A066178 -id:A066178 - cp's OEIS frontend

A251710 7-step Fibonacci sequence starting with (0,0,0,0,0,1,0).

0, 0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 32, 63, 126, 251, 500, 996, 1984, 3952, 7872, 15681, 31236, 62221, 123942, 246888, 491792, 979632, 1951392, 3887103, 7742970, 15423719, 30723496, 61200104, 121908416, 242837200, 483723008, 963558913, 1919374856, 3823325993

Offset: 0

Arie Bos, Dec 07 2014

a(n+7) equals the number of n-length binary words avoiding runs of zeros of lengths 7i+6, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

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

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251711, A251712, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 0 0 1 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 0, 0, 1, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).
G.f.: x^5*(x-1)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025
a(n) = A066178(n-5)-A066178(n-6). - R. J. Mathar, Mar 28 2025

A251711 7-step Fibonacci sequence starting with (0,0,0,0,1,0,0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 16, 31, 62, 124, 247, 492, 980, 1952, 3888, 7745, 15428, 30732, 61217, 121942, 242904, 483856, 963824, 1919903, 3824378, 7618024, 15174831, 30227720, 60212536, 119941216, 238918608, 475917313, 948010248, 1888402472, 3761630113

Offset: 0

Arie Bos, Dec 07 2014

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

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251712, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 0 1 0 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 0, 1, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).
G.f.: x^4*(-1+x+x^2)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025
a(n) = A066178(n-4)-A066178(n-5)-A066178(n-6). - R. J. Mathar, Mar 28 2025

A251712 7-step Fibonacci sequence starting with (0,0,0,1,0,0,0).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 4, 8, 15, 30, 60, 120, 239, 476, 948, 1888, 3761, 7492, 14924, 29728, 59217, 117958, 234968, 468048, 932335, 1857178, 3699432, 7369136, 14679055, 29240152, 58245336, 116022624, 231112913, 460368648, 917037864, 1826706592, 3638734129

Offset: 0

Arie Bos, Dec 07 2014

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

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 1 0 0 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 1, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

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

A251713 7-step Fibonacci sequence starting with (0,0,1,0,0,0,0).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 112, 223, 444, 884, 1761, 3508, 6988, 13920, 27728, 55233, 110022, 219160, 436559, 869610, 1732232, 3450544, 6873360, 13691487, 27272952, 54326744, 108216929, 215564248, 429396264, 855341984, 1703810608, 3393929729

Offset: 0

Arie Bos, Dec 07 2014

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

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251712, A251714.

(see www.jsoftware.com) First construct the generating matrix
   [M=: (#.@}: + {:)\"1&.|: <:/~i.7
 1  1  1  1  1  1  1
 1  2  2  2  2  2  2
 2  3  4  4  4  4  4
 4  6  7  8  8  8  8
 8 12 14 15 16 16 16
16 24 28 30 31 32 32
32 48 56 60 62 63 64
Given that matrix, one can produce the first 7*150 numbers by
, M(+/ . *)^:(i.150) 0 0 1 0 0 0 0x

LinearRecurrence[Table[1, {7}], {0, 0, 1, 0, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

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

A251714 7-step Fibonacci sequence starting with (0,1,0,0,0,0,0).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 96, 191, 380, 757, 1508, 3004, 5984, 11920, 23744, 47297, 94214, 187671, 373834, 744664, 1483344, 2954768, 5885792, 11724287, 23354360, 46521049, 92668264, 184591864, 367700384, 732446000, 1459006208, 2906288129

Offset: 0

Arie Bos, Dec 07 2014

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

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251712, A251713.

NB. see A251713 for the program and apply it to 0 1 0 0 0 0 0.

LinearRecurrence[Table[1, {7}], {0, 1, 0, 0, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

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

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

A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Nov 02 2014

			[n\k] [0][1][2][3][4] [5] [6] [7]  [8]  [9] [10]  [11]  [12]
   [0] 1, 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0
   [1] 1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1
   [2] 1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144,  233  [A000045]
   [3] 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274,  504,  927  [A000073]
   [4] 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,  773, 1490  [A000078]
   [5] 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464,  912, 1793  [A001591]
   [6] 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492,  976, 1936  [A001592]
   [7] 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000  [A066178]
   [8] 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028  [A079262]
   [.] .  .  .  .  .   .   .   .    .    .    .     .     .
  [oo] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048  [A011782]
.
As a triangular array, starts:
  1,
  1, 0,
  1, 1, 0,
  1, 1, 1, 0,
  1, 1, 2, 1, 0,
  1, 1, 2, 3, 1, 0,
  1, 1, 2, 4, 5, 1, 0,
  1, 1, 2, 4, 7, 8, 1, 0,
  1, 1, 2, 4, 8, 13, 13, 1, 0,
  1, 1, 2, 4, 8, 15, 24, 21, 1, 0,
  ...

Stefano Spezia, First 140 antidiagonals of the array, flattened
Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.

Cf. A247505, A011782, A000045, A000073, A000078, A001591, A001592, A066178, A079262.

Cf. A048887, A092921, A144406.

