A000931 -id:A000931 - cp's OEIS frontend

A103377 a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

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

Offset: 1

Jonathan Vos Post, Feb 15 2005

k=9 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, k=7 case is A103375, k=8 case is A103376, k=10 case is A103378 and k=11 case is A103379. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=9 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^10 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0757660660868371580595995241652758206925302476392 = A230163. Note that x = (1 + x)^(1/10) = (1 + (1 + (1 + ...)^(1/10))^(1/10))^(1/10). The sequence of prime values in this k=9 case is A103387; The sequence of semiprime values in this k=9 case is A103397.

In analogy to the Fibonacci sequence, one might prefer to start this sequence with offset 0. - M. F. Hasler, Sep 19 2015

			a(83) = 257 because a(83) = a(83-9) + a(83-10). a(74) + a(73) = 129 + 128. This sequence has as elements 5, 17 and 257, which are all Fermat Primes.

A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (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) 291-306
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61(1988)1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1,1).

Cf. A000045, A000931, A079398, A103372-A103376, A103378-A103380, A103387, A103397.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* Charles R Greathouse IV, Jan 11 2013 *)

Vec((1+x+x^2)*(1+x^3+x^6)/(1-x^9-x^10)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 1 and for n>10: a(n) = a(n-9) + a(n-10).

O.g.f.: -x*(x^2+x+1)*(x^6+x^3+1)/(-1+x^9+x^10). - R. J. Mathar, May 02 2008

Edited by R. J. Mathar, May 02 2008

Edited by M. F. Hasler, Sep 19 2015

A099098 Quadrisection of a Padovan sequence.

Original entry on oeis.org

1, 1, 4, 12, 37, 114, 351, 1081, 3329, 10252, 31572, 97229, 299426, 922111, 2839729, 8745217, 26931732, 82938844, 255418101, 786584466, 2422362079, 7459895657, 22973462017, 70748973084, 217878227876, 670976837021, 2066337330754

Offset: 0

Paul Barry, Sep 29 2004

Quadrisection of sequence with g.f. 1/(1-x^2-x^3), or A000931(n+3).

			1 + x + 4*x^2 + 12*x^3 + 37*x^4 + 114*x^5 + 351*x^6 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,3,1).

Bisection of A005251.

LinearRecurrence[{2,3,1},{1,1,4},40] (* Harvey P. Dale, Aug 23 2011 *)

G.f.: (1-x-x^2)/(1-2x-3x^2-x^3);
a(n)=sum{k=0..2n, binomial(k, 4n-2k)};
a(n)=2a(n-1)+3a(n-2)+a(n-3);
a(n)=A000931(4n+3).
a(n) = Sum [k=0..n, C(2n-k, 2k) ].

A104769 Expansion of g.f. -x/(1+x-x^3).

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -2, 2, -1, -1, 3, -4, 3, 0, -4, 7, -7, 3, 4, -11, 14, -10, -1, 15, -25, 24, -9, -16, 40, -49, 33, 7, -56, 89, -82, 26, 63, -145, 171, -108, -37, 208, -316, 279, -71, -245, 524, -595, 350, 174, -769, 1119, -945, 176, 943, -1888, 2064, -1121, -767, 2831, -3952

Offset: 0

Creighton Dement, Mar 24 2005

Generating floretion is "jesright".

Pisano period lengths: 1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168,.. (which differs from A104217 for example at index 23). - R. J. Mathar, Aug 10 2012

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1).

Apart from signs, essentially the same as A050935 and A078013.
Cf. A247917 (negative).
Cf. A000931, A057597.

LinearRecurrence[{-1, 0, 1}, {0, -1, 1}, 61] (* or *)
CoefficientList[Series[-x/(1 + x - x^3), {x, 0, 60}], x] (* Michael De Vlieger, Jul 02 2021 *)

a(n)=([0,1,0;0,0,1;1,0,-1]^n*[0;-1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

a(n) = -A247917(n-1).
Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 0, a(1) = -1, a(2) = 1.
a(n+1) - a(n) = ((-1)^(n+1))*a(n+5).
a(n) = ((-1)^n)*A050935(n+1) = ((-1)^n)*A078013(n+2).
a(n) = A104771(n) - A104770(n).

Edited by Ralf Stephan, Apr 05 2009

A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 240, 1680, 15120, 181440, 2903040, 60963840, 1706987520, 63158538240, 3094768373760, 201159944294400, 17299755209318400, 1972172093862297600, 297797986173206937600, 59559597234641387520000

Offset: 1

John Lien, Feb 17 2007

Ian Stewart, Tales of a Neglected Number
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Eric Weisstein's World of Mathematics, Padovan Sequence
E. Wilson, The Scales of Mt. Meru

Cf. A000931, A003266, A060006, A253924.

