A086088 -id:A086088 - cp's OEIS frontend

A323120 Number of compositions of n where the difference between largest and smallest parts equals three.

0, 0, 2, 3, 12, 23, 58, 123, 266, 557, 1168, 2400, 4911, 9982, 20196, 40643, 81499, 162899, 324678, 645404, 1280042, 2533565, 5005449, 9872464, 19442206, 38234945, 75096856, 147324049, 288705896, 565201569, 1105475313, 2160338661, 4218403428, 8230986025

Offset: 3

Alois P. Heinz, Jan 05 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..3509

Column k=3 of A214258.

a(n) ~ c * d^n, where d = A086088 = 1.92756197548292530426190586173662216869855... is the real root of the equation d^4 - d^3 - d^2 - d - 1 = 0 and c = 0.566342887702651534849280952809003081138499070062015666102331676335... is the real root of the equation 563*c^4 - 563*c^3 + 174*c^2 - 22*c + 1 = 0. - Vaclav Kotesovec, Jan 07 2019

A374752 Decimal expansion of phi_4, a limit point of the set of Pisot numbers in (1,2).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2024

			1.933184981899520446791424030335631586375183784479...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche, Christiane Frougny, and Kevin G. Hare, On Univoque Pisot Numbers, Mathematics of Computation, Vol. 76, No. 259, July 2007, pp. 1639-1660 (arXiv version).
Kevin G. Hare and Nikita Sidorov, Conjugates of Pisot numbers, International Journal of Number Theory, Vol. 17, No. 06 (2021), pp. 1307-1321 (arXiv version).
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number.
Index entries for algebraic numbers, degree 5.

Cf. A001622, A058265, A086088, A109134, A275828, A374751.

First[RealDigits[Root[#^5 - 2*#^4 + # - 1 &, 1], 10, 100]]

Equals the real root of x^5 - 2*x^4 + x - 1.

Using the notation of Hare and Sidorov (2021, see Theorem 3.1), phi_1 = psi_1 (A001622) < phi_2 (A109134) < psi_2 (A058265) < phi_3 (A275828) < chi (A374751) < psi_3 (A086088) < phi_4 (this constant) < ... < psi_r < phi_(r+1) < ... < 2.

A202805 a(n) is the largest k in an n_nacci(k) sequence (Fibonacci(k) for n=2, tribonacci(k) for n=3, etc.) such that n_nacci(k) >= 2^(k-n-1).

Original entry on oeis.org

6, 12, 25, 48, 94, 184, 363, 719, 1430, 2851, 5691, 11371, 22728, 45443, 90870, 181724, 363429, 726839, 1453658, 2907295, 5814566, 11629107

Offset: 2

Frank M Jackson, Dec 24 2011

From Frank M Jackson, Jul 02 2023: (Start)
Define the n_nacci sequence, basically row n in A092921, with an offset of 0, n_nacci(k) = 0 for 0 <= k <= n-2 and n_nacci(n-1) = 1. Thereafter, n_nacci(k) for k >= n continues as the sum of its previous n terms.
This means that n_nacci(k) = 2^(k-n) for n <= k <= 2n-1. In the limit as n tends to infinity the n_nacci sequence after an initial large set of zeros followed by 1 has successive terms of ascending powers of 2.
As the n-acci constants, (A001622, A058265, A086088, A103814,...) are smaller than 2, for each n_nacci sequence there is a largest k such that n_nacci(k) >= 2^(k-n-1). (End)

			For n=3, the tribonacci sequence is 0,0,1,1,2,4,7,...,149,274,504,... and the 13th term is 504 < 512 so a(n)=12 because 274 is greatest term >= 2^(12-3-1) = 256.

