A060006 -id:A060006 - cp's OEIS frontend

A086106 Decimal expansion of positive root of x^4 - x^3 - 1 = 0.

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

Offset: 1

Eric W. Weisstein, Jul 09 2003

Also the growth constant of the Fibonacci 3-numbers A003269 [Stakhov et al.]. - R. J. Mathar, Nov 05 2008

			1.380277569...
The four solutions are the present one, -A230151, and the two complex ones 0.2194474721... - 0.9144736629...*i and its complex conjugate. - _Wolfdieter Lang_, Aug 19 2022

Iain Fox, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See the beta(3) constant pp. 3-4.
A. Stakhov and B. Rozin, Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos, Solit. Fractals 27 (2006), 1162-1177.
Eric Weisstein's World of Mathematics, Pisot Number.
Index entries for algebraic numbers, degree 4

Cf. -A230151 (other real root).

Cf. A060006.

RealDigits[Root[ -1 - #1^3 + #1^4 &, 2], 10, 110][[1]]

polrootsreal( x^4-x^3-1)[2] \\ Charles R Greathouse IV, Apr 14 2014

default(realprecision, 20080); x=solve(x=1, 2, x^4 - x^3 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b086106.txt", n, " ", d));  \\ Iain Fox, Oct 23 2017

Equals (1 + (A^2 + sqrt(A^4 - 16*u*A^2 + 2*A))/A)/4 with A = sqrt(8*u + 3/2), u = (-(Bp/2)^(1/3) + (Bm/2)^(1/3)*(1 - sqrt(3)*i)/2 - 3/8)/6, with Bp = 27 + 3*sqrt(3*283), Bm = 27 - 3*sqrt(3*283), and i = sqrt(-1). (Standard computation of a quartic.) The other (negative) real root -A230151 is obtained by using in the first formula the negative square root. The other two complex roots are obtained by replacing A by -A in these two formulas. - Wolfdieter Lang, Aug 19 2022

A306734 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 8, 9, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 14, 15, 15, 15, 15, 15, 15, 16

Offset: 0

Ilya Gutkovskiy, Mar 06 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A053256, A053258, A053261, A333179, A333180.

nmax = 90; CoefficientList[Series[Sum[x^(k^2) Product[(1 + x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 10 2020 *)

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.424889520435345887204307524... = sqrt((23 + (10051/2 - (1173*sqrt(69))/2)^(1/3) + ((23/2)*(437 + 51*sqrt(69)))^(1/3))/69)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 6*s - 23*s^2 + 23*s^3 = 0. - Vaclav Kotesovec, Mar 11 2020

Limit_{n->infinity} a(n) / A333179(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885... - Vaclav Kotesovec, Mar 11 2020

A164001 Spiral of triangles around a hexagon.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426

Offset: 1

Omar E. Pol, Oct 27 2009

a(n) is the side length of the n-th triangle in a spiral around a hexagon with side length = 1.
Sequence very similar to A134816, but without repeated terms. Records in A134816. Also records in A000931, the Padovan sequence.
Column k=2 of A242464 (with offset 0). - Alois P. Heinz, May 19 2014
a(n) is the number of bitstrings of length n-1 without two consecutive 0's or three consecutive 1's. - Zachary Stier, Mar 16 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

LinearRecurrence[{0,1,1},{1,2,3,4},50] (* Harvey P. Dale, Jul 08 2017 *)

If n < 5 then a(n) = n, otherwise a(n) = a(n-2) + a(n-3).
G.f.: -x - 1 + (-x^2 - 2*x - 1)/(-1 + x^2 + x^3). a(n) = A000931(n+4) + A000931(n+5) = A000931(n+7), n > 1. - R. J. Mathar, Oct 29 2009
a(n) ~ 1.67873... * 1.32471...^(n-1) where 1.32471... is the real root of x^3 - x - 1 = 0 (see A060006), and 1.67873... is the real root of 23*x^3 - 46*x^2 + 13*x - 1 = 0. - Ricardo Bittencourt, May 14 2023

A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Iain Fox, Oct 10 2017

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

The ratio of successive terms of A122115 converges to this number.

			1.570147312196054362910665...

Iain Fox, Table of n, a(n) for n = 1..20000
Hans Walser, Funktionenfolge [in German]
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 5

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A122115, A293560 (inverse).