From R. J. Mathar, Sep 14 2010: (Start)
A000931 := proc(n) option remember; if n = 0 then 1; elif n <=2 then 0; else procname(n-2)+procname(n-3) ; end if; end proc:
A126772 := proc(n) mul( A000931(i),i=5..n+4) ; end proc: seq(A126772(n),n=1..40) ; (End)

Rest[FoldList[Times,1,LinearRecurrence[{0,1,1},{1,1,1},30]]] (* Harvey P. Dale, Apr 29 2013 *)

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = 1.324717957244746... (see A060006) is the root of the equation r^3 = r + 1, d = 0.393641282401116385386658448446561... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0, c = 1.25373683131537208838997864311903035079685338006712312402418098138010834953... (see A253924). - Vaclav Kotesovec, Jan 26 2015

More terms from R. J. Mathar, Sep 14 2010

A203181 T(n,k) is the number of n X k 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 4, 7, 4, 3, 4, 6, 17, 17, 6, 4, 5, 10, 41, 59, 41, 10, 5, 7, 18, 97, 205, 205, 97, 18, 7, 9, 30, 235, 724, 952, 724, 235, 30, 9, 12, 50, 607, 2466, 4654, 4654, 2466, 607, 50, 12, 16, 86, 1415, 8948, 23083, 32411, 23083, 8948, 1415, 86, 16, 21, 146

Offset: 1

R. H. Hardin, Dec 30 2011

Table starts
.1..1...2....2......3.......4........5.........7..........9..........12
.1..1...2....4......6......10.......18........30.........50..........86
.2..2...7...17.....41......97......235.......607.......1415........3486
.2..4..17...59....205.....724.....2466......8948......30945......108083
.3..6..41..205....952....4654....23083....115377.....551208.....2757161
.4.10..97..724...4654...32411...223567...1625772...10889470....76035931
.5.18.235.2466..23083..223567..2208945..22411843..216858412..2141041521
.7.30.607.8948.115377.1625772.22411843.323885934.4389100997.61921804090

			Some solutions for n=5 k=3
..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..0....0..1..2
..1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1
..2..1..2....2..1..0....2..1..2....0..1..0....0..1..2....0..1..2....0..1..0
..0..1..2....0..1..2....0..1..0....1..2..1....1..2..0....1..2..0....1..2..1
..1..0..1....1..0..1....1..0..1....2..0..1....0..1..1....0..1..1....2..0..1

R. H. Hardin, Table of n, a(n) for n = 1..364

Column 1 is A000931(n+5). Column 2 is A203175.

A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5a(n-1) - 4a(n-2) + a(n-3).

Original entry on oeis.org

0, 1, 1, 1, 2, 7, 28, 114, 465, 1897, 7739, 31572, 128801, 525456, 2143648, 8745217, 35676949, 145547525, 593775046, 2422362079, 9882257736, 40315615410, 164471408185, 670976837021, 2737314167775, 11167134898976

Offset: 0

N. J. A. Sloane

nonn

Old name was "Take every 5th term of Padovan sequence A000931".

Lim_{n -> infinity} a(n+1)/a(n) = p^5 = 4.0795956..., where p is the plastic constant (A060006). - Jianing Song, Feb 04 2019

Index entries for linear recurrences with constant coefficients, signature (5, -4, 1).

Cf. A000931, A012814, A012864, A060006, A176476.

A012855 := proc(n,A,B,C) option remember; if n = 0 then A elif n = 1 then B elif n = 2 then C else 5*procname(n-1,A,B,C)-4*procname(n-2,A,B,C)+procname(n-3,A,B,C); fi; end; [ seq(A012855(i,0,1,1),i = 0..40) ]; # R. J. Mathar, Dec 30 2011

CoefficientList[Series[(4x^2-x)/(x^3-4x^2+5x-1),{x,0,40}],x] (* or *) LinearRecurrence[{5,-4,1},{0,1,1},40] (* Harvey P. Dale, Mar 28 2013 *)

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

a(n) = A000931(5*n-12) for n >= 3. - Alois P. Heinz, Feb 04 2019
G.f. (4x^2 - x)/(x^3 - 4x^2 + 5x - 1). For n > 2, a(n) = 1 + Sum_{k=0..n-3} A012814(k). - Ralf Stephan, Jan 15 2004
a(n) = 1 + A176476(n-3) = 1 + Sum_{k=0..n-3} A000931(5*k+2) for n >= 3. - Jianing Song, Feb 04 2019

Edited by N. J. A. Sloane, Feb 06 2019 at the suggestion of Jianing Song, replacing imprecise definition with formula from Harvey P. Dale, Mar 28 2013

A017818 Expansion of 1/(1-x^3-x^4-x^5).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 3, 3, 4, 6, 8, 10, 13, 18, 24, 31, 41, 55, 73, 96, 127, 169, 224, 296, 392, 520, 689, 912, 1208, 1601, 2121, 2809, 3721, 4930, 6531, 8651, 11460, 15182, 20112, 26642, 35293, 46754, 61936, 82047

