A060006 -id:A060006 - cp's OEIS frontend

A368747 Self-describing bit sequences from the beta transform.

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 26, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 98, 100, 101, 104, 105, 106, 108, 109, 112, 113, 114, 115, 116, 117, 118, 120

Offset: 0

Linas Vepstas, Feb 06 2024

nonn

A bit sequence b_0, b_1, ..., b_k of the binary representation of an odd integer 2n+1 is self-describing if the largest real root beta of the monic polynomial p_n(x) = x^(k+1) - b_0 * x^k - b_1 * x^(k-1) - ... - b_k regenerates the same bit sequence when the beta transform t(x) = (beta * x) mod 1 is iterated for x=1, the generated bit being zero or one, depending on whether the modulo was taken or not. Not all integers n generate such self-describing polynomials; the sequence given here begins the list of valid self-describing polynomials.
The number of such valid polynomials of degree m is given by Moreau's necklace counting function A001037.
The bit sequences are not Lyndon words, and cannot be rotated, although there are the same number of them (given by the necklace function).
The bit sequences are not isomorphic to the irreducible polynomials over the field F_2 of two elements, although there are the same number of them (given by the necklace function).

			n=1 generates p_1(x) = x^2 - x - 1 whose largest real root is the golden mean A000045. Iteration of the golden mean under the beta transform terminates after two steps, and requires modulo-one to be applied at each step, thus giving the bit sequence 11.
n=2 generates a polynomial whose largest root is the limit of Narayana's A058265.
n=3 ... is the tribonacci limit A058265.
n=4 ... is the 2nd Pisot number A086106.
n=5 is not valid (not self-describing).
n=6 ... is A109134.
n=7 ... is the tetranacci limit A086088.
n=8 ... is the silver (plastic) number A060006.
n=9 is not valid (not self-describing).
n=10 ... is a Pisot number A293506.
n=11 is not valid (not self-describing).
Sequences corresponding to larger values of n are not (currently) in the OEIS, except when n = 2^m - 1, which are limits to the generalized Fibonacci numbers.

Linas Vepstas, Table of n, a(n) for n = 0..10000
Linas Vepstas, On the Beta Transform, arXiv:1812.10593 [math.DS], 2018.

The binary representation for every integer 2n+1 encodes a polynomial p_n(x) but not all such polynomials have (positive, real) roots r_n that are self-describing. An integer n is valid if it is self-describing; the validity filter is theta_n(r_n) = 1 where theta_n(x) is recursively defined as theta_n(x) = theta_{n/2}(x) * (x < r_{n/2}) if n is even, and theta_n(x) = theta_{(n-1)/2}(x) if n is odd. The sequence starts with theta_0(x) = 1.

A370202 a(n) = a(n-3) + a(n-2) + gcd(a(n-2), a(n-1)) with a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 7, 17, 15, 25, 37, 41, 63, 79, 105, 143, 185, 249, 329, 435, 579, 767, 1015, 1347, 1783, 2363, 3131, 4147, 5495, 7279, 9643, 12775, 16923, 22419, 29701, 39343, 52121, 69045, 91465, 121171, 160511, 212637, 281683, 373149, 494321, 654833, 867471

Offset: 1

Eli Jaffe, Feb 11 2024

The ratio between consecutive terms (a(n)/a(n-1)) appears to approach the plastic constant A060006.

Eli Jaffe, Table of n, a(n) for n = 1..1000

Cf. A060006, A083658.

from math import gcd
def terms(n):
  nums = [1,1,1]
  for i in range(n-3):
    new_num = nums[i] + nums[i+1] + gcd(nums[i+1], nums[i+2])
    nums.append(new_num)
  return nums

A097600 A Binet like formula using the Akiyama-Thurston tile roots for a Minimal Pisot theta0 sequence.

Original entry on oeis.org

1, 0, 1, 2, 2, 3, 4, 5, 7, 10, 13, 18, 23, 31, 41, 55, 73, 97, 129, 170, 226, 299, 397, 526, 696, 923, 1223, 1620, 2146, 2843, 3766, 4989, 6610, 8756, 11599, 15366, 20356, 26966, 35723, 47323, 62689, 83046, 110013, 145736, 193059, 255749, 338796

Offset: 1

Roger L. Bagula, Sep 20 2004

Let r1 = -0.662358978622373051..-0.562279512062301289..*i, r2 = complex-conjugate(r1), and r3 = 1.3247179572.. = A060006 be the three roots of the polynomial x^3-x-1. i is the imaginary unit. Then f(n) = (r3^n-r2^n-r2^(5*n))/(r3-r2-r2^5) is a sequence of numbers, approximately f(1) = 1, f(2) = 0.756+0.786*i, f(3) = 1.263+0.017*i, f(4) = 2.1929+0.704*i, f(5) = 2.205+0.6866*i etc. a(n) is floor(Re(f(n)).

Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

Cf. A001644.

NSolve[x^3-x-1==0, x] r1=-0.662358978622373051`-0.562279512062301289` I r2=-0.662358978622373051`+0.562279512062301289` I r3=1.32471795724474605` (* Binet like formula for the Minimal Pisot*) f[n_]=(r3^n-((r2^n)+(r2^(5*n))))/(r3-r2-r2^5) a=Table[Floor[Re[f[n]]], {n, 1, 50}]

A191909 Decimal expansion of the limit of the square root of the ratio of consecutive Padovan numbers.

Original entry on oeis.org

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

Offset: 0

Fabrice Auzanneau, Jun 19 2011

This is the square root of the inverse of the plastic number A060006: 1.32471795724...
This is the positive root of x^6 + x^4 - 1 = 0 and the square root of A075778.
An algebraic integer of degree 6 and minimal polynomial x^6 + x^4 - 1. - Charles R Greathouse IV, Apr 21 2016

			0.868836961832709301806569964191...

Wikipedia, Plastic number
Index entries for algebraic numbers, degree 6

Cf. A000931, A060006.

RealDigits[x/.FindRoot[x^6+x^4==1,{x,.8},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 17 2014 *)

polrootsreal(x^6+x^4-1)[2] \\ Charles R Greathouse IV, Apr 21 2016

A193599 Indices n such that Padovan(n) > R^n/(2*R+3) where R is the only real root of the polynomial x^3-x-1.

Original entry on oeis.org

0, 3, 5, 6, 8, 10, 11, 13, 16, 18, 21, 23, 24, 26, 28, 29, 31, 34, 36, 39, 41, 42, 44, 46, 47, 49, 52, 54, 55, 57, 59, 60, 62, 65, 67, 70, 72, 73, 75, 77, 78, 80, 83, 85, 88, 90, 91, 93, 95, 96, 98, 101, 103, 106, 108, 109, 111, 114, 116, 119, 121, 122, 124

Offset: 0

Francesco Daddi, Jul 31 2011

nonn

R is plastic number (A060006).

			For n=24, Padovan(24) = 151 > 150.99309... = R^24/(2*R+3).

Cf. A000931, A060006.

lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, 0, lim}]; p = CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim+1], p[[#]] > powers[[#]] &] - 1 (* T. D. Noe, Aug 01 2011 *)

A193600 Indices n such that Padovan(n) < r^n/(2*r+3) where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 14, 15, 17, 19, 20, 22, 25, 27, 30, 32, 33, 35, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 58, 61, 63, 64, 66, 68, 69, 71, 74, 76, 79, 81, 82, 84, 86, 87, 89, 92, 94, 97, 99, 100, 102, 104, 105, 107, 110, 112, 113, 115, 117, 118, 120, 123

Offset: 1

Francesco Daddi, Jul 31 2011

nonn

R is the so-called plastic number (A060006). Padovan(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r (real), s, t are the three roots of x^3-x-1. Also Padovan(n) is asymptotic to r^n / (2*r+3).

			For n=25, Padovan(25) = A000931(25) = 200 < 200.023... = r^25/(2*r+3).

Cf. A000931, A060006.

lim=200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, lim}]; p = Rest[CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]]; Select[Range[lim], p[[#]] <= powers[[#]] &] (* T. D. Noe, Aug 01 2011 *)

A193627 Indices n such that Perrin(n) < r^n where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

1, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 24, 25, 27, 29, 30, 32, 35, 37, 38, 40, 42, 43, 45, 48, 50, 53, 55, 56, 58, 60, 61, 63, 66, 68, 71, 73, 74, 76, 78, 79, 81, 84, 86, 89, 91, 92, 94, 97, 99, 102, 104, 105, 107, 109, 110, 112, 115, 117, 120, 122, 123, 125

Offset: 1

Francesco Daddi, Aug 01 2011

nonn

r is the so-called plastic number (A060006).
Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.
Also Perrin(n) is asymptotic to r^n.
To calculate r^n (for n>2) we can observe that: r^n=s(n)*r^2+t(n)*r+u(n) where s(3)=0, t(3)=1, u(3)=1; s(n+1)=t(n), t(n+1)=s(n)+u(n), u(n+1)=s(n). - Francesco Daddi, Aug 02 2011

			For n=27 Perrin(27) = A001608(27) = 1983 < 1983.044... = r^27

Cf. A001608, A060006, A051016.

lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n], {n, 0, lim}]; p = CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim + 1], p[[#]] <= powers[[#]] &] - 1 (* T. D. Noe, Aug 02 2011 *)

A193640 Indices n such that Perrin(n) > r^n where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

0, 2, 3, 5, 8, 10, 13, 15, 16, 18, 20, 21, 23, 26, 28, 31, 33, 34, 36, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 62, 64, 65, 67, 69, 70, 72, 75, 77, 80, 82, 83, 85, 87, 88, 90, 93, 95, 96, 98, 100, 101, 103, 106, 108, 111, 113, 114, 116, 118, 119, 121, 124

Offset: 0

Francesco Daddi, Aug 02 2011

nonn

r is the so-called plastic number (A060006).
Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.
Also Perrin(n) is asymptotic to r^n.
To calculate r^n (for n>2) we can observe that: r^n=s(n)*r^2+t(n)*r+u(n) where s(3)=0, t(3)=1, u(3)=1; s(n+1)=t(n), t(n+1)=s(n)+u(n), u(n+1)=s(n).