First@ RealDigits[Root[#^5 - #^4 - #^2 - 1 &, 1], 10, 87] (* Michael De Vlieger, Oct 23 2017 *)

solve(x=1, 2, x^5 - x^4 - x^2 - 1) \\ Michel Marcus, Oct 11 2017

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^4 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293506.txt", n, " ", d));

More terms from Andrey Zabolotskiy, Oct 12 2017

A160155 Decimal expansion of the one real root of x^5-x-1.

Original entry on oeis.org

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

Offset: 1

Harry J. Smith, May 03 2009

The other (complex) roots are 0.181232444469875383... + 1.08395410131771066...*i, and -0.764884433600584726... + 0.352471546031726249...*i, together with their complex conjugates. - Wolfdieter Lang, Dec 15 2022

This quintic is in some sense the smallest and/or simplest algebraic equation for which there is no explicit expression for the roots. (The "equivalent" quintic x^5 - x + 1 has the opposite real root, x = -1.1673..., while x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1).) - M. F. Hasler, Jul 12 2025

			1.16730397826141868425604589985484218072056037152548903914008244927565...

Harry J. Smith, Table of n, a(n) for n = 1..20000
David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
Wikipedia, Abel-Ruffini theorem, created Nov. 27, 2002, retrieved July 12, 2025.
Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394.
Index entries for algebraic numbers, degree 5

Cf. A039922 (continued fraction), A001622 (golden ratio phi = root of x^2 - x - 1), A060006 (plastic constant, root of x^3 - x - 1), A060007 (root of x^4 - x - 1).

RealDigits[Root[x^5-x-1, x, 1], 10, 105] // First (* Jean-François Alcover, Jul 09 2015 *)

localprec(20080); r=real(polroots('x^5 - 'x - 1)[1]); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b160155.txt", n, " ", d)) \\ Edited by M. F. Hasler, Jul 12 2025

polrootsreal(x^5-x-1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/5))^(1/5))^(1/5))^(1/5))^(1/5). - Ilya Gutkovskiy, Dec 15 2017

A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228, 128800, 170624, 226029, 299425, 396654, 525455

Offset: 0

N. J. A. Sloane

Limit_{n->infinity} a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006). - Gerald McGarvey, Sep 19 2004
a(n) is the number of distinct even run-types taken over nonempty subsets of [n+1]. The run-type of a set of positive integers is the sequence of lengths when the set is decomposed into maximal runs of consecutive integers and it is even if all its entries are even. For example, the set {2,3,5,6,9,10,11} has run-type (2,2,3) and a(6)=6 counts (2),(4),(6),(2,2),(2,4),(4,2). - David Callan, Jul 14 2006
Partial sums of the sequence obtained by deleting the first 2 terms of A000931. Example: 0+1+0+1+1 = 3 = a(4). - David Callan, Jul 14 2006
One less than the sequence obtained by deleting the first 7 terms of A000931. - Ira M. Gessel, May 02 2007
This sequence counts ordered partitions of (n-1) into parts less than or equal to 3, in which the order of 1's are unimportant. Alternately, the order of 2's and 3's are important (see example). - David Neil McGrath, Apr 26 2015
Interleaving of A289692 and A077855. - Bruce J. Nicholson, Apr 09 2018

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...
a(7)=8, with (n-1)=6. The partially ordered partitions of 6 are (33),(321,312,132=one),(231,213,123=one),(3111,1311,1131,1113=one),(222),(2211,1122,1221,2112,1212,2121=one),(21111,12111,11211,11121,11112=one),(111111). - _David Neil McGrath_, Apr 26 2015

Robert Israel, Table of n, a(n) for n = 0..7360
O. Bouillot, The Algebra of Multitangent Functions, arXiv:1404.0992 [math.NT], 2014.
O. Bouillot, The Algebra of Multitangent Functions, Journal of Algebra, Volume 410, 15 July 2014, Pages 148-238.
J. H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1070
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Cf. A000931, A054405, A060006, A077905.

Cf. A289692, A077855.

[0,1] cat [ n le 4 select (n) else Self(n-1)+Self(n-2)-Self(n-4): n in [1..45] ]; // Vincenzo Librandi, Apr 27 2015

f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2)-a(n-4),seq(a(i)=[0,1,1,2][i+1],i=0..3)},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, May 04 2015

a[ n_] := If[ n < 0, SeriesCoefficient[ -x^3 / (1 - x^2 - x^3 + x^4), {x, 0, -n}], SeriesCoefficient[ x / (1 - x - x^2 + x^4), {x, 0, n}]]; (* Michael Somos, Nov 29 2013 *)
LinearRecurrence[{1, 1, 0, -1}, {0, 1, 1, 2}, 50] (* Vincenzo Librandi, Apr 27 2015 *)

