A072877 -id:A072877 - cp's OEIS frontend

A065918 Decimal expansion of log(2 + sqrt(3)).

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

Offset: 1

Frank Ellermann, Dec 08 2001

x = 2^n - 1 is prime if and only if x divides cosh(2^(n - 2)*log(2 + sqrt(3))).

			1.316957896924816708625046347307968444...

Harry J. Smith, Table of n, a(n) for n = 1..2000
Chris Caldwell, Primality Proving, Arndt's theorem.
Index entries for transcendental numbers.

Cf. A001075, A019973, A072877.

First@ RealDigits[Log[2 + Sqrt@ 3], 10, 102] (* Michael De Vlieger, May 12 2019 *)

default(realprecision, 2080); x=log(2 + sqrt(3)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065918.txt", n, " ", d)) \\ Harry J. Smith, Nov 04 2009

acosh(2) \\ Charles R Greathouse IV, Jan 07 2016

Equals arccosh(2) since arccosh(x) = log(x + sqrt(x^2 - 1)). - Stanislav Sykora, Nov 01 2013
Equals arctanh(sqrt(3)/2). - Amiram Eldar, Feb 09 2024
Equals log(4) - Sum_{k>=1} (2*k - 1)!!/(k*k!*2^(3*k + 1)). - Antonio Graciá Llorente, Feb 14 2024
Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(3*n - 1/2)*(2*n + 1)). - Antonio Graciá Llorente, Nov 13 2024

A072876 a(1) = a(2) = a(3) = a(4) = 1 and a(n) = (a(n-1)*a(n-3) + a(n-2)^3)/a(n-4) for n >= 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 11, 49, 739, 41926, 36876163, 1504578225617, 67856786028033600651, 81238311359334144709516343054051, 8472940010945536421401513734595877223414710434640386

Offset: 1

Benoit Cloitre, Jul 28 2002

A variation of a Somos-4 sequence with a(n-2)^3 in place of a(n-2)^2.

Seiichi Manyama, Table of n, a(n) for n = 1..21

Cf. A002390, A006720, A072877, A111459.

Nest[Append[#, (#[[-1]]*#[[-3]] + #[[-2]]^3)/#[[-4]] ] &, {1, 1, 1, 1}, 11] (* Michael De Vlieger, May 12 2019 *)
RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==(a[n-1]a[n-3]+a[n-2]^3)/ a[n-4]},a,{n,20}] (* Harvey P. Dale, May 15 2019 *)

Lim_{n->infinity} (log(log a(n)))/n = log((1+sqrt(5))/2) or about 0.48. See A002390. However, convergence is extremely slow. - Andrew Hone, Nov 15 2005
From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
  a(n) = ceiling(e^(c0*phi^n + d0/phi^n)) if n is even,
         ceiling(e^(c1*phi^n + d1/phi^n)) if n is odd,
where
  phi = (1 + sqrt(5))/2,
   c0 =  0.087172479898911051233710515749226588954735607680...
   c1 =  0.087662681482404614007222542134598226046349621976...
   d0 = -9.574280373370101810186207466479291633433387765559...
   d1 = -4.425515288739040257644546086989175506652492968654...
(End)

A111459 Generalized Somos-4 sequence with a(n-2)^2 replaced by a(n-2)^5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 35, 313, 26261407, 1001689887346, 356879751557595054813966522072161803, 3221974575788016845202611315068840860244866942009716269469

Offset: 0

Andrew Hone, Nov 15 2005

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..14
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics 28 (2002), 119-144.
D. Gale, Tracking the Automatic Ant, Springer (1998), pp. 1-5.
D. Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
D. Gale, Somos sequence update, Mathematical Intelligencer 13 (4) (1991), 49-50.

Cf. A006720, A072876, A072877.

L[0]:=0; L[1]:=0; L[2]:=0; L[3]:=0; for n from 0 to 4000 do L[n+4]:=evalf(ln(1+exp(L[n+3]+L[n+1]-5*L[n+2]))+5*L[n+2]-L[n]): od: for n from 3990 to 4000 do print(evalf(ln(L[n+4])/(n+4))): od: #Note: this calculates L[n]=ln(a[n]) and illustrates slow convergence of ln(ln(a[n]))/n to 0.783...

a(n) = (a(n-1)*a(n-3) + a(n-2)^5)/a(n-4) for n >= 4 with a(0) = a(1) = a(2) = a(3) = 1. As n tends to infinity, log(log(a(n)))/n tends to (1/2)*log((5 + sqrt(21))/2) or about 0.783.

A217787 a(n) = (a(n-1)*a(n-3) + 1) / a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 14, 19, 43, 67, 91, 206, 321, 436, 987, 1538, 2089, 4729, 7369, 10009, 22658, 35307, 47956, 108561, 169166, 229771, 520147, 810523, 1100899, 2492174, 3883449, 5274724, 11940723, 18606722, 25272721, 57211441, 89150161, 121088881

Offset: 0

Michael Somos, Mar 25 2013

This sequence is similar to A005246 whose recursion is a(n) = (a(n-1)*a(n-2) + 1) / a(n-3). - Michael Somos, Feb 10 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 9*x^7 + 14*x^8 + 19*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..4411
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016. See (better) also. See Example 6.2.18.
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-1).

Cf. A003501.

Cf. A005246, A048736, A006720, A072876, A072877, A111459.

[n le 3 select 1 else (Self(n)*Self(n-2)+1)/Self(n-3): n in [0..40]]; // Bruno Berselli, Mar 25 2013

a[ n_] := With[{m = If [n < 0, 3 - n, n]}, SeriesCoefficient[ (1 + x + x^2 - 4 x^3 - 3 x^4 - 2 x^5) / (1 - 5 x^3 + x^6), {x, 0, m}]]; (* Michael Somos, Jan 18 2015 *)
LinearRecurrence[{0,0,5,0,0,-1},{1,1,1,1,2,3},40] (* Harvey P. Dale, Nov 20 2016 *)

{a(n) = if( n<0, n = 3-n); polcoeff( (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6) + x * O(x^n), n)};

G.f.: (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6).
a(n) = a(3-n) for all n in Z.
a(n+3) + a(n-3) = 5*a(n) for all n in Z.
a(n+1) + a(n-1) = a(n) * (2 + [n mod 3 == 0]) for all n in Z.
a(n+3k)+a(n-3k) = A003501(k)*a(n) for n>=3k. - Bruno Berselli, Mar 25 2013

cp's OEIS Frontend

A065918 Decimal expansion of log(2 + sqrt(3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A072876 a(1) = a(2) = a(3) = a(4) = 1 and a(n) = (a(n-1)*a(n-3) + a(n-2)^3)/a(n-4) for n >= 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A111459 Generalized Somos-4 sequence with a(n-2)^2 replaced by a(n-2)^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A217787 a(n) = (a(n-1)*a(n-3) + 1) / a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula