A060006 -id:A060006 - cp's OEIS frontend

A113788 Number of irreducible multiple zeta values at weight n.

0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 21, 28, 34, 45, 56, 73, 92, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185

Offset: 1

R. J. Mathar, Jan 27 2006

n * a(n) is the Möbius transform of the Perrin sequence A001608.

Number of unlabeled (i.e., defined up to a rotation) maximal independent sets of the n-cycle graph having n isomorphic representatives. - Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Danny Rorabaugh, Table of n, a(n) for n = 1..8000
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 1st line of Table 1 (p. 6).
R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 [math.CO], 2007-2008.
R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, JIS 11 (2008), #08.5.7.
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, UTAS-PHYS-96-44; arXiv:hep-th/9609128, 1996.
D. J. Broadhurst and D. Kreimer, Associated multiple zeta values with positive knots via Feynman diagrams up to 9 knots, Phys. Lett B, 393 (1997), 403-412.
M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.

Cf. A001608, A127687, A125951.

A113788 := proc(n::integer)
    local resul,d;
    resul :=0;
    for d from 1 to n do
        if n mod d = 0 then
            resul := resul +numtheory[mobius](n/d)*A001608(d);
        fi;
    od:
    RETURN(resul/n);
end: # R. J. Mathar, Apr 25 2006

(* p = A001608 *) p[n_] := p[n] = p[n-2] + p[n-3]; p[0] = 3; p[1] = 0; p[2] = 2; a[n_] := (1/n)*Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jul 16 2012, from first formula *)

z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z**2 + z**3))
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 24 2020

a(n) = (1/n) * Sum_{d|n} mu(n/d)*Perrin(d), where Perrin(d) = A001608 starting with 0, 2, 3, ... .
a(n) = Sum_{d|n} mu(n/d)*A127687(d) = (1/n) * Sum_{d|n} mu(n/d)*A001608(d). - Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007
For p an odd prime, a(p) = Sum_{i=0..floor((p-3)/6)} (A(i)+B(i)-1)!/(A(i)!*B(i)!), where A(i) = (p-3)/2 - 3*i, and B(i) = 1 + 2*i. - Richard Turk, Sep 08 2015
a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019

A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 240, 1680, 15120, 181440, 2903040, 60963840, 1706987520, 63158538240, 3094768373760, 201159944294400, 17299755209318400, 1972172093862297600, 297797986173206937600, 59559597234641387520000

Offset: 1

John Lien, Feb 17 2007

Ian Stewart, Tales of a Neglected Number
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Eric Weisstein's World of Mathematics, Padovan Sequence
E. Wilson, The Scales of Mt. Meru

Cf. A000931, A003266, A060006, A253924.

From R. J. Mathar, Sep 14 2010: (Start)
A000931 := proc(n) option remember; if n = 0 then 1; elif n <=2 then 0; else procname(n-2)+procname(n-3) ; end if; end proc:
A126772 := proc(n) mul( A000931(i),i=5..n+4) ; end proc: seq(A126772(n),n=1..40) ; (End)

Rest[FoldList[Times,1,LinearRecurrence[{0,1,1},{1,1,1},30]]] (* Harvey P. Dale, Apr 29 2013 *)

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = 1.324717957244746... (see A060006) is the root of the equation r^3 = r + 1, d = 0.393641282401116385386658448446561... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0, c = 1.25373683131537208838997864311903035079685338006712312402418098138010834953... (see A253924). - Vaclav Kotesovec, Jan 26 2015

More terms from R. J. Mathar, Sep 14 2010

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)); }

A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5a(n-1) - 4a(n-2) + a(n-3).

Original entry on oeis.org

0, 1, 1, 1, 2, 7, 28, 114, 465, 1897, 7739, 31572, 128801, 525456, 2143648, 8745217, 35676949, 145547525, 593775046, 2422362079, 9882257736, 40315615410, 164471408185, 670976837021, 2737314167775, 11167134898976

Offset: 0

N. J. A. Sloane

nonn

Old name was "Take every 5th term of Padovan sequence A000931".

Lim_{n -> infinity} a(n+1)/a(n) = p^5 = 4.0795956..., where p is the plastic constant (A060006). - Jianing Song, Feb 04 2019

Index entries for linear recurrences with constant coefficients, signature (5, -4, 1).

Cf. A000931, A012814, A012864, A060006, A176476.

A012855 := proc(n,A,B,C) option remember; if n = 0 then A elif n = 1 then B elif n = 2 then C else 5*procname(n-1,A,B,C)-4*procname(n-2,A,B,C)+procname(n-3,A,B,C); fi; end; [ seq(A012855(i,0,1,1),i = 0..40) ]; # R. J. Mathar, Dec 30 2011

CoefficientList[Series[(4x^2-x)/(x^3-4x^2+5x-1),{x,0,40}],x] (* or *) LinearRecurrence[{5,-4,1},{0,1,1},40] (* Harvey P. Dale, Mar 28 2013 *)

a(n) = my(v=vector(n+1), u=[0,1,1]); for(k=1, n+1, v[k]=if(k<=3, u[k], 5*v[k-1] - 4*v[k-2] + v[k-3])); v[n+1] \\ Jianing Song, Feb 04 2019

a(n) = A000931(5*n-12) for n >= 3. - Alois P. Heinz, Feb 04 2019
G.f. (4x^2 - x)/(x^3 - 4x^2 + 5x - 1). For n > 2, a(n) = 1 + Sum_{k=0..n-3} A012814(k). - Ralf Stephan, Jan 15 2004
a(n) = 1 + A176476(n-3) = 1 + Sum_{k=0..n-3} A000931(5*k+2) for n >= 3. - Jianing Song, Feb 04 2019

Edited by N. J. A. Sloane, Feb 06 2019 at the suggestion of Jianing Song, replacing imprecise definition with formula from Harvey P. Dale, Mar 28 2013

A051016 Numbers n for which r^n-floor(r^n) < 1/2, where r is the real root of x^3-x-1.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, 19, 22, 24, 25, 27, 29, 30, 32, 35, 37, 38, 40, 42, 43, 45, 48, 50, 53, 55, 56, 58, 60, 61, 63, 66, 68, 71, 73, 74, 76, 78, 79, 81, 84, 86, 89, 91, 92, 94, 97, 99, 102, 104, 105, 107, 109, 110, 112, 115, 117, 120, 122, 123, 125, 127

Offset: 1

Eric W. Weisstein

nonn

For large powers, r^n is very close to an integer.

Eric Weisstein's World of Mathematics, Pisot Number

Cf. A051017, A060006 (r = 1.32471...).

Flatten[ With[ {r = Root[ -1 - #1 + #1^3 &, 1 ]}, Position[ Table[ r^n - Floor[ r^n ], {n, 1, 200} ], _?(#1 < 1/2 & ), 1 ] ] ]

Corrected by Don Reble, May 04 2006

A051017 Numbers n for which r^n-floor(r^n) > 1/2, where r is the real root of x^3-x-1.

Original entry on oeis.org

2, 9, 10, 13, 15, 16, 18, 20, 21, 23, 26, 28, 31, 33, 34, 36, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 62, 64, 65, 67, 69, 70, 72, 75, 77, 80, 82, 83, 85, 87, 88, 90, 93, 95, 96, 98, 100, 101, 103, 106, 108, 111, 113, 114, 116, 118, 119, 121, 124, 126, 129, 131

Offset: 1

Eric W. Weisstein

nonn

For large powers, r^n is very close to an integer.

Eric Weisstein's World of Mathematics, Pisot Number

Cf. A051016, A060006.

Flatten[ With[ {r = Root[ -1 - #1 + #1^3 &, 1 ]}, Position[ Table[ r^n - Floor[ r^n ], {n, 1, 200} ], _?(#1 > 1/2 & ), 1 ] ] ]

Corrected by Don Reble, May 04 2006

A230163 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=10.

Original entry on oeis.org

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

Offset: 1

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.

			1.0757660660868371580595995241652758206925302476392032794...

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 10.

Cf. A001622, A060006, A060007, A160155, A230159-A230162.

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

polrootsreal(x^10-x-1)[2] \\ Charles R Greathouse IV, Feb 11 2025

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset: 0

Vaclav Kotesovec, Mar 17 2020

nonn

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

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

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.

Cf. A192918, A376841.

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

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

cp's OEIS Frontend

A113788 Number of irreducible multiple zeta values at weight n.

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A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

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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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PARI

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

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PARI

A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3).

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A051016 Numbers n for which r^n-floor(r^n) < 1/2, where r is the real root of x^3-x-1.

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A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5a(n-1) - 4a(n-2) + a(n-3).

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).