A060006 -id:A060006 - cp's OEIS frontend

A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)(1 + x^k)^n) ).

1, 1, 2, 4, 7, 14, 25, 44, 79, 137, 237, 408, 689, 1162, 1946, 3231, 5342, 8776, 14340, 23326, 37758, 60847, 97670, 156145, 248697, 394719, 624343, 984360, 1547187, 2424581, 3788730, 5904230, 9176723, 14226914, 22002523, 33947526, 52258177, 80268131, 123028407

Offset: 0

Paul D. Hanna, Nov 02 2012

nonn

Compare to the dual g.f. of A219229:

exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*x^7 +...
where
log(A(x)) = x/1*((1+x*(1+x))*(1+x^2*(1+x^2))*(1+x^3*(1+x^3))*...) +
x^2/2*((1+x^2*(1+x)^2)*(1+x^4*(1+x^2)^2)*(1+x^6*(1+x^3)^2)*...) +
x^3/3*((1+x^3*(1+x)^3)*(1+x^6*(1+x^2)^3)*(1+x^9*(1+x^3)^3)*...) +
x^4/4*((1+x^4*(1+x)^4)*(1+x^8*(1+x^2)^4)*(1+x^12*(1+x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*x^10/10 +...

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Paul D. Hanna)

Cf. A219229, A218575, A218552, A218153.

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,(1+x^(m*k)*(1+x^k+x*O(x^n))^m )))),n)}
for(n=0,50,print1(a(n),", "))

Conjecture: a(n) ~ c * d^n, where d = A060006 = 1.3247179572447... is the real root of the equation d*(d^2-1) = 1 and c = 43328430766.390... . - Vaclav Kotesovec, Apr 09 2016

A230159 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=6.

Original entry on oeis.org

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

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=6.

			1.1347241384015194926054460545064728402796672263828014859...

Paolo P. Lava, Table of n, a(n) for n = 1..1000 (corrected by _Sean A. Irvine_)

Cf. A001622, A060006, A060007, A160155, A230160-A230163.

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);

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

A230160 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=7.

Original entry on oeis.org

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

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=7.

			1.1127756842787054706297040205710929356068592718552836814...

Paolo P. Lava, Table of n, a(n) for n = 1..1000 (corrected by _Sean A. Irvine_)

Cf. A001622, A060006, A060007, A160155, A230159, A230161-A230163.

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^7 - x - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230161 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=8.

Original entry on oeis.org

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

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=8.

			1.0969815577985598179082789671675370895925301082127867138...

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001622, A060006, A060007, A160155, A230159, A230160, A230162, A230163.

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^8 - x - 1, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230162 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=9.

Original entry on oeis.org

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

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=9.

			1.0850702454914508283368958640973142340506536310308965814...

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001622, A060006, A060007, A160155, A230159-A230161, A230163.

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[(#^9-#-1)&, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[x/.FindRoot[x^9-x-1==0,{x,1},WorkingPrecision->100]][[1]] (* Harvey P. Dale, Jul 31 2017 *)

A253924 Decimal expansion of Padovan factorial constant.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 26 2015

The Padovan factorial constant is associated with the Padovan factorial A126772.

			1.25373683131537208838997864311903035079685338006712312402418098138...

Eric Weisstein's World of Mathematics, Padovan Sequence

Cf. A000931, A060006, A126772.

Equals limit n->infinity A126772(n) / (d^(n/2) * r^(n^2/2)), where r = 1.324717957244746025960908854478... (see A060006) is the root of the equation r^3 = r + 1, d = 0.3936412824011163853866584484465616545579227324... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0.

A333180 G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 6, 3, 3, 3, 4, 4, 4, 8, 8, 8, 8, 8, 4, 9, 9, 5, 10, 10, 15, 15, 15, 15, 15, 15, 16, 16, 11, 17, 17, 18, 24, 24, 24, 30, 30, 30, 30, 31, 31, 31, 32, 26, 33, 34, 41, 41, 42, 49, 49, 56, 56, 56, 64, 64, 57, 65, 58, 59, 67, 68

Offset: 0

Vaclav Kotesovec, Mar 10 2020

nonn

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

Cf. A268188, A306734, A333181.

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

a(n) ~ c * A333198^sqrt(n), where c = 0.3836313809149103736315...

Limit_{n->infinity} a(n) / A333181(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A006888 a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, 2444729709746709953, 2298386861814452020993305, 185187471463742319884263934176321, 5618934645754484318302453706799174724040986

Offset: 0

Robert Munafo

Tends towards something like 1.60119...^(1.3247...^n) where 1.3247... = (1/2+sqrt(23/108))^(1/3)+(1/2-sqrt(23/108))^(1/3) is the smallest Pisot-Vijayaraghavan number A060006. Any four consecutive terms are pairwise coprime. - Henry Bottomley, Sep 25 2002

			From _Muniru A Asiru_, Jan 28 2018: (Start)
a(3) = a(2) + a(1) * a(0) = 1 + 1 * 1 = 2.
a(4) = a(3) + a(2) * a(1) = 2 + 1 * 1 = 3.
a(5) = a(4) + a(3) * a(2) = 3 + 2 * 1 = 5.
a(6) = a(5) + a(4) * a(3) = 5 + 3 * 2 = 11.
a(7) = a(6) + a(5) * a(4) = 11 + 5 * 3 = 26.
...
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..30

a := [1,1,1];; for n in [4..35] do a[n] := a[n-1] + a[n-2] * a[n-3]; od; a; # Muniru A Asiru, Jan 28 2018

a := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 1 elif n>=3 then procname(n-1) + procname(n-2) * procname(n-3) fi; end:
seq(a(n), n=0..35); # Muniru A Asiru, Jan 28 2018

a=1;b=1;c=1;lst={a,b,c};Do[d=a*b+c;AppendTo[lst,d];a=b;b=c;c=d,{n,2*4!}];lst  (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
Nest[Append[#, Last[#] + Times @@ #[[-3 ;; -2]]] &, {1, 1, 1}, 17] (* Michael De Vlieger, Jan 23 2018 *)
nxt[{a_,b_,c_}]:={b,c,c+b*a}; NestList[nxt,{1,1,1},20][[All,1]] (* Harvey P. Dale, Feb 03 2021 *)

Limit_{n->infinity} a(n)/(a(n-1)*a(n-5)) = 1 agrees with lim_{n->infinity} a(n) = c^(P^n) (c=1.60119..., P=PisotV) since PisotV is real root of x^3-x-1 and thus a root of x^5-x^4-1 because x^5-x^4-1 = (x^3-x-1)*(x^2-x+1) and c^(P^n)/(c^(P^(n-1))*c^(P^(n-5))) = c^(P^(n-5)*(P^5-P^4-1)). - Gerald McGarvey, Aug 14 2004

More terms from Michel ten Voorde Apr 11 2001
Typo in Mathematica code corrected by Vincenzo Librandi, Jun 09 2013
Definition clarified by Matthew Conroy, Jan 23 2018

A205579 a(n) = round(r^n) where r is the smallest Pisot number (real root r=1.3247179.. of x^3-x-1).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367, 486, 644, 853, 1130, 1497, 1983, 2627, 3480, 4610, 6107, 8090, 10717, 14197, 18807, 24914, 33004, 43721, 57918, 76725, 101639, 134643, 178364, 236282, 313007, 414646, 549289, 727653, 963935, 1276942, 1691588, 2240877, 2968530, 3932465

Offset: 0

Joerg Arndt, Jan 29 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pisot Number.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Cf. A112639 (definition using floor() instead of round()).
Cf. A060006 (decimal expansion of r=1.32471795724475...).
Cf. A051016, A051017, A169985.

CoefficientList[Series[(1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3),{x,0,100}],x] (* Vincenzo Librandi, Aug 19 2012 *)
r = Root[x^3-x-1, 1]; Table[Round[r^i], {i,0,100 }] (* Jwalin Bhatt, Mar 27 2025 *)

default(realprecision, 110);
default(format, "g.15");
r=real(polroots(x^3-x-1)[1])
v=vector(66, n, round(r^(n-1)) )

Vec((1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3)+O(x^66))

G.f.: (1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3).
From Jwalin Bhatt, Mar 26 2025: (Start)
a(n) = round(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n).
a(n) = a(n-2) + a(n-3) for n>=13. (End)

A231691 Cardinalities of the symmetric operad of dotted red and white trees.

Original entry on oeis.org

1, 6, 74, 1476, 41032, 1464672, 63865328, 3290120832, 195537380704, 13169097667584, 991181618539136, 82450282595311104, 7511417235983147008, 743790032122343820288, 79541198937597284060672, 9136079502141558495310848, 1121720442822518015112749056, 146607501639123412303738884096, 20322509742114322789584125210624, 2978025324234142178848508363882496

Offset: 1

N. J. A. Sloane, Nov 14 2013

nonn

			A(x) = x + 6*x^2/2! + 74*x^3/3! + 1476*x^4/4! + 41032*x^5/5! + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..300
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Cf. A052878, A231690.

S:= series(RootOf(y=-x-ln((1+x)/(1+3*x+x^2)),x),y,21):
seq(coeff(S,y,n)*n!,n=1..21); # Robert Israel, Sep 27 2018

terms = 20; (CoefficientList[InverseSeries[Log[x^2 + 3x + 1] - Log[1+x] - x + O[x]^(terms+1)], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 16 2018, after Gheorghe Coserea *)

N=21; x = 'x + O('x^N); Vec(serlaplace(serreverse(log(x^2+3*x+1) - log(1+x) - x))) \\ Gheorghe Coserea, Jan 18 2017

E.g.f. A(x) satisfies -A(x) - g(-A(x)) = x where g is the E.g.f. of A052878. - Gheorghe Coserea, Jan 18 2017, edited by Robert Israel, Sep 27 2018

a(n) ~ sqrt((5 + 7*s + 3*s^2) / (7 + 13*s + 5*s^2)) * n^(n-1) / ((log((1+3*s+s^2)/(1+s))-s)^(n - 1/2) * exp(n)), where s = A060006 - 1 = -1 + (27/2 - 3*sqrt(69)/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3). - Vaclav Kotesovec, Apr 21 2020

Offset changed and more terms from Gheorghe Coserea, Jan 15 2017

cp's OEIS Frontend

A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).

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A230159 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=6.

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A230160 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=7.

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A230161 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=8.

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A230162 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=9.

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A253924 Decimal expansion of Padovan factorial constant.

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A333180 G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).

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A006888 a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.

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A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(nk)(1 + x^k)^n) ).