A060006 -id:A060006 - cp's OEIS frontend

A333181 G.f.: Sum_{k>=1} (k * x^(k(k+1)) Product_{j=1..k} (1 + x^j)).

0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 3, 3, 3, 6, 3, 3, 3, 0, 4, 4, 4, 8, 8, 8, 8, 8, 4, 4, 9, 5, 5, 10, 10, 15, 15, 15, 15, 15, 15, 10, 16, 11, 11, 17, 12, 18, 24, 24, 24, 30, 30, 30, 30, 24, 31, 31, 25, 26, 26, 27, 34, 41, 35, 42, 49, 49, 56, 56, 56, 56, 64

Offset: 0

Vaclav Kotesovec, Mar 10 2020

nonn

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

Cf. A333153, A333179, A333180.

nmax = 100; CoefficientList[Series[Sum[n*x^(n*(n+1))*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)]; 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 = 0.2895947615240435716456...

Limit_{n->infinity} A333180(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 21 2021

			0.3411639019140096636847418698555241284...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347177.

RealDigits[((29 + 3 Sqrt[93])^(1/3) - 2^(1/3))/(2 (3 (9 + Sqrt[93]))^(1/3)), 10, 110][[1]]

((29 + 3*sqrt(93))^(1/3) - 2^(1/3))/(2*(3*(9 + sqrt(93)))^(1/3)) \\ Michel Marcus, Aug 21 2021

Equals -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3)/sqrt(3). - Vaclav Kotesovec, Sep 29 2024

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]]

A374003 Decimal expansion of the positive real root of x^8 - x^7 - x^6 + x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 25 2024

Tenth smallest Pisot-Vijayaraghavan number.

			1.5736789683935169887742514186293214678127040615079...

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, A374002, A293506.

First[RealDigits[Root[#^8 - #^7 - #^6 + #^2 - 1 &, 2], 10, 100]]

A072117 Continued fraction expansion of smallest Pisot-Vijayaraghavan number (positive root of x^3 = x + 1 ).

Original entry on oeis.org

1, 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, 3, 1, 8, 2, 1, 1, 14, 1, 1, 2, 1, 1, 1, 3, 1, 10, 4, 40, 1, 1, 2, 4, 9, 1, 1, 3, 3, 3, 2, 1, 17, 7, 5, 1, 1, 4, 1, 1, 3, 5, 1, 2, 6, 2, 2, 1, 1, 1, 4, 1, 3, 1, 2, 6, 5, 6, 49, 3, 7, 1, 4, 2, 12, 4, 31, 1, 47, 1, 1, 1, 8, 2, 2, 1, 2, 1

Offset: 0

Benoit Cloitre, Jun 19 2002

			This number = 1.32471795724474602....

Cf. A060006 (decimal expansion).

PARI
```
\p150 contfrac(solve(X=1,2,X^3-X-1))
```

Offset changed by Andrew Howroyd, Jul 06 2024

A176476 Partial sums of A012814.

Original entry on oeis.org

0, 1, 6, 27, 113, 464, 1896, 7738, 31571, 128800, 525455, 2143647, 8745216, 35676948, 145547524, 593775045, 2422362078, 9882257735, 40315615409, 164471408184, 670976837020, 2737314167774, 11167134898975, 45557394660800, 185855747875875, 758216295635151

Offset: 0

Carmine Suriano, Apr 18 2010

Old name was "a(n) is the minimum integer that can be expressed as the sum of n Padovan numbers (see A000931)".

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

			a(5) = A000931(2) + A000931(7) + A000931(12) + A000931(17) + A000931(22) + A000931(27) = 0 + 1 + 5 + 21 + 86 + 351 = 464.

Michael De Vlieger, Table of n, a(n) for n = 0..1638
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (6,-9,5,-1).

Cf. A000931, A012814, A012855, A060006.

LinearRecurrence[{6,-9,5,-1},{0,1,6,27},30] (* Harvey P. Dale, Feb 08 2025 *)

a(n) = my(v=vector(n+1), u=[0,1,6,27]); for(k=1, n+1, v[k]=if(k<=4, u[k], 5*v[k-1] - 4*v[k-2] + v[k-3] + 1)); v[n+1] \\ Jianing Song, Feb 04 2019

a(n) = A012855(n+3) - 1. a(n) = 6*a(n-1) - 9*a(n-2) + 5*a(n-3) - a(n-4). - R. J. Mathar, Oct 18 2010
G.f.: x/(1 - 6*x + 9*x^2 - 5*x^3 + x^4). - Colin Barker, Feb 03 2012
From Jianing Song, Feb 04 2019: (Start)
a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n) + 1.
a(n) = Sum_{k=0..n} A012814(k) = Sum_{k=0..n} A000931(5*k+2). (End)

