A010465 -id:A010465 - cp's OEIS frontend

A154248 a(n) = ( (7 + sqrt(7))^n - (7 - sqrt(7))^n )/(2*sqrt(7)).

1, 14, 154, 1568, 15484, 150920, 1462552, 14137088, 136492048, 1317130976, 12707167648, 122580846080, 1182430803904, 11405635719296, 110016806306176, 1061198588076032, 10236074368205056, 98734700455677440

Offset: 1

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

Lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(7) = 9.6457513110....

G. C. Greubel, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (14,-42).

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

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

a:= n-> (<<0|1>, <-42|14>>^n)[1, 2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 22 2013

Join[{a=1,b=14},Table[c=14*b-42*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{14,-42},{1,14},25] (* or *) Table[( (7 + sqrt(7))^n - (7 - sqrt(7))^n )/(2*sqrt(7)), {n,1,25}] (* G. C. Greubel, Sep 08 2016 *)

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 14*a(n-1) - 42*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 14x + 42x^2). (End)
E.g.f.: (1/sqrt(7))*exp(7*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 08 2016

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

Edited by Klaus Brockhaus, Oct 06 2009

A176445 Decimal expansion of sqrt(1295).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of sqrt(1295) is 35 followed by (repeat 1, 70).

sqrt(1295) = sqrt(5)*sqrt(7)*sqrt(37).

			sqrt(1295) = 35.98610843089316319412...

Cf. A002163, A010465, A010491, A176444.

RealDigits[Sqrt[1295],10,120][[1]] (* Harvey P. Dale, Apr 19 2019 *)

A178039 Decimal expansion of sqrt(44310).

Original entry on oeis.org

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

Offset: 3

Klaus Brockhaus, May 17 2010

Continued fraction expansion of sqrt(44310) is 210 followed by (repeat 2, 420).

sqrt(44310) = sqrt(2)*sqrt(3)*sqrt(5)*sqrt(7)*sqrt(211).

			sqrt(44310) = 210.49940617493437480150...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A002163 (decimal expansion of sqrt(5)), A010465 (decimal expansion of sqrt(7)), A178040 (decimal expansion of sqrt(211)), A178038 (decimal expansion of (161+sqrt(44310))/259).

A320839 Factorial expansion of sqrt(7) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

2, 1, 0, 3, 2, 2, 6, 4, 6, 2, 3, 11, 2, 8, 11, 8, 16, 5, 5, 16, 19, 5, 1, 14, 16, 7, 14, 10, 27, 12, 10, 29, 28, 19, 16, 3, 6, 4, 28, 33, 24, 21, 42, 10, 2, 45, 3, 34, 4, 1, 46, 48, 8, 5, 41, 20, 53, 17, 31, 50, 10, 6, 56, 27, 29, 18, 15, 11, 19, 49, 37, 64, 56, 51, 34, 21, 3, 27, 15, 61

Offset: 1

G. C. Greubel, Dec 10 2018

nonn

			sqrt(7) = 2 + 1/2! + 0/3! + 3/4! + 2/5! + 2/6! + 6/7! + 4/8! + 6/9! + ...

Index entries for factorial base representation

Cf. A010465 (decimal expansion), A010121 (continued fraction).

Cf. A009949 (sqrt(2)), A067881 (sqrt(3)), A068446 (sqrt(5)).

Digits:=200: a:=n->`if`(n=1,floor(sqrt(7)),floor(factorial(n)*sqrt(7))-n*floor(factorial(n-1)*sqrt(7))): seq(a(n),n=1..90); # Muniru A Asiru, Dec 10 2018

A004570 Expansion of sqrt(7) in base 3.

Original entry on oeis.org

2, 1, 2, 2, 1, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0, 1, 1, 1, 1, 2, 0, 1, 0, 0, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 2, 0, 1, 2, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 2, 0, 2, 2, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 2, 2, 2, 1, 2, 2, 1

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

d:= 7; m:=3; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[7],3,120][[1]] (* Harvey P. Dale, May 02 2012 *)

A004571 Expansion of sqrt(7) in base 4.

