A010465 -id:A010465 - cp's OEIS frontend

A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).

1, 8, 55, 368, 2449, 16280, 108199, 719072, 4778785, 31758632, 211059991, 1402652240, 9321678001, 61949553848, 411701328775, 2736064645568, 18183205205569, 120841059834440, 803079631825399, 5337067516093232

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Second binomial transform of A109115.

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(7) = 6.6457513110....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (8,-9).

Equals (A094432 without initial term 0)/3.

Cf. A010465 (decimal expansion of square root of 7), A109115.

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 08 2016

Table[Simplify[((4+Sqrt[7])^n -(4-Sqrt[7])^n)/(2*Sqrt[7])], {n, 30}] (* or *) LinearRecurrence[{8, -9},{1, 8}, 30] (* G. C. Greubel, Sep 07 2016 *)
Rest@ CoefficientList[Series[x/(1 -8x +9x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 08 2016 *)

my(x='x+O('x^30)); Vec( x/(1-8*x+9*x^2) ) \\ G. C. Greubel, May 21 2019

[lucas_number1(n,8,9) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 9*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 9*x^2). (End)
a(n) = b such that (3^(n-1)/2)*Integral_{x=0..Pi/2} (sin(n*x))/(4/3-cos(x)) dx = c + b*log(2). - Francesco Daddi, Aug 02 2011
E.g.f.: (1/sqrt(7))*exp(4*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

A171537 Decimal expansion of sqrt(3/7).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -2 -1/2 | 7/2 -5/2>.

			sqrt(3/7) = 0.6546536707079771437982924562...

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[N[Sqrt[3/7],300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

equals A002194/A010465 = 3/A010477.

A176322 Decimal expansion of sqrt(1365).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of sqrt(1365) is (repeat 1, 17, 2, 17, 1, 72) preceded by 36.

			36.94590640382233199163...

G. C. Greubel, Table of n, a(n) for n = 2..1001

Cf. A002194 (sqrt(3)), A002163 (sqrt(5)), A010465 (sqrt(7)), A010470 (sqrt(13)).

Cf. A176321 ((35+sqrt(1365))/14).

SetDefaultRealField(RealField(120)); Sqrt(1365); // G. C. Greubel, Nov 26 2019

evalf( sqrt(1365), 120); # G. C. Greubel, Nov 26 2019

RealDigits[Sqrt[1365],10,120][[1]] (* Harvey P. Dale, Oct 09 2017 *)

default(realprecision, 120); sqrt(1365) \\ G. C. Greubel, Nov 26 2019

numerical_approx(sqrt(1365), digits=120) # G. C. Greubel, Nov 26 2019

Equals sqrt(3)*sqrt(5)*sqrt(7)*sqrt(13).

A176443 Decimal expansion of sqrt(469).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of sqrt(469) is A040447.

			sqrt(469) = 21.65640782770771520178...

Cf. A010465 (decimal expansion of sqrt(7)), A010519 (decimal expansion of sqrt(67)), A176442 (decimal expansion of (21+sqrt(469))/6), A040447.

RealDigits[Sqrt[469],10,120][[1]] (* Harvey P. Dale, May 28 2025 *)

A242703 Decimal expansion of sqrt(7)/2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, May 20 2014

Absolute value of the imaginary part of any of the nontrivial divisors of 2 in O_Q(sqrt(-7)).
The incircle of a triangle with sides of lengths 4, 5, 6 units respectively has a radius of sqrt(7)/2.
With a different offset, decimal expansion of 5 * sqrt(7).
From Wolfdieter Lang, Nov 18 2017: (Start)
In a regular hexagon inscribed in a circle with a radius of 1 unit the three distinct distances between any vertex and the middle of the sides are 1/2, sqrt(7)/2 and sqrt(13)/2.
The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2). The convergents are given in A294972/A294973. (End)

			1.32287565553229529525...

Iain Fox, Table of n, a(n) for n = 1..20000 (first 1000 terms from Ivan Panchenko)

Cf. A010527, A020837, A040022, A294972/A294973.

Mathematica
```
RealDigits[Sqrt[7]/2, 10, 100][[1]]
```

{ default(realprecision, 20080); x=sqrt(7)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b242703.txt", n, " ", d)); } \\ Iain Fox, Nov 18 2017

(1/2 - sqrt(-7)/2)(1/2 + sqrt(-7)/2) = 2.

