A230151 -id:A230151 - 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

A014097 a(n) = a(n-1)+a(n-4).

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 19, 26, 34, 47, 66, 92, 126, 173, 239, 331, 457, 630, 869, 1200, 1657, 2287, 3156, 4356, 6013, 8300, 11456, 15812, 21825, 30125, 41581, 57393, 79218, 109343, 150924, 208317, 287535

Offset: 1

David Broadhurst

Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 4 sites wide.

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.

Indranil Ghosh, Table of n, a(n) for n = 1..7130
D. J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, arXiv:hep-th/9612012, 1996.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A020999, A000204, A001609, A000079, A003269, A003520, A005708, A005709, A005710.

LinearRecurrence[{1,0,0,1},{1,1,1,5},40] (* Harvey P. Dale, Mar 06 2016 *)

a(n):=sum(binomial(n-3*j,n-4*j)*n/(n-3*j),j,0,(n-1)/3); /* Vladimir Kruchinin, Mar 25 2016 */

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^(n-1)*[1;1;1;5])[1,1] \\ Charles R Greathouse IV, Sep 09 2016

G.f.: -x*(1+4*x^3)/(-1+x+x^4). a(n)= 4*A003269(n)-3*A003269(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - Vladimir Kruchinin, Mar 25 2016
From Greg Dresden, Aug 23 2019: (Start)
a(n) = r1^n + r2^n + r3^n + r4^n, where {r1,r2,r3,r4} are the four roots of x^4-x^3-1=0, see A086106, A230151.
a(n) = round(r^n) for n>21 and r the positive real root of x^4-x^3-1.
(End)

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-4.

			0.8566748838545028748523248153124343698313999454937526255...

Paolo P. Lava, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)
Index entries for algebraic numbers, degree 5

Cf. A075778, A230151, A230153-A230158, A377080.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-4);

Root[x^5 + x^4 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230158 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=10.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-10.

			0.9360691110777583783971914875702962034360714782064850849...

Paolo P. Lava, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 10.

Cf. A075778, A230151-A230157.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-10);

Root[x^11 + x^10 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

polrootsreal(x^11+x^10-1)[1] \\ Charles R Greathouse IV, Feb 07 2025

A230154 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=6.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-6.

			0.8986537126286992932608757220468058862604482200934396966...

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A075778, A230151, A230152, A230153, A230155, A230156, A230157, A230158, A230160, A377081.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-6);

RealDigits[x/.FindRoot[x^7+x^6==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 30 2013 *)

Equals 1/A230160. - Hugo Pfoertner, Oct 15 2024

A230156 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=8.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-8.

			0.9215993196339830062994303152019693942536803842533707898...

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A075778, A230151-A230155, A230157, A230158.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-8);

Root[x^9 + x^8 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230157 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=9.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-9.

			0.9295701282320228642044130369144641254353258530020248336...

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A075778, A230151-A230156, A230158.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-9);

Root[x^10 + x^9 - 1, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230153 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=5.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-5.

			0.8812714616335695944076491628413720252791939793788952636...

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A075778, A230151, A230152, A230154-A230158.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-5);

Root[x^6 + x^5 - 1, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230155 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=7.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-7.

			0.9115923534820549186286736724940501773758846943614139469...

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A075778, A230151-A230154, A230156-A230158.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-7);

Root[x^8 + x^7 - 1, 2] // RealDigits[#, 10, 130]& // First (* Jean-François Alcover, Feb 18 2014 *)

cp's OEIS Frontend

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

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A014097 a(n) = a(n-1)+a(n-4).

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A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

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A230158 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=10.

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A230154 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=6.

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A230156 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=8.

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A230157 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=9.

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