A079262 -id:A079262 - cp's OEIS frontend

A253705 Indices of primes in the 8th-order Fibonacci number sequence, A079262.

9, 17, 25, 125, 350, 1322, 108935, 199528

Offset: 1

Robert Price, Jan 09 2015

a(9) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413, A060455, A079262.

a={0,0,0,0,0,0,0,1}; step=8; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Flatten[Position[LinearRecurrence[{1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,1},200000],?PrimeQ]]-1 (* The program takes a long time to run *) (* _Harvey P. Dale, Apr 26 2018 *)

lista(nn) = {gf = x^7/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8); for (n=0, nn, if (isprime(polcoeff(gf+O(x^(n+1)), n)), print1(n, ", ")););} \\ Michel Marcus, Jan 12 2015

A104415 Number of prime factors, with multiplicity, of the nonzero octanacci numbers A079262.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 3, 1, 4, 5, 6, 6, 11, 7, 9, 1, 4, 3, 6, 8, 7, 8, 11, 10, 2, 2, 8, 4, 9, 7, 11, 11, 12, 3, 2, 4, 5, 6, 9, 10, 11, 12, 2, 4, 10, 5, 10, 9, 17, 12, 10, 4, 4, 4, 9, 11, 8, 8, 12, 12, 4, 4, 10, 11, 9, 11, 15, 13, 9, 5, 6, 5, 9, 6, 9, 9

Offset: 1

Jonathan Vos Post, Mar 06 2005

			a(0)=a(1)=0 because the first two nonzero octanacci numbers are both 1, which has zero prime divisors.
a(2)=1 because the 3rd nonzero octanacci number is 2, a prime, with only one prime divisor.
a(3)=2 because the 4th nonzero octanacci number is 4 = 2^2 which has (with multiplicity) 2 prime divisors (which happen to be equal).
a(4)=3 because the 5th nonzero octanacci number is 8 = 2^3.
a(10)=3 because A079262(10) = 255 = 3 * 5 * 17 which has 3 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..262

Cf. A001222, A079262, A104411, A104412, A104413, A104414.

a(n) = A001222(A079262(n+6)).

Offset corrected and more terms added by Amiram Eldar, Sep 08 2019

A253706 Primes in the 8th-order Fibonacci numbers A079262.

Original entry on oeis.org

2, 509, 128257, 133294824621464999938178340471931877, 4596852049500861351052672455121859744010232939954169259264638023409631672658340253083284317818242062413

Offset: 1

Robert Price, Jan 09 2015

nonn

a(6) is too large to display here. It has 395 digits and is the 1322nd term in A079262.

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705.

a={0,0,0,0,0,0,0,1}; step=8; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst

lista(nn) = {gf = x^7/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8); for (n=0, nn, if (isprime(p=polcoeff(gf+O(x^(n+1)), n)), print1(p, ", ")););} \\ Michel Marcus, Jan 12 2015

A092921 Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 2, 1, 1, 0, 1, 5, 4, 2, 1, 1, 0, 1, 8, 7, 4, 2, 1, 1, 0, 1, 13, 13, 8, 4, 2, 1, 1, 0, 1, 21, 24, 15, 8, 4, 2, 1, 1, 0, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 0, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 0, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 0

Offset: 0

Ralf Stephan, Apr 17 2004

For all k >= 1, the k-generalized Fibonacci number F(k,n) satisfies the recurrence obtained by adding more terms to the recurrence of the Fibonacci numbers.
The number of tilings of an 1 X n rectangle with tiles of size 1 X 1, 1 X 2, ..., 1 X k is F(k,n).
T(k,n) is the number of 0-balanced ordered trees with n edges and height k (height is the number of edges from root to a leaf). - Emeric Deutsch, Jan 19 2007
Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

			From _Peter Luschny_, Apr 03 2021: (Start)
Array begins:
                n = 0  1  2  3  4  5   6   7   8    9   10
  -------------------------------------------------------------
  [k=1, mononacci ] 0, 1, 1, 1, 1, 1,  1,  1,  1,   1,   1, ...
  [k=2, Fibonacci ] 0, 1, 1, 2, 3, 5,  8, 13, 21,  34,  55, ...
  [k=3, tribonacci] 0, 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, ...
  [k=4, tetranacci] 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...
  [k=5, pentanacci] 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ...
  [k=6]             0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ...
  [k=7]             0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ...
  [k=8]             0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, ...
  [k=9]             0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
Note that the first parameter in F(k, n) refers to rows, and the second parameter refers to columns. This is always the case. Only the usual naming convention for the indices is not adhered to because it is common to call the row sequences k-bonacci numbers. (End)
.
From _Peter Luschny_, Aug 12 2015: (Start)
As a triangle counting compositions of n with largest part k:
  [n\k]| [0][1] [2] [3] [4][5][6][7][8][9]
   [0] | [0]
   [1] | [0, 1]
   [2] | [0, 1,  1]
   [3] | [0, 1,  1,  1]
   [4] | [0, 1,  2,  1,  1]
   [5] | [0, 1,  3,  2,  1, 1]
   [6] | [0, 1,  5,  4,  2, 1, 1]
   [7] | [0, 1,  8,  7,  4, 2, 1, 1]
   [8] | [0, 1, 13, 13,  8, 4, 2, 1, 1]
   [9] | [0, 1, 21, 24, 15, 8, 4, 2, 1, 1]
For example for n=7 and k=3 we have the 7 compositions [3, 3, 1], [3, 2, 2], [3, 2, 1, 1], [3, 1, 3], [3, 1, 2, 1], [3, 1, 1, 2], [3, 1, 1, 1, 1]. (End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014
Eric S. Egge, Restricted permutations related to Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0109219 [math.CO], 2001.
Eric S. Egge, Restricted 3412-Avoiding Involutions, arXiv:math/0307050 [math.CO], 2003.
Eric S. Egge and Toufik Mansour, Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0203226 [math.CO], 2002.
Eric S. Egge and Toufik Mansour, 231-avoiding involutions and Fibonacci numbers, arXiv:math/0209255 [math.CO], 2002.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Abraham Flaxman, Aram W. Harrow, and Gregory B. Sorkin, Strings with Maximally Many Distinct Subsequences and Substrings, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
Ivan Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
H. Gabai, Generalized Fibonacci k-sequences, Fib. Quart., 8 (1970), 31-38.
R. Kemp, Balanced ordered trees, Random Structures and Alg., 5 (1994), pp. 99-121.
E. P. Miles jr., Generalized Fibonacci numbers and associated matrices, The Amer. Math. Monthly, 67 (1960) 745-752.
M. D. Miller, On generalized Fibonacci numbers, The Amer. Math. Monthly, 78 (1971) 1108-1109.
Michel Mollard, p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes, arXiv:2507.16387 [math.CO], 2025. See p. 3.
Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.
Hsin-Po Wang and Chi-Wei Chin, On Counting Subsequences and Higher-Order Fibonacci Numbers, arXiv:2405.17499 [cs.IT], 2024. See p. 2.

Columns converge to A166444: each column n converges to A166444(n) = 2^(n-2).
Rows 1-8 are (shifted) A057427, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
Essentially a reflected version of A048887.
See A048004 and A126198 for closely related arrays.
Cf. A066099.

F:= proc(k, n) option remember; `if`(n<2, n,
      add(F(k, n-j), j=1..min(k,n)))
    end:
seq(seq(F(k, d+1-k), k=1..d+1), d=0..12);  # Alois P. Heinz, Nov 02 2016
# Based on the above function:
Arow := (k, len) -> seq(F(k, j), j = 0..len):
seq(lprint(Arow(k, 14)), k = 1..10); # Peter Luschny, Apr 03 2021

F[k_, n_] := F[k, n] = If[n<2, n, Sum[F[k, n-j], {j, 1, Min[k, n]}]];
Table[F[k, d+1-k], {d, 0, 12}, {k, 1, d+1}] // Flatten (* Jean-François Alcover, Jan 11 2017, translated from Maple *)

F(k,n)=if(n<2,if(n<1,0,1),sum(i=1,k,F(k,n-i)))

T(m,n)=!!n*(matrix(m,m,i,j,j==i+1||i==m)^(n+m-2))[1,m] \\ M. F. Hasler, Apr 20 2018

F(k,n) = if(n==0,0, polcoeff(lift(Mod('x, Pol(vector(k+1,i, if(i==1,1,-1))))^(n+k-2)), k-1)); \\ Kevin Ryde, Jun 05 2020

# As a triangle of compositions of n with largest part k.
C = lambda n,k: Compositions(n, max_part=k, inner=[k]).cardinality()
for n in (0..9): [C(n,k) for k in (0..n)] # Peter Luschny, Aug 12 2015

F(k,n) = F(k,n-1) + F(k,n-2) + ... + F(k,n-k); F(k,1) = 1 and F(k,n) = 0 for n <= 0.
G.f.: x/(1-Sum_{i=1..k} x^i).
F(k,n) = 2^(n-2) for 1 < n <= k+1. - M. F. Hasler, Apr 20 2018
F(k,n) = Sum_{j=0..floor(n/(k+1))} (-1)^j*((n - j*k) + j + delta(n,0))/(2*(n - j*k) + delta(n,0))*binomial(n - j*k, j)*2^(n-j*(k+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960