Equals A010465/2. - R. J. Mathar, Feb 23 2021

A023119 Signature sequence of sqrt(7) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3, 11, 6, 14, 1, 9, 4, 12, 7, 15, 2, 10, 5, 13, 8, 16, 3, 11, 6, 14, 1, 9, 17, 4, 12, 7, 15, 2, 10, 18, 5, 13, 8, 16, 3, 11, 19, 6, 14, 1, 9, 17, 4, 12, 20, 7, 15, 2

Offset: 1

Clark Kimberling

nonn

C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
C. Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A010465.

A176461 Decimal expansion of sqrt(105).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of sqrt(105) is A040094.

			sqrt(105) = 10.24695076595959838322...

Cf. A002194, A002163, A010465, A176460, A040094.

RealDigits[Sqrt[105],10,120][[1]] (* Harvey P. Dale, Aug 03 2016 *)

A178309 Decimal expansion of sqrt(25277).

Original entry on oeis.org

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

Offset: 3

Klaus Brockhaus, May 24 2010

Continued fraction expansion of sqrt(25277) is 158 followed by (repeat 1, 78, 2, 78, 1, 316).

sqrt(25277) = sqrt(7)*sqrt(23)*sqrt(157).

			sqrt(25277) = 158.98742088605626537906...

Daniel Starodubtsev, Table of n, a(n) for n = 3..10000

Cf. A010465 (decimal expansion of sqrt(7)), A010479 (decimal expansion of sqrt(23)), A178310 (decimal expansion of sqrt(157)), A178308 (decimal expansion of (111+sqrt(25277))/158).

RealDigits[Sqrt[25277],10,120][[1]] (* Harvey P. Dale, Oct 12 2011 *)

A274520 a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(7).

Original entry on oeis.org

0, 2, 4, 20, 64, 248, 880, 3248, 11776, 43040, 156736, 571712, 2083840, 7597952, 27698944, 100985600, 368164864, 1342243328, 4893475840, 17840411648, 65041678336, 237125826560, 864501723136, 3151758405632, 11490527150080, 41891604733952, 152726372368384

Offset: 0

Ilya Gutkovskiy, Jun 26 2016

Number of zeros in substitution system {0 -> 111, 1 -> 1001} at step n from initial string "1" (see example).

			Evolution from initial string "1": 1 -> 1001 -> 10011111111001 -> 1001111111100110011001100110011001100110011111111001 -> ...
Therefore, number of zeros at step n:
a(0) = 0;
a(1) = 2;
a(2) = 4;
a(3) = 20, etc.

Ilya Gutkovskiy, Illustration (substitution system {0 -> 111, 1 -> 1001})
Eric Weisstein's World of Mathematics, Substitution System
Index entries for linear recurrences with constant coefficients, signature (2,6)

Cf. A010465, A083099.

Mathematica
```
LinearRecurrence[{2, 6}, {0, 2}, 27]
```

a(n)=([0,1; 6,2]^n*[0;2])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

O.g.f.: 2*x/(1 - 2*x - 6*x^2).
E.g.f.: 2*exp(x)*sinh(sqrt(7)*x)/sqrt(7).
Dirichlet g.f.: (PolyLog(s,1+sqrt(7)) - PolyLog(s,1-sqrt(7)))/sqrt(7), where PolyLog(s,x) is the polylogarithm function.
a(n) = 2*a(n-1) + 6*a(n-2).
a(n) = 2*A083099(n).
Lim_{n->infinity} a(n+1)/a(n) = 1 + sqrt(7) = 1 + A010465.

A004569 Expansion of sqrt(7) in base 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 2

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 2..10000 (terms 2..1000 from Ivan Panchenko)
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

d:= 7; m:=2; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

convert(evalf(sqrt(7),100),binary,100); # Wesley Ivan Hurt, Jan 28 2017

RealDigits[Sqrt[7],2,120][[1]] (* Harvey P. Dale, Jan 18 2012 *)

cp's OEIS Frontend

A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).

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A171537 Decimal expansion of sqrt(3/7).

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A176322 Decimal expansion of sqrt(1365).

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A176443 Decimal expansion of sqrt(469).

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A242703 Decimal expansion of sqrt(7)/2.

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A023119 Signature sequence of sqrt(7) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A176461 Decimal expansion of sqrt(105).

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A178309 Decimal expansion of sqrt(25277).