A230162 -id:A230162 - cp's OEIS frontend

A103376 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n-8) + a(n-9).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 257, 265, 293

Offset: 1

Jonathan Vos Post, Feb 05 2005

k=8 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), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, k=5 case is A103373, k=6 case is A103374 and k=7 case is A103375.
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=8 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^9 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0850702454914508283368958640973142340506536310308 = A230162. Note that x = (1 + x)^(1/9) = (1 + (1 + (1 + ...)^(1/9))^(1/9))^(1/9).
The sequence of prime values in this k=8 case is A103386; The sequence of semiprime values in this k=8 case is A103396.

			a(93) = 1200 because a(93) = a(93-8) + a(93-9) = a(85) + a(84) = 642 + 558.

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

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 (0,0,0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103372-A103375, A103377-A103380, A103386, A103396.

k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 76]
LinearRecurrence[{0,0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1,1},80] (* Harvey P. Dale, May 07 2015 *)

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0,0]^(n-1)*[1;1;1;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)/(1-x^8-x^9). - R. J. Mathar, Dec 14 2009

a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1, a(9)=1, a(n)=a(n-8)+a(n-9). - Harvey P. Dale, May 07 2015

Edited by Ray Chandler, Feb 10 2005

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

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 *)

cp's OEIS Frontend

A103376 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n-8) + a(n-9).

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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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Author

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Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica