A086106 -id:A086106 - cp's OEIS frontend

A293557 Decimal expansion of real root of x^7 - x^6 - x^5 + x^2 - 1.

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

Offset: 1

Iain Fox, Oct 11 2017

This root is also the seventh smallest of the Pisot numbers.

			1.545215649732755243252550...

Iain Fox, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 7.

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A374002, A293506, A374003.

First[RealDigits[Root[#^7 - #^6 - #^5 + #^2 - 1 &, 1], 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

solve(x=1, 2, x^7 - x^6 - x^5 + x^2 - 1) \\ Michel Marcus, Oct 13 2017

{ default(realprecision, 20080); x=solve(x=1, 2, x^7 - x^6 - x^5 + x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293557.txt", n, " ", d)); }

A333833 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 2 and |p(i) - p(i-2)| <= 3.

Original entry on oeis.org

1, 1, 2, 6, 12, 14, 18, 28, 42, 56, 74, 102, 144, 200, 274, 376, 520, 720, 994, 1370, 1890, 2610, 3604, 4974, 6864, 9474, 13078, 18052, 24916, 34390, 47468, 65520, 90436, 124826, 172294, 237814, 328250, 453076, 625370, 863184, 1191434, 1644510, 2269880, 3133064

Offset: 0

Alois P. Heinz, Apr 07 2020

			a(5) = 14: 12345, 12354, 12435, 12453, 13245, 21345, 31245, 35421, 45321, 53421, 54213, 54231, 54312, 54321.
a(6) = 18: 123456, 123465, 123546, 123564, 124356, 132456, 213456, 213465, 312456, 465321, 564312, 564321, 645321, 653421, 654213, 654231, 654312, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..7141
Alois P. Heinz, Animation of a(10) = 74 permutations
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A003274, A086106, A174700, A263690, A263696, A302510, A307269, A328648, A338614.

Join[{1, 1, 2, 6, 12}, LinearRecurrence[{1, 0, 0, 1}, {14, 18, 28, 42}, 40]] (* Jean-François Alcover, Oct 26 2021 *)

G.f.: -(2*x^8+4*x^7+2*x^6+x^5+5*x^4+4*x^3+x^2+1)/(x^4+x-1).
a(n) = 2*A302510(n-2) for n >= 6.
Limit_{n-> infinity} a(n+1)/a(n) = A086106.

A374002 Decimal expansion of the positive real root of x^6 - 2*x^5 + x^4 - x^2 + x - 1.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 25 2024

Eighth smallest Pisot-Vijayaraghavan number.

			1.561752067720297294702995364060723780790847286947...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number.
Index entries for algebraic numbers.

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A293506, A374003.

First[RealDigits[Root[#^6 - 2*#^5 + #^4 - #^2 + # - 1 &, 2], 10, 100]]

A374003 Decimal expansion of the positive real root of x^8 - x^7 - x^6 + x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 25 2024

Tenth smallest Pisot-Vijayaraghavan number.

			1.5736789683935169887742514186293214678127040615079...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number.
Index entries for algebraic numbers.

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A374002, A293506.

First[RealDigits[Root[#^8 - #^7 - #^6 + #^2 - 1 &, 2], 10, 100]]

A356032 Decimal expansion of the positive real root of x^4 + x - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Aug 27 2022

The other real (negative) root is -A060007.
One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

			r = 0.724491959000515611588372282187036565786494481350011017270...

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)

solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022

polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).

Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025

A168639 Expansion of x(1 + x^2 - x^3) / ( (1-x)(1-x-x^4) ).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 15, 21, 29, 40, 56, 78, 108, 149, 206, 285, 394, 544, 751, 1037, 1432, 1977, 2729, 3767, 5200, 7178, 9908, 13676, 18877, 26056, 35965, 49642, 68520, 94577, 130543, 180186, 248707, 343285, 473829, 654016, 902724, 1246010, 1719840, 2373857, 3276582

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

R. Pallu de la Barrière, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-1).

Cf. A003269, A086106, A098578.

R:=PowerSeriesRing(Integers(), 60);
[0] cat Coefficients(R!( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) )); // G. C. Greubel, Apr 20 2025

LinearRecurrence[{2,-1,0,1,-1}, {0,1,2,4,5}, 60] (* G. C. Greubel, Jul 28 2016 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,-1,2]^n*[0;1;2;4;5])[1,1] \\ Charles R Greathouse IV, Jul 29 2016

def A168639_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) ).list()
print(A168639_list(60)) # G. C. Greubel, Apr 20 2025

Lim_{n -> oo} a(n+1)/a(n) = 1.38027756909761411567330169182..., see A086106.
a(n) = 2*a(n-1) -a(n-2) +a(n-4) -a(n-5). - R. J. Mathar, Dec 02 2009
a(n) = A098578(n) - A098578(n-3) + A098578(n-2). - R. J. Mathar, May 23 2013
a(n) = A003269(n+4) + A003269(n+2) - A003269(n+1) - 1. - G. C. Greubel, Apr 20 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.

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A293557 Decimal expansion of real root of x^7 - x^6 - x^5 + x^2 - 1.

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A333833 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 2 and |p(i) - p(i-2)| <= 3.

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A374002 Decimal expansion of the positive real root of x^6 - 2*x^5 + x^4 - x^2 + x - 1.

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A374003 Decimal expansion of the positive real root of x^8 - x^7 - x^6 + x^2 - 1.

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A356032 Decimal expansion of the positive real root of x^4 + x - 1.

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A168639 Expansion of x*(1 + x^2 - x^3) / ( (1-x)*(1-x-x^4) ).

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A368747 Self-describing bit sequences from the beta transform.

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A168639 Expansion of x(1 + x^2 - x^3) / ( (1-x)(1-x-x^4) ).