A001592 -id:A001592 - cp's OEIS frontend

A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).

0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, 324528, 1277104, 5025200, 19770800, 77789489, 306071370, 1204272270, 4738336974, 18643463374, 73354544590, 288620849614, 1135607911375, 4468164041216, 17580442344960

Offset: 0

Jonathan Vos Post, May 19 2005

Primes include: a(6) = 2. Semiprimes include a(7) = 6 = 2 * 3, a(8) = 22 = 2 * 11, a(9) = 86 = 2 * 43, a(11) = 1366 = 2 * 683, a(19) = 77789489 = 3989 * 19501, a(23) = 18643463374 = 2 * 9321731687,

			a(0) = 0 = 0^2
a(1) = 0 = 0^2 + 0^2
a(2) = 0 = 0^2 + 0^2 + 0^2
a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
a(4) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(5) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(6) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(7) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(8) = 22 = 0^2 + 0^2 +0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1).

Cf. A001592, A107239-A107243, A107245-A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1,1},{0,0,0,0,0,1},50]^2] (* Harvey P. Dale, Jan 19 2012 *)
LinearRecurrence[{3, 2, 4, 6, 14, 28, -67, -9, -8, 28, -8, -12, 20, 5, 5, -10, 0, 2, -2, 0, -1, 1},{0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, 324528, 1277104, 5025200, 19770800, 77789489, 306071370, 1204272270},29] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_6(0)^2 + F_6(1)^2 + ... F_6(n)^2, where F_6(n) = A001592(n). a(0) = 0, a(n+1) = a(n) + A001592(n).

a(n)= 3*a(n-1) +2*a(n-2) +4*a(n-3) +6*a(n-4) +14*a(n-5) +28*a(n-6) -67*a(n-7) -9*a(n-8) -8*a(n-9) +28*a(n-10) -8*a(n-11) -12*a(n-12) +20*a(n-13) +5*a(n-14) +5*a(n-15) -10*a(n-16) +2*a(n-18) -2*a(n-19) -a(n-21) +a(n-22). [From R. J. Mathar, Aug 11 2009]

A104413 Number of prime factors, with multiplicity, of the hexanacci numbers A001592.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 3, 3, 4, 4, 5, 6, 10, 2, 2, 7, 5, 8, 7, 10, 3, 2, 6, 6, 6, 7, 11, 2, 5, 3, 4, 5, 10, 8, 1, 1, 5, 3, 7, 8, 15, 5, 3, 3, 12, 9, 9, 9, 3, 2, 9, 9, 8, 9, 13, 6, 4, 3, 7, 8, 9, 9, 5, 6, 5, 5, 6, 6, 13, 6, 4, 6, 10, 9, 7, 9, 3, 4, 9, 7, 8, 9, 14, 3, 3, 5, 7, 4, 14, 10, 1, 3, 10, 5, 9, 10, 14, 6

Offset: 5

Jonathan Vos Post, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 5..244

Cf. A001592, A001222, A104411, A104412.

PrimeOmega[LinearRecurrence[{1, 1, 1, 1, 1, 1}, {1, 1, 2, 4, 8, 16}, 100]] (* Amiram Eldar, May 16 2021 *)

a(n) = A001222(A001592(n)).

Offset changed to 5 by Georg Fischer, Dec 19 2020

More terms from Joerg Arndt, Dec 19 2020

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

Original entry on oeis.org

3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148

Offset: 1

T. D. Noe, Apr 22 2005

No other n < 30000.
This sequence uses the convention of the Noe and Post reference. Their indexing scheme differs by 4 from the indices in A001592. Sequence A249635 lists the indices of the same primes (A105759) using the indexing scheme as defined in A001592. - Robert Price, Nov 02 2014 [Edited by M. F. Hasler, Apr 22 2018]
a(13) > 3*10^5. - Robert Price, Nov 02 2014

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.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105759 (prime Fibonacci 6-step numbers), A249635 (= a(n) + 4), A001592.
Cf. A000045, A000073, A000078 (and A001631), A001591, A122189 (or A066178),  A079262, A104144,  A122265, A168082, A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
Cf. A005478, A092836, A104535, A105757, A105761, ... (primes in these sequence).
Cf. A001605, A303263, A303264 (and A104534 and A247027), A248757 (and A105756), ... (indices of primes in A000045, A000073, A000078, ...).

a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

a(n) = A249635(n) - 4. A105759(n) = A001592(A249635(n)) = A001592(a(n) + 4). - M. F. Hasler, Apr 22 2018

a(10)-a(12) from Robert Price, Nov 02 2014

Edited by M. F. Hasler, Apr 22 2018

A105759 Prime Fibonacci 6-step numbers, A001592.