Offset: 0

N. J. A. Sloane

Compositions of n into parts 3, 4, and 5. - David Neil McGrath, Jul 28 2014

The number of ways a T2 triangle can cover a row length of T1(n) triangles. - Craig Knecht, Mar 06 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, Predators and altruists arriving on jammed Riviera, arXiv:2401.01225 [math.CO], 2024. See p. 14.
Craig Knecht, The number of ways a T2 triangle can cover a row of T1 triangles.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^3-x^4-x^5))); // Vincenzo Librandi, Jun 27 2013

I:=[1,0,0,1,1]; [n le 5 select I[n] else Self(n-3)+Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Jun 27 2013

CoefficientList[Series[1 / (1 - x^3 - x^4 - x^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 27 2013 *)
LinearRecurrence[{0,0,1,1,1},{1,0,0,1,1},50] (* Harvey P. Dale, Oct 03 2020 *)

a(n) = (1/10)*(2*A001608(n) + 2*A000931(n+2) + (-1)^floor(n/2) - 3(-1)^floor((n-1)/2)). - Ralf Stephan, Jun 09 2005
a(n) = a(n-5) + a(n-4) + a(n-3). - Jon E. Schoenfield, Aug 07 2006
a(2n+3) = A060945(n). - Yasuyuki Kachi, Jul 06 2024

A047350 Numbers that are congruent to {1, 2, 4} mod 7.

Original entry on oeis.org

1, 2, 4, 8, 9, 11, 15, 16, 18, 22, 23, 25, 29, 30, 32, 36, 37, 39, 43, 44, 46, 50, 51, 53, 57, 58, 60, 64, 65, 67, 71, 72, 74, 78, 79, 81, 85, 86, 88, 92, 93, 95, 99, 100, 102, 106, 107, 109, 113, 114, 116, 120, 121, 123, 127, 128, 130, 134, 135, 137, 141

Offset: 1

N. J. A. Sloane

a(n+1) = a(n) + (a(n) mod 7). - Ben Paul Thurston, Jan 09 2008
Also defined by: a(1)=1, and a(n) = smallest number larger than a(n-1) such that a(n)^3 - a(n-1)^3 is divisible by 7. - Zak Seidov, Apr 21 2009
Union of A047353 and A017029. - R. J. Mathar, Apr 28 2009
Indices of the even numbers in the Padovan sequence. - Francesco Daddi, Jul 31 2011
Euler's problem (see Link lines, English translation by David Zao): Finding the values of a so that the form a^3-1 is divisible by 7. The three residuals that remain after the division of any square by 7 are 1, 2 and 4. Hence the values are 7n+1, 7n+2, 7n+4. - Bruno Berselli, Oct 24 2012

Leonhard Euler, The Euler Archive, Theoremata circa divisores numerorum (E134), Novi Commentarii academiae scientiarum imperialis Petropolitanae, Volume 1 (1750), p. 40 (Theorem II, example 2).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A000931, A017029, A047353, A134720.

[n : n in [0..150] | n mod 7 in [1, 2, 4]]; // Wesley Ivan Hurt, Jun 13 2016

A047350:=n->(21*n-21-6*cos(2*n*Pi/3)+4*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047350(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016

Select[Range[0, 150], MemberQ[{1, 2, 4}, Mod[#, 7]] &] (* Wesley Ivan Hurt, Jun 13 2016 *)

a(n)=n\3*7+[-3,1,2][n%3+1] \\ Charles R Greathouse IV, Jul 31 2011

From R. J. Mathar, Apr 28 2009: (Start)
G.f.: x*(1 + x + 2*x^2 + 3*x^3)/((1 + x + x^2)*(x-1)^2).
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
a(n) = a(n-3) + 7 for n > 3. (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = (21*n - 21 - 6*cos(2*n*Pi/3) + 4*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 7k-3, a(3k-1) = 7k-5, a(3k-2) = 7k-6. (End)
a(n) = 4*n - 3 - 2*floor(n/3) - 3*floor((n+1)/3). - Ridouane Oudra, Nov 23 2022

A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 512

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=7 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 and k=6 case is A103374.
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=7 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^8 - x - 1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059....
The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395.

			a(30) = 12 because a(30) = a(30-7) + a(30-8) = a(24) + a(23) = 7 + 5 = 12.

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

Cf. A000045, A000931, A079398, A103372-A103374, A103376-A103380, A103385, A103395.

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 73]
LinearRecurrence[{0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1},80]

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

G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x^7+x^8). - R. J. Mathar, Dec 14 2009

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

Corrected (one more 8 inserted) by R. J. Mathar, Dec 14 2009

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).

Original entry on oeis.org

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

cp's OEIS Frontend

A103377 a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

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A099098 Quadrisection of a Padovan sequence.

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A104769 Expansion of g.f. -x/(1+x-x^3).

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A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

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A203181 T(n,k) is the number of n X k 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

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A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3).

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A017818 Expansion of 1/(1-x^3-x^4-x^5).

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A012855 a(0) = 0, a(1) = 1, a(2) = 1; thereafter a(n) = 5a(n-1) - 4a(n-2) + a(n-3).