New name, more terms and a(0) = 0 prepended by Jianing Song, Feb 04 2019

A248049 a(n) = (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4) with a(0) = 2, a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

2, 1, 1, 1, 2, 6, 24, 240, 3960, 184800, 33033000, 26125799700, 219429008298500, 31064340573760168675, 206377779224083011749949745, 245390990689739612867279321757020455, 230795626149641527446533813473152766756062242744

Offset: 0

Michael Somos, Sep 30 2014

nonn

It seems that degrees of factors when using [2,1,1,y] as initial condition are given by A233522. - F. Chapoton, May 21 2020
It seems also that degrees (w.r.t. x) of factors when using [2,1,x,y] as initial condition are given by A247907. - F. Chapoton, Jan 03 2021
Somos conjectures that log(a(n)) ~ 1.25255*c^n, where c = A060006. - Bill McEachen, Oct 11 2022

Robert Israel, Table of n, a(n) for n = 0..26
Math Stack Exchange, Is OEIS A248049 an Integer Sequence

Cf. A233522, A060006.

a[0]:= 2: a[1]:= 1: a[2]:= 1: a[3]:= 1:
for n from 4 to 20 do
a[n] := (a[n-1] + a[n-2]) * (a[n-2] + a[n-3]) / a[n-4]
od:
seq(a[i],i=0..20); # Robert Israel, Mar 18 2020

{a(n) = if( n<0, n=4-n); if( n<4, (n==0)+1, (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4))};

a(n) = a(4-n) for all n in Z.

a(n) * a(n+4) = (a(n+1) + a(n+2)) * (a(n+2) + a(n+3)) for all n in Z.

A254232 Product of Perrin numbers A001608(2) * ... * A001608(n).

Original entry on oeis.org

2, 6, 12, 60, 300, 2100, 21000, 252000, 4284000, 94248000, 2733192000, 106594488000, 5436318888000, 369669684384000, 33270271594560000, 3959162319752640000, 625547646520917120000, 130739458122871678080000, 36214829900035454828160000

Offset: 2

Vaclav Kotesovec, Jan 27 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..127

Cf. A001608, A003266, A060006, A126772, A135407, A254231.

Table[Product[SeriesCoefficient[(3-x^2)/(1-x^2-x^3),{x,0,k}],{k,2,n}], {n,2,20}]

A254232_list, a, b, c, d = [2], 3, 0, 2, 2
for _ in range(200):
    a, b, c = b, c, a+b
    d *= c
    A254232_list.append(d) # Chai Wah Wu, Jan 28 2015

a(n) ~ c * r^(n*(n+1)/2), where r = A060006 = 1.324717957244746025960908854478... is the root of the equation r^3 = r + 1, c = 0.81845731383668335747954234022593868885066763327809025622515304041339344876... .

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.

A329974 Beatty sequence for the real solution x of 1/x + 1/(1+x+x^2) = 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87

Offset: 1

Clark Kimberling, Jan 02 2020

Let x be the real solution of 1/x + 1/(1+x+x^2) = 1. Then (floor(n*x)) and (floor(n*(x^2 + x + 1))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A329825, A060006, A329975 (complement).

Solve[1/x + 1/(1 + x + x^2) == 1, x]
u = 1/3 (27/2 - (3 Sqrt[69])/2)^(1/3) + (1/2 (9 + Sqrt[69]))^(1/3)/3^(2/3);
u1 = N[u, 150]
RealDigits[u1, 10][[1]]  (* A060006 *)
Table[Floor[n*u], {n, 1, 50}]              (* A329974 *)
Table[Floor[n*(1 + u + u^2)], {n, 1, 50}]  (* A329975 *)
Plot[1/x + 1/(1 + x + x^2) - 1, {x, -2, 2}]

a(n) = floor(n*x), where x = 1.324717... is the constant in A060006.

cp's OEIS Frontend

A333181 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

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A347178 Decimal expansion of imaginary part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

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A374002 Decimal expansion of the positive real root of x^6 - 2*x^5 + x^4 - x^2 + x - 1.

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Mathematica

A374003 Decimal expansion of the positive real root of x^8 - x^7 - x^6 + x^2 - 1.

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Mathematica

A072117 Continued fraction expansion of smallest Pisot-Vijayaraghavan number (positive root of x^3 = x + 1 ).

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PARI

Extensions

A176476 Partial sums of A012814.

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A248049 a(n) = (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4) with a(0) = 2, a(1) = a(2) = a(3) = 1.

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Maple

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A254232 Product of Perrin numbers A001608(2) * ... * A001608(n).

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A333181 G.f.: Sum_{k>=1} (k * x^(k(k+1)) Product_{j=1..k} (1 + x^j)).

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