A092526 -id:A092526 - cp's OEIS frontend

A088559 Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.

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

Offset: 0

Benoit Cloitre, Nov 19 2003

Arise in a study of AGM (arithmetic-geometric mean) and HGM (harmonic-geometric mean) - like sequences. Let u(k+1)=sqrt(u(k)*v(k)); v(k+1)=v(k)+u(k) and r(k+1)=sqrt(r(k)*s(k)); s(k+1)=1/(1/r(k)+1/s(k)). Then for any positive initial values u(0),v(0),r(0),s(0) limit k-->oo u(k)/v(k)= limit k-->oo s(k)/r(k)=R^2.
From Wolfdieter Lang, Nov 07 2022: (Start)
This equals r0 - 2/3 where r0 is the real root of y^3 - (1/3)*y - 29/27.
The other roots of x^3 + 2*x^2 + x - 1 are (-2 + w1*((29 + 3*sqrt(93))/2)^(1/3) + w2*((29 - 3*sqrt(93))/2)^(1/3))/3 = -1.2327856159... + 0.7925519925...*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(29/2)) + sqrt(3)*sinh((1/3)*arccosh(29/2))*i)/3, and its complex conjugate. (End)

			0.465571231876768026656731225219939108025577568472285701643183111249262996685...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for algebraic numbers, degree 3

Cf. A092526, A144229, A358181.

Root[x^3 + 2x^2 + x - 1, 1] // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)

allocatemem(932245000); default(realprecision, 20080); x=10*solve(x=0, 1, x^3 + 2*x^2 + x - 1); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b088559.txt", n, " ", d)); \\ Harry J. Smith, Jun 21 2009

polrootsreal(x^3 + 2*x^2 + x - 1)[1] \\ Charles R Greathouse IV, Mar 03 2016

R^2=0.46557123187676... 1+R^2=1.46557123187676... = A092526 constant.
From Vaclav Kotesovec, Dec 18 2014: (Start)
Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) - 2/3.
Equals 2*cos(arccos(29/2)/3)/3 - 2/3.
Equals A092526 - 1.
(End)
From Wolfdieter Lang, Nov 07 2022: (Start)
Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3)  + ((29 + 3*sqrt(93))/2)^(-1/3))/3.
Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3)  + ((29 - 3*sqrt(93))/2)^(1/3))/3.
Also with hperbolic cosh and arccosh instead of cos and arccos above.
(End)

A293508 Decimal expansion of the positive real root of x^6 - x^5 - x^4 + x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Iain Fox, Oct 11 2017

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

			1.501594803539087366377783...

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 6.

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

First@ RealDigits[Root[#^6 - #^5 - #^4 + #^2 - 1 &, 2], 10, 105] (* Michael De Vlieger, Oct 23 2017 *)

solve(x=1, 2, x^6 - x^5 - x^4 + x^2 - 1) \\ Michel Marcus, Oct 11 2017

{ default(realprecision, 20080); x=solve(x=1, 2, x^6 - x^5 - x^4 + x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293508.txt", n, " ", d)); }

A293509 Decimal expansion of real root of x^5 - x^3 - x^2 - x - 1.

Original entry on oeis.org

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

Offset: 1

Iain Fox, Oct 11 2017

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

			1.53415774491426691543597007610937570188254503851659513536853186300806302321...

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 5

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

RealDigits[ Solve[ x^5 - x^3 - x^2 - x - 1 == 0, x, WorkingPrecision -> 111][[-1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Nov 04 2017 *)

solve(x=1, 2, x^5 - x^3 - x^2 - x - 1) \\ Michel Marcus, Oct 13 2017

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^3 - x^2 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293509.txt", n, " ", d)); \\ Iain Fox, Oct 23 2017

polrootsreal(x^5 - x^3 - x^2 - x - 1)[1] \\ Charles R Greathouse IV, Nov 04 2017

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

Original entry on oeis.org

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)); }

A069241 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| <= 2.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 17, 28, 44, 68, 104, 157, 235, 350, 519, 767, 1131, 1665, 2448, 3596, 5279, 7746, 11362, 16662, 24430, 35815, 52501, 76956, 112797, 165325, 242309, 355135, 520490, 762830, 1117997, 1638520, 2401384, 3519416, 5157972, 7559393, 11078847

Offset: 0

Don Knuth, Apr 13 2002

Equivalently, the number of bandwidth-at-most-2 arrangements of a straight line of n vertices.

			For example, the six Hamiltonian paths when n=4 are 1234, 1243, 1324, 1342, 2134, 3124.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Cf. A003274, A000930, A092526, A263719, A302119.

a:= n-> (Matrix([[1,1,1,0,1]]). Matrix(5, (i,j)-> if i=j-1 then 1 elif j=1 then [3,-3,2,-2,1][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..50); # Alois P. Heinz, Sep 09 2008

a[0] = a[1] = a[2] = 1; a[3] = 3; a[4] = 6; a[n_] := a[n] = 3a[n-1] - 3a[n-2] + 2a[n-3] - 2a[n-4] + a[n-5]; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Feb 13 2015 *)
CoefficientList[Series[(3+x+x^2)/(1-x-x^3)-(2-x)/(1-x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{3,-3,2,-2,1},{1,1,1,3,6},60] (* Harvey P. Dale, Apr 07 2019 *)

a(n) = A003274(n)/2, n > 1.
a(n) = 3*s(n) + s(n-1) + s(n-2) - 2 - n, where s(n) = A000930(n).
G.f.: (3+x+x^2)/(1-x-x^3) - (2-x)/(1-x)^2.
Lim_{n->infinity} a(n+1)/a(n) = A092526 = 1/A263719. - Alois P. Heinz, Apr 15 2018

A224868 a(1) = greatest k such that H(k) - H(4) < 1/3 + 1/4; a(2) = greatest k such that H(k) - H(a(1)) < H(a(1)) - H(4); and for n > 2, a(n) = greatest k such that H(k) - H(a(n-1)) > H(a(n-1)) - H(a(n-2)), where H = harmonic number.

Original entry on oeis.org

7, 11, 17, 26, 39, 58, 86, 127, 187, 275, 404, 593, 870, 1276, 1871, 2743, 4021, 5894, 8639, 12662, 18558, 27199, 39863, 58423, 85624, 125489, 183914, 269540, 395031, 578947, 848489, 1243522, 1822471, 2670962, 3914486, 5736959, 8407923, 12322411, 18059372

Offset: 1

Clark Kimberling, Jul 23 2013

Suppose that x and y are positive integers and that x <=y. Let c(1) = y and c(2) = greatest k such that H(k) - H(y) < H(y) - H(x); for n > 2, let c(n) = greatest such that H(k) - H(c(n-1)) < H(c(n-1)) - H(c(n-2)). Then 1/x + ... + 1/c(1) > 1/(c(1)+1) + ... + 1/(c(2)) > 1/(c(2)+1) + ... + 1/(c(3)) > ... The decreasing sequences H(c(n)) - H(c(n-1)) and c(n)/c(n-1) converge. For what choices of (x,y) is the sequence c(n) linearly recurrent?

For A224868, (x,y) = (3,4); it appears that the sequence a(n) is linearly recurrent with signature (2,-1,1,-1). Possibly the constant at A202537 is the limit of the sequences H(c(n))-H(c(n-1)). Possibly the constant at A092526 is the limit of c(n)/c(n-1).

			The first three values (a(1),a(2),a(3)) = (7,11,17) match the beginning of the following inequality chain (and partition of {1/m: m>=3}):
1/3+1/4 > 1/5+1/6+1/7 > 1/8+1/9+1/10+1/11 > 1/12+ ... +1/17 > ...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A224820.

z = 100; h[n_] := h[n] = HarmonicNumber[N[n, 500]]; x = 3; y = 4; a[1] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[y] - h[x - 1], {w, 1}, WorkingPrecision -> 400]]; a[2] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[a[1]] - h[y], {w, a[1]}, WorkingPrecision -> 400]]; Do[s = 0; a[t] = -1 + Ceiling[w /. FindRoot[h[w] == 2 h[a[t - 1]] - h[a[t - 2]], {w, a[t - 1]}, WorkingPrecision -> 400]], {t, 3, z}]; m = Map[a, Range[z]] (* A224868 *)
N[Table[h[a[t]] - h[a[t - 1]], {t, 2, z, 25}], 5]  (* A202537? *)
N[Table[a[n]/a[n - 1], {n, 2, z, 25}], 5]  (* A092526? *)
(* Peter J. C. Moses, Jul 23 2013 *)

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (conjectured).

G.f.: (7 - 3 x + 2 x^2 - 4 x^3)/(1 - 2 x + x^2 - x^3 + x^4) (conjectured).

