A160155 -id:A160155 - cp's OEIS frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613, 4163, 4824, 5684, 6697, 7776

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=4 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1) and k=3 case is A079398 (offset so as to begin 1,1,1,1).
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
For this k=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0, A160155.
The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.

			a(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.

Zanten, A. J. van, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

Indranil Ghosh, Table of n, a(n) for n = 1..14857
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988), 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1).

Cf. A000931, A079398, A103373-A103380, A103382, A103392.

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]
LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},70] (* Harvey P. Dale, Apr 22 2015 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,1,0,0,0]^(n-1)*[1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f. -x*(1+x)*(1+x^2) / ( -1+x^4+x^5 ). - R. J. Mathar, Aug 26 2011

a(n) = A124789(n-2)+A124798(n-1). - R. J. Mathar, Jun 30 2020

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A060007 Decimal expansion of the positive real root of x^4 - x - 1.

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Mar 14 2001

Original name: Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
v_2 = A001622 - 1. [Corrected by M. F. Hasler, Jul 12 2025]
v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - M. F. Hasler, Jul 12 2025
A Perron number of the 4th degree polynomial (see Boys and Wu). - R. J. Mathar, Mar 19 2011
This number is not ruler-and-compass constructible because x^4-x-1 and its resolvent x^3+4x+1 are irreducible over the rationals. - Jean-François Alcover, Aug 31 2015
The other (negative) real root -0.724491959... is -A356032. The first of the pair of complex conjugate roots is obtained by negating in the formula for v_4 below sqrt(2*u) and sqrt(u), giving -0.2481260628... - 1.0339820609...*i. - Wolfdieter Lang, Aug 27 2022
The sequence a(n) = v_4^((n^2-n)/2) satisfies the Somos-4 recursion a(n+2)*a(n-2) = a(n+1)*a(n-1) + a(n)^2 for all n in Z. - Michael Somos, Mar 24 2023

			v_4 = 1.220744084605759475361685349...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
F. Rothelius, Formulae
Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394
Index entries for algebraic numbers, degree 4

Cf. A001622 (golden ratio, root of x^2 - x - 1), A060006 (plastic number, root of x^3 - x - 1), A202540 (log thereof), A160155 (root of x^5 - x - 1), A356032 (root of x^4 + x - 1), A006720, A298813.

r:=(108+12*sqrt(849))^(1/3): (sqrt(12/sqrt(-8/r+r/6)+48/r-r) + sqrt(-48/r+r))/(2*sqrt(6)): evalf(%,105); # Vaclav Kotesovec, Oct 12 2013

RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 11 2012 *)
Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 04 2013 *)

default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* Michael Somos, Mar 22 2023 */

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - Ilya Gutkovskiy, Dec 15 2017
v_4 = (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)), ep = (-1 + sqrt(3)*i)/2 and i = sqrt(-1).
For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - Wolfdieter Lang, Aug 27 2022
Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - Antonio Graciá Llorente, Dec 13 2024
Equals exp(A202540) = sqrt(A298813). - Hugo Pfoertner, Dec 14 2024

More terms from Benoit Cloitre, Jan 11 2003

Simplified definition from M. F. Hasler, Jul 12 2025

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 *)

A358939 Decimal expansion of the real root of x^5 + x^3 - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Dec 15 2022

The other (complex) roots are 0.217853219392291296... + 1.16695124566484991...*i and -0.636663106805772389... + 0.664701565064356279...*i, together with their conjugates.

			0.83761977482696218499752729419180609392505451858960237912530556912378529...

Cf. A160155, A230152, A358940, A358941, A358942.

RealDigits[x /. FindRoot[x^5 + x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

A358940 Decimal expansion of the real root of x^5 - x^3 - 1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Dec 12 2022

The other (complex) roots are 0.340794866197006415... + 0.785423103049449080...*i and -0.959047717892755927... + 0.428365956254189316...*i, and their conjugates.

			1.23650570339149902433757480097614678268104294354611496776617384170726143...

Cf. A160155, A230152, A358939, A358941, A358942.

RealDigits[x /. FindRoot[x^5 - x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

A358941 Decimal expansion of the real root of x^5 + x^2 - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Dec 15 2022

The other (complex) roots are 0.464912201602897854... + 1.07147384027026940...*i, and -0.869277501842593861 + 0.388269406599740355...*i, and their conjugates.

Cf. A160155, A230152, A358939, A358940, A358942.

RealDigits[x /. FindRoot[x^5 + x^2 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

solve(x=0, 1, x^5 + x^2 - 1) \\ Michel Marcus, Dec 19 2022

0.808730600479392013738554526511400064951377351559313075548116401836543340...

cp's OEIS Frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

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A060007 Decimal expansion of the positive real root of x^4 - 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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