Original entry on oeis.org

2, 13435170943, 26649774581, 610186256014622144673892607

Offset: 1

T. D. Noe, Apr 22 2005

nonn

The next term has 196 digits. - Harvey P. Dale, Apr 23 2025

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
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105758 (indices of prime Fibonacci 6-step numbers).

a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 1000}]; lst
Select[LinearRecurrence[{1,1,1,1,1,1},{1,0,0,0,0,0},100],PrimeQ] (* Harvey P. Dale, Apr 23 2025 *)

A249635 Indices of prime Fibonacci 6-step numbers, A001592.

Original entry on oeis.org

7, 40, 41, 96, 664, 6095, 8419, 11471, 13690, 38835, 49832, 97152

Offset: 1

Robert Price, Nov 02 2014

This sequence is similar to A105758 but uses the indexing scheme defined by A001592, whose indices start with 0.

a(13) > 3*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
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A001592, A105758.

a = {0, 0, 0, 0, 0, 1}; For[n = 6, n ≤ 1000, n++, sum = Plus @@ a;
If[PrimeQ[sum], Print[n]]; a = RotateLeft[a]; a[[6]] = sum]

a(n) = A105758 (n) + 4.

A381508 Pisano period of Hexanacci numbers (A001592) mod n.

Original entry on oeis.org

1, 7, 728, 14, 208, 728, 342, 28, 2184, 1456, 354312, 728, 9520, 2394, 1456, 56, 709928, 2184, 5227320, 1456, 124488, 354312, 279840, 728, 1040, 9520, 6552, 2394, 243880, 1456, 71040, 112, 4606056, 4969496, 35568, 2184, 20362908, 5227320, 123760, 1456, 201840

Offset: 1

Martin Guerra and Doron Zeilberger, Apr 24 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..222
Martin Guerra and Doron Zeilberger, Maple program

Cf. A001175, A001592.

# load programs from linked file:
seq(Pis([[0$5, 1],[1$6]],n,400000), n=1..16);

from math import lcm
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A381508(n):
    if n == 1:
        return 1
    f = factorint(n).items()
    if len(f) > 1:
        return lcm(*(A381508(a**b) for a,b in f))
    else:
        k, x = 1, (0,0,0,0,1,1)
        while x != (0,0,0,0,0,1):
            k += 1
            x = x[1:]+(sum(x) % n,)
        return k # Chai Wah Wu, Apr 25 2025

a(17)-a(41) from Alois P. Heinz, Apr 25 2025

A000383 Hexanacci numbers with a(0) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 11, 21, 41, 81, 161, 321, 636, 1261, 2501, 4961, 9841, 19521, 38721, 76806, 152351, 302201, 599441, 1189041, 2358561, 4678401, 9279996, 18407641, 36513081, 72426721, 143664401, 284970241, 565262081, 1121244166, 2224080691, 4411648301

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..3358 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook)
B. G. Baumgart, Letter to the editor Part 1 Part 2 Part 3, Fib. Quart. 2 (1964), 260, 302.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).

Cf. A060455.
Cf. A001592 (Hexanacci numbers with a(0) = ... = a(4) = 0 and a(5)=1).
Cf. A247192 (indices of primes in this sequence).
Cf. A249413 (primes in this sequence).

A000383:=(-1+z**2+2*z**3+3*z**4+4*z**5)/(-1+z**2+z**3+z**4+z**5+z+z**6); # Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[1$6]]). Matrix(6, (i,j)-> if (i=j-1) or j=1 then 1 else 0 fi)^n)[1,6]: seq(a(n), n=0..28); # Alois P. Heinz, Aug 26 2008

LinearRecurrence[{1,1,1,1,1,1},{1,1,1,1,1,1},50] (* Harvey P. Dale, Oct 30 2013 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,1,1,1,1]^n*[1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Sep 24 2015

G.f. ( -1+x^2+2*x^3+3*x^4+4*x^5 ) / ( -1+x+x^2+x^3+x^4+x^5+x^6 ). - R. J. Mathar, Oct 11 2011

A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536, 1023897200, 2043730736

Offset: 0

Michael Joseph Halm, Feb 04 2003

a(n+7) is the number of compositions of n into parts <= 8. - Joerg Arndt, Sep 24 2020

			a(16) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255.