Original entry on oeis.org

2, 2, 2, 1, 1, 1, 0, 3, 3, 3, 3, 1, 1, 0, 3, 2, 2, 1, 1, 3, 3, 0, 1, 3, 1, 0, 3, 1, 2, 3, 3, 0, 1, 3, 0, 3, 2, 2, 2, 1, 3, 3, 2, 1, 2, 0, 1, 3, 3, 3, 0, 0, 3, 1, 3, 2, 2, 0, 2, 0, 0, 3, 1, 1, 1, 1, 0, 2, 2, 1, 3, 1, 3, 3, 3, 3, 2, 1, 3, 2, 3, 1, 3, 2, 0, 1, 0, 0, 1, 1, 1, 3, 1, 3, 3, 3, 0, 2, 0

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

d:= 7; m:=4; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[7], 4, 100][[1]] (* G. C. Greubel, Mar 25 2018 *)

A004572 Expansion of sqrt(7) in base 5.

Original entry on oeis.org

2, 3, 1, 0, 3, 2, 4, 4, 1, 3, 0, 0, 3, 3, 1, 4, 3, 3, 1, 4, 1, 1, 2, 4, 0, 4, 0, 4, 0, 4, 4, 0, 1, 4, 2, 4, 1, 3, 2, 4, 4, 0, 0, 1, 0, 3, 3, 4, 0, 3, 0, 4, 4, 3, 3, 2, 2, 0, 1, 4, 1, 0, 2, 3, 2, 0, 3, 1, 2, 0, 0, 0, 3, 0, 0, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 4, 1, 3, 0, 2, 2, 3, 2, 2, 3, 4, 3, 3, 0

Offset: 1

N. J. A. Sloane

Matthew House, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465 (same in base 10).

d:= 7; m:=5; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[N[Sqrt@ 7, 120], 5] // First (* Michael De Vlieger, Aug 06 2015 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A004573 Expansion of sqrt(7) in base 6.

Original entry on oeis.org

2, 3, 5, 1, 2, 5, 2, 1, 0, 1, 2, 2, 3, 2, 1, 2, 5, 0, 0, 3, 5, 0, 3, 5, 0, 5, 2, 1, 2, 5, 3, 3, 0, 4, 4, 5, 1, 1, 5, 5, 2, 3, 4, 5, 2, 1, 5, 0, 4, 4, 1, 0, 3, 0, 5, 5, 2, 5, 0, 1, 1, 2, 3, 1, 5, 4, 5, 3, 0, 1, 5, 5, 3, 5, 0, 3, 1, 2, 3, 5, 5, 1, 0, 2, 0, 3, 3, 1, 4, 0, 5, 5, 2, 2, 2, 4, 4, 2, 0

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

d:= 7; m:=6; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

RealDigits[Sqrt[7], 6, 100][[1]] (* G. C. Greubel, Mar 26 2018 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A004574 Expansion of sqrt(7) in base 7.

Original entry on oeis.org

2, 4, 3, 4, 3, 3, 0, 6, 6, 5, 4, 2, 4, 6, 0, 3, 3, 2, 3, 3, 3, 2, 6, 1, 3, 6, 2, 4, 6, 6, 1, 5, 6, 6, 2, 3, 0, 4, 2, 2, 0, 0, 2, 3, 1, 3, 5, 2, 5, 2, 0, 6, 4, 4, 3, 1, 5, 6, 1, 4, 5, 1, 0, 6, 0, 2, 2, 3, 3, 2, 1, 3, 5, 5, 3, 5, 5, 4, 1, 6, 2, 1, 2, 2, 1, 5, 3, 5, 6, 0, 3, 5, 2, 0, 1, 0, 5, 1, 6

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Vincenzo Librandi)
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

Prune(Reverse(IntegerToSequence(Isqrt(7*7^200), 7))); // Vincenzo Librandi, Apr 28 2017

RealDigits[Sqrt[7], 7, 100][[1]] (* G. C. Greubel, Mar 25 2018 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A004576 Expansion of sqrt(7) in base 9.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010465.

d:= 7; m:=9; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[7],9,105][[1]] (* Harvey P. Dale, Dec 07 2011 *)

Corrected and extended by Harvey P. Dale, Dec 07 2011

cp's OEIS Frontend

A154248 a(n) = ( (7 + sqrt(7))^n - (7 - sqrt(7))^n )/(2*sqrt(7)).

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A176445 Decimal expansion of sqrt(1295).

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A178039 Decimal expansion of sqrt(44310).

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A320839 Factorial expansion of sqrt(7) = Sum_{n>=1} a(n)/n!.

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Maple

A004570 Expansion of sqrt(7) in base 3.

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A004571 Expansion of sqrt(7) in base 4.

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A004572 Expansion of sqrt(7) in base 5.

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A004573 Expansion of sqrt(7) in base 6.

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A004574 Expansion of sqrt(7) in base 7.

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