Cf. A000045, A000073, A000078.

nAcci := proc(n,k)
    option remember ;
    if k <= n-2 then
        0;
    elif k = n-1 then
        1;
    else
        add( procname(n,i),i=k-n..k-1) ;
    end if;
end proc:
A202805 := proc(n)
    local k ;
    for k from n do
        if nAcci(n,k) < 2^(k-n-1) then
            return k-1;
        end if;
    end do:
end proc:
for n from 2 do
    print(n,A202805(n)) ;
end do: # R. J. Mathar, Mar 11 2024

fib[n_, m_] := (Block[{nacci}, (Do[nacci[g]=0, {g, 0, m - 2}];
nacci[m-1]=1;nacci[p_] := (nacci[p]=Sum[nacci[h], {h, p-m, p-1}]);nacci[n])]);
crossover[q_] := (Block[{$RecursionLimit=Infinity}, (k=0;While[fib[k+q+1, q]>=2^k, k++];k+q)]);
Table[crossover[j], {j, 2, 12}]

def nacci(n): # generator of n_nacci terms
    window = [0]*(n-1) + [1]
    yield from window
    while True:
        an = sum(window)
        yield an
        window = window[1:] + [an]
def a(n):
    pow2 = 1
    for k, t in enumerate(nacci(n)):
        if k > n + 1: pow2 <<= 1
        if 0 < t < pow2: return k-1
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Jan 29 2025

Edited by N. J. A. Sloane, May 20 2023
There seems to be an error in the Comment. See "History" tab. - N. J. A. Sloane, Jun 24 2023
Removed musing about what might define "complete" sequences. - R. J. Mathar, Mar 11 2024
a(17)-a(23) from Michael S. Branicky, Jan 29 2025

A368747 Self-describing bit sequences from the beta transform.

Original entry on oeis.org

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.

A382626 Decimal expansion of the smallest (in absolute value) root of 1-x-x^2-x^3-x^4.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Apr 01 2025

The base of the asymptotic growth derived from Binet's formula for 4-nacci sequences.

The 4 roots are -1.2906..., -0.11407..+-1.2167..*i, and this.

			0.51879006367588422190745389443...

Cf. A086088, A001622 (Fibonacci), A192918 (3-nacci), A382627 (5-nacci).

Equals 1/A086088.

A125898 Floor((quadronacci ratio)^n).

Original entry on oeis.org

1, 3, 7, 13, 26, 51, 98, 190, 367, 708, 1364, 2630, 5071, 9775, 18841, 36318, 70007, 134942, 260110, 501380, 966441, 1862874, 3590806, 6921503, 13341626, 25716810, 49570746, 95550687, 184179871, 355018115, 684319420, 1319068095, 2542585503

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

			Quadronacci ratio is root 1.92756... (A086088) of polynomial x^4-x^3-x^2-x-1.

Cf. A125890-A125899.

With[{c=x/.FindRoot[x^4-x^3-x^2-x-1==0,{x,1.9}, WorkingPrecision->100]}, Floor[c^Range[40]]] (* Harvey P. Dale, Mar 05 2012 *)

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

Original entry on oeis.org

3, 3, 4, 5, 7, 7, 10, 13, 14, 14, 19, 23, 23, 31, 34, 34, 46, 50, 60, 65, 73, 79, 88, 92, 107, 113, 126, 139, 149, 168, 182, 198, 210, 227, 244, 265, 276, 292, 317, 340, 369, 384, 408, 436, 444, 480, 516, 540, 565, 606, 628, 669, 704, 735, 759, 810, 829, 895, 925

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Mar 23 2014

nonn

The n-bonacci constant is a unique root x_1>1 of the equation x^n-x^(n-1)-...-x-1=0. So, for n=2 we have Fibonacci constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265); for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.

			Let n=2, then c_2 = phi (Fibonacci constant). We have round(c_2^2)=3 is not == 1 (mod 4), round(c_2^3)=4 is not == 1 (mod 6), while round(c_2^5)=11 == 1 (mod 10) and one can prove that for p>=5, we have round(c_2^p) == 1 (mod 2*p). Since 5=prime(3), then a(2)=3.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Cf. A001622, A007619, A007663, A058265, A086088, A103814, A118427, A118428, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565, A239566.

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A323120 Number of compositions of n where the difference between largest and smallest parts equals three.

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A374752 Decimal expansion of phi_4, a limit point of the set of Pisot numbers in (1,2).

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A202805 a(n) is the largest k in an n_nacci(k) sequence (Fibonacci(k) for n=2, tribonacci(k) for n=3, etc.) such that n_nacci(k) >= 2^(k-n-1).

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A368747 Self-describing bit sequences from the beta transform.

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A382626 Decimal expansion of the smallest (in absolute value) root of 1-x-x^2-x^3-x^4.

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A125898 Floor((quadronacci ratio)^n).

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Mathematica

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

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