T. D. Noe, Table of n, a(n) for n=0..207
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.
Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
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.
Fred J. Rispoli, Fibonacci Polytopes and Their Applications, Fib. Q., 43,3 (2005), 227-233.
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).

Cf. A066178, A001592, A001591, A001630, A000073, A000045.
Row 8 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Cf. A253706, A253705. Primes and indices of primes in this sequence.

for j from 0 to 6 do a[j]:=0 od: a[7]:=1: for n from 8 to 45 do a[n]:=sum(a[n-i],i=1..8) od:seq(a[n],n=0..45); # Emeric Deutsch, Apr 16 2005

LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=8},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: x^7/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8). - Emeric Deutsch, Apr 16 2005
a(1)..a(9) = 1, 1, 2, 4, 8, 16, 32, 64, 128. a(10) and following are given by 63*2^(n-8)+(1/2+sqrt(5/4))^(n-6)/sqrt(5)-(1/2-sqrt(5/4))^(n-6)/sqrt(5). Offset 10. a(10)=255. - Al Hakanson (hawkuu(AT)gmail.com), Feb 14 2009
Another form of the g.f.: f(z) = (z^7 - z^8)/(1 - 2*z + z^9), then a(n) = Sum_{i=0..floor((n-7)/9)} (-1)^i*binomial(n-7-8*i,i)*2^(n-7-9*i) - Sum_{i=0..floor((n-8)/9)} (-1)^i*binomial(n-8-8*i,i)*2^(n-8-9*i) with Sum_{i=m..n} alpha(i) = 0 for m>n. - Richard Choulet, Feb 22 2010
Sum_{k=0..7*n} a(k+b)*A171890(n,k) = a(8*n+b), b>=0.
a(n) = 2*a(n-1) - a(n-9). - Vincenzo Librandi, Dec 20 2010

Corrected by Joao B. Oliveira (oliveira(AT)inf.pucrs.br), Nov 25 2004

A066178 Number of binary bit strings of length n with no block of 8 or more 0's. Nonzero heptanacci numbers, A122189.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600

Offset: 0

Len Smiley, Dec 14 2001

Analogous bit string description and o.g.f. (1-x)/(1-2x+x^{k+1}) works for nonzero k-nacci numbers.

Compositions of n into parts <= 7. - Joerg Arndt, Aug 06 2012

T. D. Noe, Table of n, a(n) for n = 0..200
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.
Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Zhao Hui Du, Link giving derivation and proof of the formula
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
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
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Heptanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Cf. A000045 (k=2, Fibonacci numbers), A000073 (k=3, tribonacci) A000078 (k=4, tetranacci) A001591 (k=5, pentanacci) A001592 (k=6, hexanacci), A122189 (k=7, heptanacci).

Row 7 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 8; a[5] = 16; a[6] = 32; a[7] = 64; a[n_] := 2*a[n - 1] - a[n - 8]; Array[a, 31, 0]
CoefficientList[ Series[(1 - x)/(1 - 2 x + x^8), {x, 0, 30}], x]
LinearRecurrence[{1,1,1,1,1,1,1},{1,1,2,4,8,16,32},40] (* Harvey P. Dale, Nov 16 2014 *)

O.g.f.: 1/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
a(n) = Sum_{i=n-7..n-1} a(i).
a(n) = round((r-1)/((t+1)*r - 2*t) * r^(n-1)), where r is the heptanacci constant, the real root of the equation x^(t+1) - 2*x^t + 1 = 0 which is greater than 1. The formula could also be used for a k-step Fibonacci sequence if r is replaced by the k-bonacci constant, as in A000045, A000073, A000078, A001591, A001592. - Zhao Hui Du, Aug 24 2008
a(n) = 2*a(n-1) - a(n-8). - Vincenzo Librandi, Dec 20 2010

Definition corrected by Vincenzo Librandi, Dec 20 2010

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

cp's OEIS Frontend

A107244 Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).

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A104413 Number of prime factors, with multiplicity, of the hexanacci numbers A001592.

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A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

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A105759 Prime Fibonacci 6-step numbers, A001592.

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A249635 Indices of prime Fibonacci 6-step numbers, A001592.

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A381508 Pisano period of Hexanacci numbers (A001592) mod n.

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A000383 Hexanacci numbers with a(0) = ... = a(5) = 1.

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A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).

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