A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 11, 14, 20, 27, 39, 52, 75, 102, 145, 201, 286, 397, 565, 791, 1123, 1581, 2248, 3173, 4517, 6399, 9112, 12945, 18457, 26270, 37502, 53478, 76416, 109146, 156135, 223301, 319764, 457884, 656288, 940795, 1349671, 1936620

Offset: 1

N. J. A. Sloane, Nov 15 2001

nonn

Inverse Euler transform of A078012. (The inverse of 1-1/(p^3-p^2) is p^2(p-1)/(p^3-p^2-1) = 1-1/(1+p^2-p^3). Setting 1/p=x gives (1-x)/(1-x-x^3), the g.f. of A078012.) - R. J. Mathar, Jul 26 2010

			x^3 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + 7*x^12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, sequence gamma_{2,j}^(A).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Index entries for sequences related to Artin's conjecture

Cf. A065414.

read("transforms") ;
A078012 := proc(n) option remember; if n <3 then op(n+1,[1,0,0]) ; else procname(n-1)+procname(n-3) ; end if; end proc:
a078012 := [seq(A078012(n),n=1..80)] ; EULERi(%) ;
# R. J. Mathar, Jul 26 2010

A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];
s = {}; For[i = 1, i <= m, i++, AppendTo[s, i*a078012[[i]] - Sum[s[[d]] * a078012[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d ], 0]*s[[d]], {d, 1, i}]/i, {i, m}] (* Jean-François Alcover, Apr 15 2016, after R. J. Mathar *)

a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - Vaclav Kotesovec, Jun 13 2020

More terms from R. J. Mathar, Jul 26 2010

A048634 a(n) = a(n-1)*a(n-3) + a(n-2), with a(0)=a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 2, 3, 5, 13, 44, 233, 3073, 135445, 31561758, 96989417779, 13136731722638413, 414618347540933702027833, 40213592128486236142855326045681320, 528275171395527518169753769210241662354568290572993

Offset: 0

N. J. A. Sloane, David(AT)interface.co.uk

A048634 := proc(n) option remember; if n<=1 then 0 elif n=2 then 1 else A048634(n-1)*A048634(n-3)+A048634(n-2); fi; end;

RecurrenceTable[{a[n] == a[n-1]*a[n-3] + a[n-2], a[0] == 0, a[1] == 0, a[2] == 1}, a, {n, 0, 20}] (* Vaclav Kotesovec, Aug 16 2021 *)

a(n) ~ c^(A092526^n), where c = A344388 = 1.0574735961... (very close to A201506). - Vaclav Kotesovec, Aug 16 2021

Name clarified by Michel Marcus, Aug 16 2021

A369346 Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

Original entry on oeis.org

1, 2, 6, 1, 3, 5, 4, 22, 1, 1, 4, 1, 2, 84, 1, 3, 1, 6, 1, 3, 1, 9, 1, 1, 1, 1, 19, 3, 1, 2, 1, 5, 1, 5, 2, 2, 1, 1, 1, 1, 76, 6, 8, 1, 1, 5, 1, 5, 1, 1, 25, 1, 2, 1, 116, 2, 1, 8, 1, 1, 3, 1, 53, 5, 276, 2, 1, 1, 1, 3, 3, 2, 1, 1, 4, 13, 1, 1, 1, 4, 1, 1, 1, 9, 9, 1, 1, 9, 6, 1, 2, 32

Offset: 0

Patrick McKinley, Jan 20 2024

Patrick McKinley, Table of n, a(n) for n = 0..12174
Eric Weisstein's World of Mathematics, Supergolden Ratio.

Cf. A092526 (decimal expansion), A381124, A381125 (convergents).

ContinuedFraction[x/.First[Solve[x^3-x^2-1==0,x]],92] (* Stefano Spezia, Jan 21 2024 *)

\p100 \\ realprecision
contfrac(solve(x = 1, 2, x^3 - x^2 - 1),, 80) \\ Hugo Pfoertner, Jan 21 2024

/* The "test" calculation evaluates the cubic to confirm the calculation of the root. */
define iter(frac)
{j = 0
 while(frac > 1){
   frac -= 1;
   j+=1}
 j
 return 1/frac}
scale=12578
f=(1+(e(l(((29+3*sqrt(93))/2))/3))+(e(l(((29-3*sqrt(93))/2))/3)))/3
psi=f
test=(psi-1)*psi*psi-1
for(i=0;i<12175;i++)f=iter(f)

Offset changed by Andrew Howroyd, Feb 14 2025

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]]

cp's OEIS Frontend

A088559 Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.

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A293508 Decimal expansion of the positive real root of x^6 - x^5 - x^4 + x^2 - 1.

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A293509 Decimal expansion of real root of x^5 - x^3 - x^2 - 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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A069241 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| <= 2.

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A224868 a(1) = greatest k such that H(k) - H(4) < 1/3 + 1/4; a(2) = greatest k such that H(k) - H(a(1)) < H(a(1)) - H(4); and for n > 2, a(n) = greatest k such that H(k) - H(a(n-1)) > H(a(n-1)) - H(a(n-2)), where H = harmonic number.

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A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

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