A222134 -id:A222134 - cp's OEIS frontend

A367453 Decimal expansion of (-1 + sqrt(21))/10 = 1/A222134.

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

Offset: 0

Wolfdieter Lang, Nov 20 2023

Positive root of the minimal polynomial x^2 + 1/5 - 1/5. The negative root is -(1/5)*A222134 = -0.558257569...
c^n = A(-n) + B(-n)*phi21, and A(n) = S21(n+1) - S21(n) = A365824(n), with phi21 = A222134, and B(n) = S21(n) = A015440(n-1), where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
The formula for negative indices of S is S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.

			c = 0.3582575694955840006588047193728008488984456...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A010477, A049310, A222134, A222135, A365824.

First[RealDigits[(Sqrt[21]-1)/10,10,100]] (* Paolo Xausa, Nov 21 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^(n-1)*(quadgen(84)-1))) \\ Michal Paulovic, Nov 20 2023

c = 1/phi21 = (1/5)*(1 - phi21), with phi21 = (1 + sqrt(21))/2 = A222134, hence an algebraic number of the real quadratic number field Q(sqrt(21)) but not an algebraic integer like phi24.
Equals (A010477-1)/10. - R. J. Mathar, Nov 21 2023
Equals 2*A222135/10. - Hugo Pfoertner, Mar 21 2024

A015440 a(n) = a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 6, 11, 41, 96, 301, 781, 2286, 6191, 17621, 48576, 136681, 379561, 1062966, 2960771, 8275601, 23079456, 64457461, 179854741, 502142046, 1401415751, 3912125981, 10919204736, 30479834641, 85075858321, 237475031526, 662854323131, 1850229480761, 5164501096416

Offset: 0

Olivier Gérard

Original name: Generalized Fibonacci numbers.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 6*a(n-2) equals the number of 6-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 3, 6, 6, 1, 6, 21, 12, 18, 3, 40, 6, 56, 21, 6, 24, 16, 18, 360, 6, .... - R. J. Mathar, Aug 10 2012
From Wolfdieter Lang, Jan 02 2024: (Start)
This sequence {a(n-1)}, with a(-1) = 0, appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with A(n) = A365824(n), as phi21^n = A(n) + a(n-1)*phi21(n), for n >= 0.
Limit_{n->oo} a(n)/a(n-1) = phi21. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), section 14.8, pp. 317-318
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1, 1994.
Index entries for linear recurrences with constant coefficients, signature (1,5).

Cf. A006130, A006131, A015441, A049310, A222134, A365824.

[n le 2 select 1 else Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012

A015440 := proc(n)
    if n <= 1 then
        1;
    else
        procname(n-1)+5*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, May 15 2016

a[n_]:=(MatrixPower[{{1,3},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1, 5}, {1, 1}, 100] (* Vincenzo Librandi, Nov 06 2012 *)

a(n)=abs([1,3;1,-2]^n*[1;1])[2,1] \\ Charles R Greathouse IV, Feb 03 2014

[lucas_number1(n,1,-5) for n in range(1, 28)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) + 5*a(n-2).
a(n) = (( (1+sqrt(21))/2 )^(n+1) - ( (1-sqrt(21))/2 )^(n+1))/sqrt(21).
a(n) = Sum_{k=0..ceiling(n/2)} 5^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
G.f.: 1/(1 - x - 5x^2). - R. J. Mathar, Sep 03 2008
a(n) = Sum_{k=0..n} A109466(n,k)*(-5)^(n-k). - Philippe Deléham, Oct 26 2008
From Jeffrey R. Goodwin, May 28 2011: (Start)
A special case of a more general class of Lucas sequences given by
U(n) = U(n-1) + (4^(m-1)-1)/3 U(n-2).
U(n) = (( (1+sqrt((4^(m)-1)/3))/2 )^(n+1) - ( (1-sqrt((4^(m)-1)/3))/2 )^(n+1))/sqrt((4^(m)-1)/3). Fix m = 2 to get the formula for the Fibonacci sequence, fix m = 3 to get the formula for a(n). (End)
G.f.: G(0)/(2-x), where G(k)= 1 + 1/(1 - x*(21*k-1)/(x*(21*k+20) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: Q(0)/x -1/x, where Q(k) = 1 + 5*x^2 + (k+2)*x - x*(k+1 + 5*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013
a(n) = (Sum_{k=1..n+1, k odd} binomial(n+1,k)*21^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014
With an initial 0 prepended, the sequence [0, 1, 1, 6, 11, 41, 96, ...] satisfies the congruences a(n*p^k) == (3|p)*(7|p)*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where (n|p) denotes the Legendre symbol. See Young, Theorem 1, Corollary 1(i). - Peter Bala, Dec 28 2022
a(n) = sqrt(-5)^(n-1)*S(n-1,1/sqrt(-5)), for n >= 0, with the Chebyshev polynomial S(n, x) (see A049310). - Wolfdieter Lang, Nov 17 2023
From Peter Bala, Jun 27 2025: (Start)
The following products telescope:
Product_{k >= 0} (1 + 5^k/a(2*k+1)) = 1 + sqrt(21).
Product_{k >= 1} (1 - 5^k/a(2*k+1)) = 1/22 * (1 + sqrt(21)).
Product_{k >= 0} (1 + (-5)^k/a(2*k+1)) = (1/21) * (21 + sqrt(21)).
Product_{k >= 1} (1 - (-5)^k/a(2*k+1)) = (1/22) * (21 + sqrt(21)). (End)
E.g.f.: exp(x/2)*(sqrt(21)*cosh(sqrt(21)*x/2) + sinh(sqrt(21)*x/2))/sqrt(21). - Stefano Spezia, Jul 04 2025

A010477 Decimal expansion of square root of 21.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {1, 1, 2, 1, 1, 8} repeated. - Harry J. Smith, Jun 03 2009

The fundamntal algebraic (integer) number in the field Q(sqrt(21)) is (1 + sqrt(21))/2 = A222134. - Wolfdieter Lang, Nov 21 2023

			4.582575694955840006588047193728008488984456576767971902607242123906868...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Frankfurter Allgemeine Zeitung, Feb 09 2011 (in German)
Index entries for algebraic numbers, degree 2

Cf. A010125 (continued fraction), A248249 (Egyptian fraction).

Cf. A020778 (reciprocal), A222134.

RealDigits[N[Sqrt[21],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(21); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010477.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

A235162 Decimal expansion of (sqrt(33) + 1) / 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 06 2014

Solution of y^2 - y - 8 = 0.
Decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
The sequence with a(1) = 2 is decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
A basis for the integers of the real quadratic number field K(sqrt(33)) is
  <1, omega(33)>, where omega(33) = (1 + sqrt(33))/2. - Wolfdieter Lang, Feb 11 2020

			3.37228132326901432992530573410946465911013222899139618384993873528...

Christopher M. Conrey, Table of n, a(n) for n = 1..10000

Cf. A001622, A010488, A209927, A222132, A222134, A223140.

val = vpa((sqrt(sym(33))+1)/2,10001); list = char(val)-'0'; list = list([1,3:end-1]); % Christopher M. Conrey, Jan 26 2022

Mathematica
```
RealDigits[(1 + Sqrt[33])/2, 10, 130]
```

A365824 a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.

Original entry on oeis.org

1, 0, 5, 5, 30, 55, 205, 480, 1505, 3905, 11430, 30955, 88105, 242880, 683405, 1897805, 5314830, 14803855, 41378005, 115397280, 322287305, 899273705, 2510710230, 7007078755, 19560629905, 54596023680, 152399173205, 425379291605, 1187375157630, 3314271615655

Offset: 0

Wolfdieter Lang, Nov 20 2023

This sequence {a(n)} appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with b(n) = A015440(n-1), with A015440(-1) = 0, as phi21^n = a(n) + b(n)*phi21(n), for n >= 0. But the later given formulas in terms of scaled Chebyshev polynomials, called here {S21(n)}, are valid also for negative n values, i.e., for nonnegative powers of 1/phi21 = (-1 + sqrt(21))/10 = 0.35825756949... = A367453.

Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(21))/2 = A222134 = 2.791287...

			phi21^2 = a(2) + b(2)*phi(n) = 5 + phi21 = 7.79128784..., a real algebraic integer in Q(sqrt(21)).
(1/phi21)^2 = a(-2) + b(-2)*phi21 = (1/25)*(6 - phi21) = 0.12834848..., a real algebraic number in Q(sqrt(21)).

