123.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

a(n) = 1 + floor((n+2)/4) - floor((n+1)/4) + 5*(floor((n+4)/4) - floor((n+3)/4)) for n > 0. - Wesley Ivan Hurt, Apr 10 2017
From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 2, a(2^e) = 6 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^s + 1/2^(2*s-2)). (End)
G.f.: (3 + x + 2*x^2 + x^3 + 3*x^4)/(1 - x^4). - Stefano Spezia, Jul 26 2025

A194395 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(14) and < > denotes fractional part.

Original entry on oeis.org

1, 5, 9, 13, 17, 21, 25, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 63, 67, 71, 75, 79, 83, 87, 121, 125, 129, 133, 137, 141, 145, 149, 151, 152, 153, 155, 156, 157, 159, 160, 161, 163, 164, 165, 167, 168, 169

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010471, A194368, A194396, A194397.

r = Sqrt[14]; 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, 200}];
Flatten[Position[t1, 1]]       (* A194395 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194396 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]       (* A194397 *)

A194396 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(14) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 38, 42, 46, 50, 54, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 94, 98, 102, 106, 110, 114, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

Every term is even; see A194368.

Cf. A010471, A194368, A194395, A194397.

r = Sqrt[14]; 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, 200}];
Flatten[Position[t1, 1]]       (* A194395 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194396 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]       (* A194397 *)

A194397 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(14) and < > denotes fractional part.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 27, 61, 65, 69, 73, 77, 81, 85, 89, 91, 92, 93, 95, 96, 97, 99, 100, 101, 103, 104, 105, 107, 108, 109, 111, 112, 113, 115, 116, 117, 119, 123, 127, 131, 135, 139, 143, 147, 181, 185, 189, 193, 197, 201, 205, 209, 211, 212, 213, 215

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A010471 (sqrt(14)), A194368, A194396, A194397.

r:= sqrt(14):
X:= 0: R:= NULL: count:= 0:
for n from 1 while count < 100 do
  X:= X + frac(1/2+n*r) - frac(n*r);
  if X > 0 then
    count:= count+1;
    R:= R, n
  fi
od:
R; # Robert Israel, Nov 25 2020

r = Sqrt[14]; 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, 200}];
Flatten[Position[t1, 1]]       (* A194395 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t2, 1]]       (* A194396 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]       (* A194397 *)

A041020 Numerators of continued fraction convergents to sqrt(14).

Original entry on oeis.org

3, 4, 11, 15, 101, 116, 333, 449, 3027, 3476, 9979, 13455, 90709, 104164, 299037, 403201, 2718243, 3121444, 8961131, 12082575, 81456581, 93539156, 268534893, 362074049, 2440979187, 2803053236, 8047085659

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,30,0,0,0,-1).

Cf. A010471, A041021.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[14],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*)

G.f.: (3+4*x+11*x^2+15*x^3+11*x^4-4*x^5+3*x^6-x^7)/(1-30*x^4+x^8). - Colin Barker, Jan 03 2012

a(n) = 30*a(n-4) - a(n-8). - Wesley Ivan Hurt, Aug 04 2025

A041021 Denominators of continued fraction convergents to sqrt(14).

Original entry on oeis.org

1, 1, 3, 4, 27, 31, 89, 120, 809, 929, 2667, 3596, 24243, 27839, 79921, 107760, 726481, 834241, 2394963, 3229204, 21770187, 24999391, 71768969, 96768360, 652379129, 749147489, 2150674107, 2899821596, 19549603683, 22449425279, 64448454241, 86897879520

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,30,0,0,0,-1).

Cf. A010471, A041020, A157878 (quadrisection), A068204 (quadrisection).

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[14],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*)
Convergents[Sqrt[14],30]//Denominator (* or *) LinearRecurrence[{0,0,0,30,0,0,0,-1},{1,1,3,4,27,31,89,120},30] (* Harvey P. Dale, Aug 17 2024 *)

G.f.: (1+1*x+3*x^2+4*x^3-3*x^4+x^5-x^6)/(1-30*x^4+x^8). - Colin Barker, Jan 03 2012

A176217 Decimal expansion of (14+4*sqrt(14))/7.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of (14+4*sqrt(14))/7 is A010712.

			(14+4*sqrt(14))/7 = 4.13808993529939507747...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010471 (decimal expansion of sqrt(14)), A010712 (repeat 4, 7).

RealDigits[(14+4Sqrt[14])/7,10,120][[1]] (* Harvey P. Dale, Jan 24 2015 *)

A177033 Decimal expansion of (2+sqrt(14))/4.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 01 2010

Continued fraction expansion of (2+sqrt(14))/4 is A068073.

			(2+sqrt(14))/4 = 1.43541434669348534639...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010471 (decimal expansion of sqrt(14)), A068073 (repeat 1, 2, 3, 2).

RealDigits[(2+Sqrt[14])/4,10,120][[1]] (* Harvey P. Dale, Jul 18 2011 *)

cp's OEIS Frontend

A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

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A248243 Egyptian fraction representation of sqrt(14) (A010471) using a greedy function.

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A010123 Continued fraction for sqrt(14).

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A194395 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(14) and < > denotes fractional part.

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A194396 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(14) and < > denotes fractional part.

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A194397 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(14) and < > denotes fractional part.

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A041020 Numerators of continued fraction convergents to sqrt(14).

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A041021 Denominators of continued fraction convergents to sqrt(14).

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A176217 Decimal expansion of (14+4*sqrt(14))/7.

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