Offset: 0

Jean Lefort (jlefort.apmep(AT)wanadoo.fr), Mar 07 2005

Sometimes called the Fibonacci 9-step numbers.
For n >= 8, this gives the number of integers written without 0 in base ten, the sum of digits of which is equal to n-7. E.g., a(11) = 8 because we have the 8 numbers: 4, 13, 22, 31, 112, 121, 211, 1111.
The offset for this sequence is fairly arbitrary. - N. J. A. Sloane, Feb 27 2009

T. D. Noe, Table of n, a(n) for n = 0..208
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1).

Cf. A000045, A000073, A000078, A001591, A001592, A066178, A079262 (Fibonacci n-step numbers).

Cf. A255529 (Indices of primes in this sequence).

for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i),i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i),i=0..floor((n-9)/10)):od:seq(k(n),n=0..50);a:=taylor((z^8-z^9)/(1-2*z+z^(10)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); # Richard Choulet, Feb 22 2010

a={1, 0, 0, 0, 0, 0, 0, 0, 0}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=9},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

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; 1,1,1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 16 2015

A104144(n,m=9)=(matrix(m,m,i,j,j==i+1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 22 2018

a(n) = Sum_{k=1..9} a(n-k) for n > 8, a(8) = 1, a(n) = 0 for n=0..7.
G.f.: x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9). - N. J. A. Sloane, Dec 04 2011
Another form of the g.f. f: f(z) = (z^8-z^9)/(1-2*z+z^(10)), then a(n) = Sum_((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i), i=0..floor((n-8)/10))-Sum_((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i), i=0..floor((n-9)/10)) with Sum_(alpha(i), i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010
From N. J. A. Sloane, Dec 04 2011: (Start)
Let b be the smallest root (in magnitude) of g(x) := 1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9, b = 0.50049311828655225605926845999420216157202861343888...
Let c = -b^8/g'(b) = 0.00099310812055463178382193226558248643030626601288701...
Then a(n) is the nearest integer to c/b^n. (End)

Edited by N. J. A. Sloane, Aug 15 2006 and Nov 11 2006
Incorrect formula deleted by N. J. A. Sloane, Dec 04 2011
Name edited by M. F. Hasler, Apr 22 2018

A123526 Octanacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 8, 15, 29, 57, 113, 225, 449, 897, 1793, 3578, 7141, 14253, 28449, 56785, 113345, 226241, 451585, 901377, 1799176, 3591211, 7168169, 14307889, 28558993, 57004641, 113783041, 227114497, 453327617, 904856058, 1806120905

Offset: 1

Danny Rorabaugh, Nov 10 2006

Robert Price, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1).

Cf. A000045, A000213, A000288, A000322, A000383, A060455, A079262.

Cf. A254412, A254413. Indices of primes and primes in this sequence.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( x*(1-x^2-2*x^3-3*x^4-4*x^5-5*x^6-6*x^7)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8) )); // G. C. Greubel, Mar 10 2021

m:=50; S:=series( x*(1-x^2-2*x^3-3*x^4-4*x^5-5*x^6-6*x^7)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8), x, m+1):
seq(coeff(S, x, j), j=1..m); # G. C. Greubel, Mar 10 2021

Module[{nn=8,lr},lr=PadRight[{},nn,1];LinearRecurrence[lr,lr,20]] (* Harvey P. Dale, Feb 04 2015 *)

Vec(x*(1-x^2-2*x^3-3*x^4-4*x^5-5*x^6-6*x^7)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8) + O(x^50)) \\ Colin Barker, Oct 19 2015

@CachedFunction
def A123526(n):
    if (n<9): return 1
    else: return sum(A(n-j) for j in (1..8))
[A123526(n) for n in [1..50]] # G. C. Greubel, Mar 10 2021

a(n)=1 for 1 <= n <= 8, a(n) = a(n-1) + a(n-2) +...+ a(n-8) for n > 8.