{a(n) = polcoeff( if( n<0, -x^3 / (1 - x^2 - x^3 + x^4), x / (1 - x - x^2 + x^4)) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Nov 29 2013 */

x='x+O('x^99); concat(0, Vec(x/((1-x)*(1-x^2-x^3)))) \\ Altug Alkan, Apr 09 2018

a(n) = A000931(n+7)-1.
a(0)=0, a(1)=1, a(2)=1 then for n>2 a(n)=ceiling(r*a(n-1)) where r is the positive root of x^3-x-1=0. - Benoit Cloitre, Jun 19 2004
G.f.: x/((1-x)*(1-x^2-x^3)). - Jon Perry, Jul 04 2004
For n>2 a(n) = floor(sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))) + 1. - Gerald McGarvey, Sep 19 2004
a(n) = Sum_{k=1..floor((n+2)/3)} binomial(floor((n+2-k)/2),k). This formula counts even run-types by length. - David Callan, Jul 14 2006
a(n) = a(n-2) + a(n-3) + 1. - Mark Dols, Feb 01 2010
a(n) + a(n+1) = A054405(n). Partial sums is A054405. - Michael Somos, Dec 01 2013
a(-3-n) = -A077905(n) for all n in Z. - Michael Somos, Sep 25 2014

A060007 Decimal expansion of the positive real root of x^4 - x - 1.

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Mar 14 2001

Original name: Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
v_2 = A001622 - 1. [Corrected by M. F. Hasler, Jul 12 2025]
v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - M. F. Hasler, Jul 12 2025
A Perron number of the 4th degree polynomial (see Boys and Wu). - R. J. Mathar, Mar 19 2011
This number is not ruler-and-compass constructible because x^4-x-1 and its resolvent x^3+4x+1 are irreducible over the rationals. - Jean-François Alcover, Aug 31 2015
The other (negative) real root -0.724491959... is -A356032. The first of the pair of complex conjugate roots is obtained by negating in the formula for v_4 below sqrt(2*u) and sqrt(u), giving -0.2481260628... - 1.0339820609...*i. - Wolfdieter Lang, Aug 27 2022
The sequence a(n) = v_4^((n^2-n)/2) satisfies the Somos-4 recursion a(n+2)*a(n-2) = a(n+1)*a(n-1) + a(n)^2 for all n in Z. - Michael Somos, Mar 24 2023

			v_4 = 1.220744084605759475361685349...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
F. Rothelius, Formulae
Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394
Index entries for algebraic numbers, degree 4

Cf. A001622 (golden ratio, root of x^2 - x - 1), A060006 (plastic number, root of x^3 - x - 1), A202540 (log thereof), A160155 (root of x^5 - x - 1), A356032 (root of x^4 + x - 1), A006720, A298813.

r:=(108+12*sqrt(849))^(1/3): (sqrt(12/sqrt(-8/r+r/6)+48/r-r) + sqrt(-48/r+r))/(2*sqrt(6)): evalf(%,105); # Vaclav Kotesovec, Oct 12 2013

RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 11 2012 *)
Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 04 2013 *)

default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* Michael Somos, Mar 22 2023 */

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - Ilya Gutkovskiy, Dec 15 2017
v_4 = (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)), ep = (-1 + sqrt(3)*i)/2 and i = sqrt(-1).
For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - Wolfdieter Lang, Aug 27 2022
Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - Antonio Graciá Llorente, Dec 13 2024
Equals exp(A202540) = sqrt(A298813). - Hugo Pfoertner, Dec 14 2024

More terms from Benoit Cloitre, Jan 11 2003

Simplified definition from M. F. Hasler, Jul 12 2025

A333179 G.f.: Sum_{k>=0} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 5, 5, 4, 4, 4, 4, 5, 6, 5, 6, 7, 7, 8, 8, 8, 8, 9, 8, 8, 8, 7, 8, 8, 8, 8, 9, 9, 10, 11, 11

Offset: 0

Vaclav Kotesovec, Mar 10 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A053261, A306734.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.3207396095989103757477946185... = sqrt((1 - (2/(23*(23 + 3*sqrt(69))))^(1/3) + ((1/2)*(23 + 3*sqrt(69)))^(1/3)/23^(2/3))/3)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 8*s - 23*s^2 + 23*s^3 = 0.

