A075778 -id:A075778 - cp's OEIS frontend

A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

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

Offset: 0

R. J. Mathar, Jan 22 2013

An algebraic number of degree 6. - Charles R Greathouse IV, Apr 14 2014

The denominator of this algebraic number is 2, since its double is an algebraic integer. - Charles R Greathouse IV, Nov 12 2014

			0.744861766619744236593170428604392367240163...

Index entries for algebraic numbers, degree 6

Cf. A075778, A210462.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210463 := proc()
        local a075778,a210462 ;
        a075778 := A075778neg() ;
        a210462 := A210462() ;
        -1/a075778-a210462^2 ;
        sqrt(%) ;
end proc:
evalf(A210463()) ;

-((2^(1/3)*(25 - 3*Sqrt[69])^(2/3) - 2)/(2*2^(2/3)*Sqrt[3]*(25 - 3*Sqrt[69])^(1/3))) // RealDigits[#, 10, 125]& // First (* Jean-François Alcover, Feb 20 2013 *)

polrootsreal(64*x^6+32*x^4+4*x^2-23)[2] \\ Charles R Greathouse IV, Apr 14 2014

Equals sqrt(1/A075778-A210462^2).

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 *)

A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 69, 82, 98, 115, 135, 158, 184, 214, 248, 286, 330, 379, 435, 497, 569, 648, 739, 840, 955, 1082, 1228, 1388, 1572, 1775, 2005, 2259, 2549, 2867, 3228, 3626, 4076, 4571, 5131, 5745, 6438, 7199, 8053, 8992, 10045, 11199

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000

Column 3 of A350889.

Cf. A168657, A237756, A350894.

CoefficientList[Series[Sum[x^(3k^2)/Product[1-x^j,{j,3k-1}],{k,64}],{x,0,64}],x] (* Stefano Spezia, Jan 22 2022 *)
Table[Count[IntegerPartitions[n],?(3#[[-1]]==Length[#]&)],{n,70}] (* _Harvey P. Dale, Jul 13 2023 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, 3*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..3*k-1} (1-x^j).

a(n) ~ c * exp(2*sqrt((5*log(A075778)^2 + 2*polylog(2, 1 - A075778))*n)) / n^(3/4), where c = (3*log(A075778)^2 + polylog(2, A075778^2))^(1/4) / (2*sqrt(3*Pi*(1 + A075778)*(2 + 3*A075778))) = 0.0582980106266835787... - Vaclav Kotesovec, Jan 24 2022, updated Oct 14 2024

A262990 G.f. A(x) satisfies: a([n/r^2]) = [x^n] A(x)^2/x and a([n/r^3]) = [x^n] A(x)^3/x^2, for n>=1, where r^2 + r^3 = 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 10, 20, 22, 40, 51, 67, 114, 126, 203, 230, 354, 468, 571, 885, 908, 1486, 1674, 2250, 3045, 3586, 5418, 5322, 8186, 9560, 12234, 16341, 17976, 27912, 26970, 38435, 46383, 57024, 76794, 80805, 125205, 116376, 165914, 201580, 232352, 322980, 332388, 508710, 456154, 645661, 800495, 886018, 1241190, 1226382, 1916682, 1693454, 2290704, 2935492

Offset: 1

Paul D. Hanna, Oct 06 2015

nonn

The integer floor values, [n/r^2] and [n/r^3] where r^2 + r^3 = 1, form Beatty sequences and thus together contain all the positive integers without repetition.
Here r = 6 / ( (108 + 12*sqrt(69))^(1/3) + (108 - 12*sqrt(69))^(1/3) ) = 0.75487766624669276.... satisfies r^2 + r^3 = 1.
Not equal to A090845.
What is the rate of growth of this sequence?

			G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 10*x^7 + 20*x^8 + 22*x^9 + 40*x^10 + 51*x^11 + 67*x^12 + 114*x^13 + 126*x^14 + 203*x^15 +...
where the terms are formed from the union of coefficients in A(x)^2 and A(x)^3.
The coefficients of A(x)^2 begin:
A^2 = [1, 2, 5, 10, 20, 40, 67, 126, 203, 354, 571, 908, 1486, 2250, 3586, 5322, 8186, 12234, 17976, 26970, 38435, 57024, 80805, 116376, 165914, 232352,...]
and form the terms of this sequence at positions [n/r^2] for n>=1:
{[n/r^2]} = [1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 49, 50, 52, 54, 56, 57, 59, ...].
The coefficients of A(x)^3 begin:
A^3 = [1, 3, 9, 22, 51, 114, 230, 468, 885, 1674, 3045, 5418, 9560, 16341, 27912, 46383, 76794, 125205, 201580, 322980, 508710, 800495, 1241190, ...]
and form the terms of this sequence at positions [n/r^3] for n>=1:
{[n/r^3]} = [2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 69, 72, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A075778.

{a(n) = local(A=vector(n+1),B=A,C=A,r=6/((108+12*sqrt(69))^(1/3)+(108-12*sqrt(69))^(1/3))); A[1]=1;A[2]=1;
for(i=1,ceil(log(#A)/log(1/r)),
B=vector(floor(#A/r^2));for(n=1,#A,m=floor(n/r^2);if(m<=#B,B[m]=Vec(Ser(A)^2)[n]));
C=vector(floor(#A/r^3));for(n=1,#A,m=floor(n/r^3);if(m<=#C,C[m]=Vec(Ser(A)^3)[n]));
A=vector(#A,n,if(C[n]==0,B[n],C[n]));); A[n]}
for(n=1,80,print1(a(n),", "))

A276519 Expansion of Product_{k>=1} 1/(1 - x^(2k) - x^(3k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 17, 19, 34, 37, 61, 75, 112, 138, 209, 256, 376, 478, 675, 866, 1222, 1566, 2175, 2830, 3873, 5055, 6900, 9011, 12213, 16045, 21599, 28429, 38191, 50290, 67341, 88884, 118669, 156751, 209018, 276200, 367734, 486376, 646688

Offset: 0

Vaclav Kotesovec, Nov 15 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A264905, A266686, A275820, A275821, A276526.

nmax=50; CoefficientList[Series[1/Product[1-x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * p / r^n, where r = A075778 = 1/A060006 = 0.7548776662466927600495... is the real root of the equation r^3 + r^2 - 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) - r^(3*n)) = 3.820450591662541853... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.

A276526 Expansion of Product_{k>=1} 1/(1 - x^(2k) + x^(3k)).

Original entry on oeis.org

1, 0, 1, -1, 2, -2, 3, -4, 7, -8, 11, -15, 22, -27, 37, -51, 70, -90, 121, -162, 220, -288, 381, -512, 688, -902, 1197, -1598, 2127, -2809, 3722, -4949, 6581, -8699, 11519, -15301, 20305, -26862, 35581, -47208, 62591, -82859, 109756, -145506, 192856, -255388

Offset: 0

Vaclav Kotesovec, Nov 16 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A264905, A266686, A275820, A275821, A276519.

nmax = 50; CoefficientList[Series[1/Product[1-x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * p / r^n, where r = -A075778 = -0.7548776662466927600495... is the real root of the equation r^3 - r^2 + 1 = 0, p = Product_{n>1} 1/(1 - r^(2*n) + r^(3*n)) = 1.9844809074648434... and c = 0.41149558866264576338190038... is the real root of the equation -1 + 8*c - 23*c^2 + 23*c^3 = 0.

cp's OEIS Frontend

A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A276519 Expansion of Product_{k>=1} 1/(1 - x^(2k) - x^(3k)).

A276526 Expansion of Product_{k>=1} 1/(1 - x^(2k) + x^(3k)).