G.f.: x*(1 -x^2 -2*x^3 -3*x^4 -4*x^5 -5*x^6 -6*x^7)/(1 -x -x^2 -x^3 -x^4 -x^5 -x^6 -x^7 -x^8). - Colin Barker, Oct 19 2015

A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521472, 1042432, 2083841, 4165637, 8327186, 16646200, 33276064, 66519472, 132973664, 265816832, 531372800, 1062224128

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 18 2006

nonn

The (1,10)-entry of the matrix M^n, where M is the 10 X 10 matrix {{0,1,0,0,0, 0,0,0,0,0},{0,0,1,0,0,0,0,0,0,0},{0,0,0,1,0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0}, {0,0,0,0,0,1,0,0,0,0},{0,0,0,0,0,0,1,0,0,0},{0,0,0,0,0,0,0,1,0,0},{0,0,0,0,0, 0,0,0,1,0},{0,0,0,0,0,0,0,0,0,1},{1,1,1,1,1,1,1,1,1,1}}.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1).

Cf. A000322, A001591, A001592, A079262.
Cf. A001591, A001592, A122189, A079262, A104144, A168082, A168083, A168084. - Richard Choulet, Feb 22 2010
Cf. A257227, A257228 for primes in this sequence.

with(linalg): p:=-1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9+x^10: M[1]:=transpose(companion(p,x)): for n from 2 to 40 do M[n]:=multiply(M[n-1],M[1]) od: seq(M[n][1,10],n=1..40);
k:=10:for n from 0 to 50 do l(n):=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))):od:seq(l(n),n=0..50);k:=10:a:=taylor((z^(k-1)-z^(k))/(1-2*z+z^(k+1)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); # Richard Choulet, Feb 22 2010

M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}; v[1] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 50}]
a={1,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]  (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=10},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

a(n) = Sum_{j=1..10} a(n-j) for n>=10; a(n) = 0 for 0<=n<=8, a(9) = 1 (follows from the minimal polynomial of M; a Maple program based on this recurrence relation is much slower than the given Maple program, based on the definition).
G.f.: -x^9/(-1+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
Another form of the g.f. f: f(z)=(z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=10. Then a(n)=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))) with k=10 and sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

Edited by N. J. A. Sloane, Oct 29 2006 and Mar 05 2011

A251672 8-step Fibonacci sequence starting with 0,0,0,0,0,0,1,0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 32, 64, 127, 254, 507, 1012, 2020, 4032, 8048, 16064, 32064, 64001, 127748, 254989, 508966, 1015912, 2027792, 4047536, 8079008, 16125952, 32187903, 64248058, 128241127, 255973288, 510930664, 1019833536, 2035619536

Offset: 0

Arie Bos, Dec 06 2014

a(n+8) equals the number of n-length binary words avoiding runs of 0's of lengths 8i+7, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1).

Other 8-step Fibonacci sequences are A079262, A105754, A251740, A251741, A251742, A251744, A251745.

LinearRecurrence[Table[1, {8}], {0, 0, 0, 0, 0, 0, 1, 0}, 43] (* Michael De Vlieger, Dec 09 2014 *)

a(n+8) = a(n) +a(n+1) +a(n+2) +a(n+3) +a(n+4) +a(n+5) +a(n+6) +a(n+7).
G.f.: x^6*(x-1)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8) . - R. J. Mathar, Mar 28 2025
a(n) = A079262(n+1)-A079262(n). - R. J. Mathar, Mar 28 2025

A107246 Sum of squares of octanacci numbers (Fibonacci 8-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, 345952, 1378208, 5490992, 21877296, 87163696, 347276080, 1383600944, 5512434480, 21962292529, 87500852554, 348615720590, 1388934122190, 5533708922574, 22047074027470

Offset: 0

Jonathan Vos Post, May 27 2005

Primes in this sequence include: a(8) = 2, a(17) = 280927. Semiprimes in this sequence include: a(9) = 6 = 2 * 3, a(10) = 22 = 2 * 11, a(11) = 86 = 2 * 43, a(13) = 1366 = 2 * 683, a(14) = 5462 = 2 * 2731, a(24) = 5512110374 = 2 * 2756055187, a(25) = 21961968423 = 3 * 7320656141, a(36) = 88177707994468342 = 2 * 44088853997234171.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 8, 14, 30, 60, 120, -266, -24, -38, -32, 120, -22, -50, -64, 136, 16, 30, 22, -68, 0, 10, 18, -28, 0, -6, -8, 14, 0, 0, -2, 2, 0, 0, 1, -1).