Limit_{n->infinity} A306734(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Aug 17 2005

The silver number (A060006) is equal to Phi*(Phi-1).
Also Phi*(Phi-1) = 1/(Phi-1). - Richard R. Forberg, Oct 08 2014
Equations to which this is a root can also be written as: x = sqrt(x + sqrt(x)); x^2 - x - sqrt(x) = 0; or this form where n = 1: x = n + 1/sqrt(x). When n = 2 then the root is 2.618033988... = A104457 = 1 + A001622 or 1 + "Golden Ratio" called phi. - Richard R. Forberg, Oct 08 2014
Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2. - Jean-François Alcover, Apr 16 2015
Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). Conjectures: the limit L(r) of c(k+1)/c(k) as k -> oo exists, L(r) is discontinuous at 5/3 (cf. A279676), and the left limit of L(r) as r->5/3 is Phi. - Clark Kimberling, Jul 11 2017
From Wolfdieter Lang, Nov 07 2022: (Start)
This equals r + 2/3 where r is the real root of y^3 - (1/3)*y - 25/27.
The other roots of x^3 - 2*x^2 + x - 1 are (2 + w1*((25 + 3*sqrt(69))/2)^(1/3) + w2*((25 - 3*sqrt(69))/2)^(1/3))/3 = 0.1225611668... + 0.7448617668...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (2 - cosh((1/3)*arccosh(25/2)) + sqrt(3)*sinh((1/3)*arccosh(25/2))*i)/3, and its complex conjugate. (End)

			1.75487766624669276004950889635852869189460661777279314398928397064...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.11, p. 340.
Martin Gardner, A Gardner's Workout, pp. 124-126, A. K. Peters MA 2001.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.
Simon Plouffe, Plouffe's Inverter .
Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Prop. 2.3 p. 744.
Yuru Zou and Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.
Index entries for algebraic numbers, degree 3.

Cf. A001622, A001685, A060006, A075778, A104457, A358181.

FindRoot[x^3 - 2x^2 + x - 1 == 0, {x, 1.75}, WorkingPrecision -> 128][[1, 2]] (* Robert G. Wilson v, Aug 19 2005 *)
Root[x^3-2x^2+x-1, x, 1] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)

d=104;default(realprecision,d);print(k=solve(x=1,2,(x-1)^2-1/x)); for(c=0,d,z=floor(k);print1(z,",",);k=10*(k-z))

polrootsreal(x^3-2*x^2+x-1)[1] \\ Charles R Greathouse IV, Aug 15 2014

Equals 1+A075778. - R. J. Mathar, Aug 20 2008
Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2. - Vaclav Kotesovec, Oct 08 2014
Equals Rho^2 where Rho is the plastic number 1.3247179572...(see A060006). - Philippe Deléham, Sep 29 2020
From Wolfdieter Lang, Nov 07 2022: (Start)
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 + 3*sqrt(69))/2)^(-1/3))/3.
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 - 3*sqrt(69))/2)^(1/3))/3.
Equals 2*(1 + cosh((1/3)*arccosh(25/2)))/3. (End)
Equals - Sum_{k>=1} Gamma(k - k/5 - 1)*Gamma(k/5 + 1)*sin(3*k*Pi/5)/(k*Pi*Gamma(k)). - Antonio Graciá Llorente, Dec 14 2024

Extended by Klaus Brockhaus and Robert G. Wilson v, Aug 19 2005

A228777 Decimal expansion of the third smallest Pisot-Vijayaraghavan number.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 04 2013

			1.443268791270373107628127607386...

Iain Fox, Table of n, a(n) for n = 1..20000 (first 1000 terms from Vincenzo Librandi).
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot number
Index entries for algebraic numbers, degree 5

Cf. A060006, A086106, A092526.

Maple
```
fsolve(x^5-x^4-x^3+x^2-1,x,1.4..1.5) ;
```

Root[Function[x, x^5-x^4-x^3+x^2-1], 1] // RealDigits[#, 10, 90]& // First (* Jean-François Alcover, Feb 20 2014 *)

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^4 - x^3 + x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b228777.txt", n, " ", d));  \\ Iain Fox, Oct 23 2017

A root of x^5-x^4-x^3+x^2-1.

cp's OEIS Frontend

A086106 Decimal expansion of positive root of x^4 - x^3 - 1 = 0.

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A306734 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j).

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A164001 Spiral of triangles around a hexagon.

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A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

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A160155 Decimal expansion of the one real root of x^5-x-1.

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A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

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A060007 Decimal expansion of the positive real root of x^4 - x - 1.

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A333179 G.f.: Sum_{k>=0} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)).