A060007 -id:A060007 - cp's OEIS frontend

A017817 a(n) = a(n-3) + a(n-4), with a(0)=1, a(1)=a(2)=0, a(3)=1.

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 4, 6, 5, 6, 10, 11, 11, 16, 21, 22, 27, 37, 43, 49, 64, 80, 92, 113, 144, 172, 205, 257, 316, 377, 462, 573, 693, 839, 1035, 1266, 1532, 1874, 2301, 2798, 3406, 4175, 5099, 6204, 7581, 9274, 11303, 13785, 16855, 20577, 25088

Offset: 0

N. J. A. Sloane

Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=3, I={0,1}. - Vladimir Baltic, Mar 07 2012
Number of compositions (ordered partitions) of n into parts 3 and 4.
For n>=2, a(n-2) is the number of ways to tile the 1xn board with dominoes and squares (ie. monominoes) such that there are either one or two squares between dominoes, no squares at either end of the board, and there is at least one domino. - Enrique Navarrete, Sep 01 2024
For n>=3, a(n-3) is the number of ways to tile the 1xn board with triominoes (ie. size 1x3) and squares (ie. size 1x1) such that there are either none or one squares between triominoes, no squares at either end of the board, and there is at least one triomino. - Enrique Navarrete, Sep 07 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 484
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
E. Wilson, The Scales of Mt. Meru
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).

A003269(n) = a(-4-n)(-1)^n.

a:=[1,0,0,1];; for n in [5..60] do a[n]:=a[n-3]+a[n-4]; od; a; # G. C. Greubel, Mar 05 2019

I:=[1,0,0,1]; [n le 4 select I[n] else Self(n-3) +Self(n-4): n in [1..60]]; // G. C. Greubel, Mar 05 2019

LinearRecurrence[{0,0,1,1}, {1,0,0,1}, 60] (* G. C. Greubel, Mar 05 2019 *)

a(n)=polcoeff(if(n<0,(1+x)/(1+x-x^4),1/(1-x^3-x^4)) +x*O(x^abs(n)), abs(n))

(1/(1-x^3-x^4)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 05 2019

G.f.: 1/(1-x^3-x^4).
a(n)/a(n-1) tends to A060007. - Gary W. Adamson, Oct 22 2006
a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k). - Seiichi Manyama, Mar 06 2019

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

A160155 Decimal expansion of the one real root of x^5-x-1.

Original entry on oeis.org

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

Offset: 1

Harry J. Smith, May 03 2009

The other (complex) roots are 0.181232444469875383... + 1.08395410131771066...*i, and -0.764884433600584726... + 0.352471546031726249...*i, together with their complex conjugates. - Wolfdieter Lang, Dec 15 2022

This quintic is in some sense the smallest and/or simplest algebraic equation for which there is no explicit expression for the roots. (The "equivalent" quintic x^5 - x + 1 has the opposite real root, x = -1.1673..., while x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1).) - M. F. Hasler, Jul 12 2025

			1.16730397826141868425604589985484218072056037152548903914008244927565...

Harry J. Smith, Table of n, a(n) for n = 1..20000
David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
Wikipedia, Abel-Ruffini theorem, created Nov. 27, 2002, retrieved July 12, 2025.
Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394.
Index entries for algebraic numbers, degree 5

Cf. A039922 (continued fraction), A001622 (golden ratio phi = root of x^2 - x - 1), A060006 (plastic constant, root of x^3 - x - 1), A060007 (root of x^4 - x - 1).

RealDigits[Root[x^5-x-1, x, 1], 10, 105] // First (* Jean-François Alcover, Jul 09 2015 *)

localprec(20080); r=real(polroots('x^5 - 'x - 1)[1]); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b160155.txt", n, " ", d)) \\ Edited by M. F. Hasler, Jul 12 2025

polrootsreal(x^5-x-1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/5))^(1/5))^(1/5))^(1/5))^(1/5). - Ilya Gutkovskiy, Dec 15 2017

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

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 *)

A298813 Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 13 2018

Let (d(n)) = (1,0,1,0,1,0,1,...), s(n) = sqrt(s(n-1) + d(n)) for n > 0, and s(0) = 1.
Then s(2n) -> 1.49021612009995..., as in A298813;
and s(2n+1) -> 1.22074408..., as in A060007.
Let (e(n)) = (0,1,0,1,0,1,0,...), t(n) = sqrt(t(n-1) + e(n)) for n > 0, and t(0) = 1.
Then t(2n) -> 1.22074408..., as in A060007;
and t(2n+1) -> 1.49021612009995..., as in A298813.
The four solutions are: x1, this one; x2, the least A072223; and the two complex ones x3=-1.007552359378... + 0.513115795597...*i and x4, its complex conjugate; Re(x3) = Re(x4) = -(x1+x2)/2; Im(x3) = -Im(x4) = sqrt(1/(x1*x2) - Re(x3)^2). - Andrea Pinos, Sep 20 2023

			1.49021612009995...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A060007, A072223, A298814, A298815.

r = x /. NSolve[x^4 - 2 x^2 - x + 1 == 0, x, 100][[4]];
RealDigits[r][[1]]; (* A298813 *)
RealDigits[Root[x^4-2x^2-x+1,2],10,120][[1]] (* Harvey P. Dale, May 02 2022 *)

solve(x=1, 2, x^4 - 2*x^2 - x + 1) \\ Michel Marcus, Nov 05 2018

Equals sqrt((1 + 2*cos(arccos(155/128)/3))/3) + sqrt(2/3 - 2*cos(arccos(155/128)/3)/3 + sqrt(3/(1 + 2*cos(arccos(155/128)/3)))/4). - Vaclav Kotesovec, Sep 21 2023

Equals sqrt(1/3 + s/9 + 1/s) + sqrt(2/3 - s/9 - 1/s + 1 / (4 * sqrt(1/3 + s/9 + 1/s))) where s = (4185/128 + sqrt(5570289/16384))^(1/3). - Michal Paulovic, Dec 30 2023

A356032 Decimal expansion of the positive real root of x^4 + x - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Aug 27 2022

The other real (negative) root is -A060007.
One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

			r = 0.724491959000515611588372282187036565786494481350011017270...

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)

solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022

polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).

Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025

A337571 Decimal expansion of the real positive solution to x^4 = x+4.

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Sep 01 2020

x = (4 + (4 + (4 + ... )^(1/4))^(1/4))^(1/4).

The negative value (-1.5337511687...) is the real negative solution to x^4 = 4-x.

			1.5337511687552...

Index entries for algebraic numbers, degree 4

Cf. A337570, A060007, A294644.

format long; solve('x^4-x-4=0'); ans(1), (eval(ans))

RealDigits[x /. FindRoot[x^4 - x - 4, {x, 1}, WorkingPrecision -> 100], 10, 90][[1]] (* Amiram Eldar, Sep 03 2020 *)

PARI
```
solve(n=0,2,n^4-n-4)
```
PARI
```
polroots(n^4-n-4)[2]
```

polrootsreal(n^4-n-4)[2] \\ Charles R Greathouse IV, Oct 27 2023

Equals sqrt(sqrt(1/s) - s/16) + sqrt(s/16) where s = (sqrt(16804864/27) + 32)^(1/3) - (sqrt(16804864/27) - 32)^(1/3). [Simplified by Michal Paulovic, Jun 22 2021]

cp's OEIS Frontend

A017817 a(n) = a(n-3) + a(n-4), with a(0)=1, a(1)=a(2)=0, a(3)=1.

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A160155 Decimal expansion of the one real root of x^5-x-1.

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

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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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