Cf. A079262, A107239-A107245, A107247-A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,1},40]^2] (* Harvey P. Dale, May 25 2014 *)
LinearRecurrence[{3, 2, 4, 8, 14, 30, 60, 120, -266, -24, -38, -32, 120, -22, -50, -64, 136, 16, 30, 22, -68, 0, 10, 18, -28, 0, -6, -8, 14, 0, 0, -2, 2, 0, 0, 1, -1},{0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, 345952, 1378208, 5490992, 21877296, 87163696, 347276080, 1383600944, 5512434480, 21962292529, 87500852554, 348615720590, 1388934122190, 5533708922574, 22047074027470, 87838639467470, 349961474550734, 1394295671696334, 5555069815204303, 22132178477202944, 88177707994792448},31] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_8(0)^2 + F_8(1)^2 + ... F_8(n)^2, where F_8(n) = A079262(n).

Corrected from a(16) on by R. J. Mathar, Aug 11 2009

A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 11, 19, 31, 49, 79, 129, 209, 337, 545, 883, 1429, 2311, 3739, 6051, 9791, 15841, 25631, 41473, 67105, 108577, 175681, 284259, 459941, 744199, 1204139, 1948339, 3152479, 5100817, 8253295, 13354113, 21607409, 34961521

Offset: 0

Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

nonn

This sequence has the same growth rate as the Fibonacci sequence, since x^4 - x^3 - x - 1 has the real roots phi and -1/phi.

The Ca1 sums, see A180662 for the definition of these sums, of triangle A035607 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 05 2011

			G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 11*x^7 + 19*x^8 + 31*x^9 + ...

S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002

Seiichi Manyama, Table of n, a(n) for n = 0..4786
K. T. Atanassov, D. R. Deford, A. G. Shannon, Pulsated Fibonacci recurrences, Fibonacci Quarterly, Vol. 52, No. 5, Dec. 2014, pp. 22-27.
Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. Fibonacci numbers A000045; Lucas numbers A000032; tribonacci numbers A000213; tetranacci numbers A000288; pentanacci numbers A000322; hexanacci numbers A000383; 7th-order Fibonacci numbers A060455; octanacci numbers A079262; 9th-order Fibonacci sequence A127193; 10th-order Fibonacci sequence A127194; 11th-order Fibonacci sequence A127624, A128429.

a:=[1,1,1,1];; for n in [5..50] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jul 15 2019

[n le 4 select 1 else Self(n-1) + Self(n-3) + Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2015

# From R. J. Mathar, Jul 22 2010: (Start)
A010684 := proc(n) 1+2*(n mod 2) ; end proc:
A000032 := proc(n) coeftayl((2-x)/(1-x-x^2),x=0,n) ; end proc:
A126116 := proc(n) ((-1)^floor(n/2)*A010684(n)+2*A000032(n))/5 ; end proc: seq(A126116(n),n=0..80) ; # (End)
with(combinat): A126116 := proc(n): fibonacci(n-1) + fibonacci(floor((n-4)/2)+1)* fibonacci(ceil((n-4)/2)+2) end: seq(A126116(n), n=0..38); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1,0,1,1},{1,1,1,1},50] (* Harvey P. Dale, Nov 08 2011 *)

Vec((x-1)*(1+x+x^2)/((x^2+x-1)*(x^2+1)) + O(x^50)) \\ Altug Alkan, Dec 25 2015

((1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 15 2019

From R. J. Mathar, Jul 22 2010: (Start)
G.f.: (1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2)).
a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5.
a(2*n) = A061646(n). (End)
From Johannes W. Meijer, Aug 05 2011: (Start)
a(n) = F(n-1) + A070550(n-4) with F(n) = A000045(n).
a(n) = F(n-1) + F(floor((n-4)/2) + 1)*F(ceiling((n-4)/2) + 2). (End)
a(n) = (1/5)*((sqrt(5)-1)*(1/2*(1+sqrt(5)))^n - (1+sqrt(5))*(1/2*(1-sqrt(5)))^n + sin((Pi*n)/2) - 3*cos((Pi*n)/2)). - Harvey P. Dale, Nov 08 2011
(-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - Michael Somos, Feb 05 2012
a(n)^2 + 3*a(n-2)^2 + 6*a(n-5)^2 + 3*a(n-7)^2 = a(n-8)^2 + 3*a(n-6)^2 + 6*a(n-3)^2 + 3*a(n-1)^2. - Greg Dresden, Jul 07 2021
a(n) = A293411(n)-A293411(n-1). - R. J. Mathar, Jul 20 2025

Edited by Don Reble, Mar 09 2007

cp's OEIS Frontend

A253705 Indices of primes in the 8th-order Fibonacci number sequence, A079262.

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A104415 Number of prime factors, with multiplicity, of the nonzero octanacci numbers A079262.

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A253706 Primes in the 8th-order Fibonacci numbers A079262.

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A092921 Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.

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A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

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A123526 Octanacci numbers.

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A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

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