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A010477 (sqrt(21)), A015440, A049310, A222134, A367453.

LinearRecurrence[{1,5},{1,0},50] (* Paolo Xausa, Nov 21 2023 *)

a(n) = abs([1, 3; 1, -2]^(n-2)*[5; 5])[2, 1] \\ Thomas Scheuerle, Nov 20 2023

a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.
G.f.: (1 - x)/(1 - x - 5*x^2).
a(n) = S21(n+1) - S21(n), for n >= 0, where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).
The above mentioned sequence {b(n)} has terms b(n) = A015440(n-1) =  S21(n), for n >= 0, with the same recurrence as {a(n)} but with b(0) = 0 and b(1) = 1, and g.f. x/(1 - x - 5*x^2).
The formula for negative indices of S is: S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.

A222135 Decimal expansion of sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 08 2013

Sequence with a(1) = 2 is decimal expansion of sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))) - A222134.

			1.791287847477920003294023596864...

Cf. A090458, A107905, A222134.

Mathematica
```
RealDigits[(Sqrt[21] - 1)/2, 10, 130]
```

Closed form: (sqrt(21) - 1)/2 = A090458-2 = A107905-3 = A222134-1.
sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))) + 1 = sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))). See A222134.
Minimal polynomial: x^2 + x - 5. - Stefano Spezia, Jul 02 2025

A178134 Sum_{m=0..(n-1)/2} A176263(n-m-1, m).

Original entry on oeis.org

0, 1, 1, 2, -3, -2, -32, -81, -311, -810, -2515, -6864, -19944, -55043, -156023, -433522, -1217427, -3391226, -9488456, -26462205, -73933535, -206293134, -576040339, -1607642688, -4488069168, -12526662167, -34967630447

Offset: 0

Roger L. Bagula, May 20 2010

The limiting ratio is (alternating) A222134, 5 times a root of the polynomial 5x^2+x-1 in the denominator of the g.f.

Index entries for linear recurrences with constant coefficients, signature (1,7,-2,-6,-4,-25,5,25).

Cf. A000800, A004148.

A178134 := proc(n)
    add( A176263(n-m-1,m), m=0..(n-1)/2) ;
end proc: # R. J. Mathar, May 15 2016

Clear[a, f, a0, t]
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
t[n_, m_, a_] := f[m + 1, a] + f[n + 1 - m, a] - f[n + 1, a];
a = 5;
a0[n_] := Sum[t[n - m - 1, m, a], {m, 0, Floor[(n - 1)/2]}];
Table[a0[n], {n, 0, 30}]

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 25,5,-25,-4,-6,-2,7,1]^n*[0;1;1;2;-3;-2;-32;-81])[1,1] \\ Charles R Greathouse IV, May 15 2016

G.f. -x*(1-6*x^2-10*x^3-5*x^4+5*x^5) / ( (x-1)*(1+x)*(5*x^2+x-1)*(5*x^4+x^2-1) ). - R. J. Mathar, Nov 05 2012

New name from R. J. Mathar, May 15 2016

A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 03 2018

For x >= 0, sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)))) = (sqrt(4*x+1) + 1)/2. This is an integer for each x such that 2*x is a term in A000217.

			3.541381265149109844499842122601033531042485047393205593209576523243166362659...

Index entries for algebraic numbers, degree 2

Cf. A000007, A001622, A000038, A209927, A222132, A222134, A223140, A235162.

evalf((sqrt(37)+1)/2,120); # Muniru A Asiru, Oct 07 2018

RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[9, {135}]], 10, 111][[1]] (* or *)
RealDigits[(Sqrt[37] + 1)/2, 10, 111][[1]]

(sqrt(37)+1)/2 \\ Altug Alkan, Oct 03 2018

Minimal polynomial: x^2 - x - 9. - Stefano Spezia, Jul 02 2025

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A367453 Decimal expansion of (-1 + sqrt(21))/10 = 1/A222134.

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A015440 a(n) = a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.

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A010477 Decimal expansion of square root of 21.

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A235162 Decimal expansion of (sqrt(33) + 1) / 2.

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A365824 a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.

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A222135 Decimal expansion of sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))).

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A178134 Sum_{m=0..(n-1)/2} A176263(n-m-1, m).

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