			For n=20 Perrin(20) = A001608(20) = 277 > 276.992... = r^20

Cf. A001608, A060006, A051017.

lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n], {n, 0, lim}]; p = CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim + 1], p[[#]] > powers[[#]] &] - 1 (* T. D. Noe, Aug 02 2011 *)

A245324 Decimal expansion of c_1, a constant associated with the computation of the maximal modulus of an algebraic integer.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 18 2014

			0.421799361484442769768076146101817449688034838616099694013595501477...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.30 Pisot-Vijayaraghavan-Salem Constants, p. 194.

David W. Boyd, The Maximal Modulus of an Algebraic Integer. Mathematics of Computation, Vol. 45, No. 171, Jul., 1985, pp. 243-249.
Eric Weisstein's MathWorld, Pisot Number
Eric Weisstein's MathWorld, Plastic Constant
Index entries for transcendental numbers

Cf. A060006 (theta0).

theta0 = Root[x^3 - x - 1, x, 1]; RealDigits[(3/2)*Log[theta0], 10, 108] // First

c_1 = (3/2)*log(theta0), where theta0 is the smallest Pisot number, which is the real root of x^3 - x - 1.

A248048 Numerator of u(n) where u(n) = (u(n-1) + u(n-2)) * (u(n-2) + u(n-3)) / u(n-4) with u(0) = -1, u(1) = u(2) = u(3) = 1.

Original entry on oeis.org

-1, 1, 1, 1, -4, -6, 30, -240, 1260, 35700, 1256640, -199020360, -202839655480, 1124753048683264, -181696576073176468304, 1026657060420588391021488976, 919637161132261232937508950440493056, 839430326120023909391651548323223480699275649536

Offset: 0

Michael Somos, Sep 30 2014

u(144) has denominator 2.
The sequence terms with u(0)=x and the same recursion is a Laurent polynomial in x. This sequence is what you get when you replace x with -1. Note that u(n) = 0 if -7 <= n <= -1. Usually u(n) for n <= -5 would not be defined using the backward recursion due to division by 0. - Michael Somos, Mar 04 2020
Again, the u(n) sequence with u(0)=x is given by u(n) = v(n)/(2^b(n-13) * x^b(n-3)) where b is A023434 and v(n) is a polynomial in x. For example, u(6) = (2*x^2 + 20*x + 48)/x^2. Also, A248049(n) has the same recursion as u(n) and initial values except A248049(0)=2. Note, log(abs(u(n))) ~ r^n where r = A060006 the plastic constant. - Michael Somos, Mar 04 2020

Cf. A023434, A060006, A248049.

{a(n) = if( n<-4, n=-8-n); if( n<0, 0, n<4, (-1)^(n==0), (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4))};

a(n) = a(-8-n) for all n in Z.
u(n) * u(n+4) = (u(n+1) + u(n+2)) * (u(n+2) + u(n+3)) for all n in Z.
a(7*n) < 0, a(7*n + 4) < 0, a(7*n + 5) < 0, a(7*n + 1) > 0, a(7*n + 2) > 0, a(7*n + 3) > 0, a(7*n + 6) > 0 for all n >= 0.

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A368747 Self-describing bit sequences from the beta transform.

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A370202 a(n) = a(n-3) + a(n-2) + gcd(a(n-2), a(n-1)) with a(1) = a(2) = a(3) = 1.

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A097600 A Binet like formula using the Akiyama-Thurston tile roots for a Minimal Pisot theta0 sequence.

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A191909 Decimal expansion of the limit of the square root of the ratio of consecutive Padovan numbers.

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A193599 Indices n such that Padovan(n) > R^n/(2*R+3) where R is the only real root of the polynomial x^3-x-1.

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A193600 Indices n such that Padovan(n) < r^n/(2*r+3) where r is the real root of the polynomial x^3-x-1.

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A193627 Indices n such that Perrin(n) < r^n where r is the real root of the polynomial x^3-x-1.

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A193640 Indices n such that Perrin(n) > r^n where r is the real root of the polynomial x^3-x-1.

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A245324 Decimal expansion of c_1, a constant associated with the computation of the maximal modulus of an algebraic integer.

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