A := (n,k) -> coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k):
seq(print(seq(A(n,k), k=0..12)), n=0..9);

A[n_, k_] := A[n, k] = If[k<0, 0, If[k==0, 1, Sum[A[n, j], {j, k-n, k-1}]]]; Table[A[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2019 *)

A(n, k) = Sum_{j=0..floor(k/(n+1))} (-1)^j*((k - j*n) + j + delta(k,0))/(2*(k - j*n) + delta(k,0))*binomial(k - j*n, j)*2^(k-j*(n+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

A111431 a(n) = Fibonacci(tribonacci(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 13, 233, 46368, 701408733, 37889062373143906, 6161314747715278029583501626149, 818706854228831001753880637535093596811413714795418360007

Offset: 0

Jonathan Vos Post, Nov 13 2005

			a(0) = Fibonacci(tribonacci(0)) = A000045(A000073(0)) = A000045(0) = 0.
a(1) = Fibonacci(tribonacci(1)) = A000045(A000073(1)) = A000045(0) = 0.
a(2) = Fibonacci(tribonacci(2)) = A000045(A000073(2)) = A000045(1) = 1.
a(3) = Fibonacci(tribonacci(3)) = A000045(A000073(3)) = A000045(1) = 1.
a(4) = Fibonacci(tribonacci(4)) = A000045(A000073(4)) = A000045(2) = 1.
a(5) = Fibonacci(tribonacci(5)) = A000045(A000073(5)) = A000045(4) = 3.
a(6) = Fibonacci(tribonacci(6)) = A000045(A000073(6)) = A000045(7) = 13.
a(7) = Fibonacci(tribonacci(7)) = A000045(A000073(7)) = A000045(13) = 233.
a(8) = A000045(A000073(8)) = A000045(24) = 46368.
a(9) = A000045(A000073(9)) = A000045(44) = 701408733.
a(10) = A000045(A000073(10)) = A000045(81) = 37889062373143906.

Alois P. Heinz, Table of n, a(n) for n = 0..16

Cf. A000045, A000073, A066178, A007570, A111425.

a:= n-> (<<0|1>, <1|1>>^((<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]))[1, 2]:
seq(a(n), n=0..13);  # Alois P. Heinz, Aug 09 2018

Fibonacci/@LinearRecurrence[{1,1,1},{0,0,1},15] (* Harvey P. Dale, Jan 04 2013 *)

a(n) = A000045(A000073(n)).

A151975 The number of ways one can flip seven consecutive tails (or heads) when flipping a coin n times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 3, 8, 20, 48, 112, 256, 576, 1279, 2811, 6126, 13256, 28512, 61008, 129952, 275712, 582913, 1228551, 2582048, 5412984, 11321744, 23631056, 49229312, 102377216, 212560127, 440668919, 912310222, 1886316324, 3895528632, 8035861664

Offset: 0

Benjamin Merkel, Aug 05 2012

a(n-1) is the number of compositions of n with at least one part >=8. - Joerg Arndt, Aug 06 2012

			a(0)=0 means that there are no cases of seven consecutive tails (or heads) in zero coin flips.  Likewise, a(1)=a(2)=...=a(6)=0.  a(7)=1 since there is exactly one case of seven consecutive tails in seven coin flips.

Colin Barker, Table of n, a(n) for n = 0..1000
Benjamin E. Merkel, Probabilities of Consecutive Events in Coin Flipping, OhioLINK, 2011
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-1,-1,-1,-2).

Cf. A050231, A050232, A050233, A143662.

N=66;  x='x+O('x^N);
gf = (1-x)/(1-2*x); /* A011782(n): compositions of n */
gf -= 1/(1 - (x+x^2+x^3+x^4+x^5+x^6+x^7)); /* A066178(n): compositions of n into parts <=7 */
v151975=Vec(gf + 'a0);  v151975[1]=0; /* kludge to get all terms */
v151975 /* show terms */
/* Joerg Arndt, Aug 06 2012 */

concat(vector(7), Vec(x^7/((2*x-1)*(x^7+x^6+x^5+x^4+x^3+x^2+x-1)) + O(x^100))) \\ Colin Barker, Oct 16 2015

a(n) = A000079(n) - A066178(n+1).

G.f.: x^7 / ((2*x-1)*(x^7+x^6+x^5+x^4+x^3+x^2+x-1)). - Colin Barker, Oct 16 2015

A303264 Indices of primes in tetranacci sequence A000078.

Original entry on oeis.org

5, 9, 13, 14, 38, 58, 403, 2709, 8419, 14098, 31563, 50698, 53194, 155184

Offset: 1

M. F. Hasler, Apr 18 2018

T = A000078 is defined by T(n) = Sum_{k=1..4} T(n-k), T(3) = 1, T(n) = 0 for n < 3.

The largest terms correspond to unproven probable primes T(a(n)).

Cf. A000045, A000073, A000078, A001591, A001592, A122189 (or A066178), ... (Fibonacci, tribonacci, tetranacci numbers).
Cf. A005478, A092836, A104535, A105757, A105759, A105761, ... (primes in Fibonacci numbers and above generalizations).
Cf. A001605, A303263, A303264, A248757, A249635, ... (indices of primes in A000045, A000073, A000078, ...).
Cf. A247027: Indices of primes in the tetranacci sequence A001631 (starting 0, 0, 1, 0...), A104534 (a variant: a(n) - 2), A105756 (= A248757 - 3), A105758 (= A249635 - 4).

a(n,N=5,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

a(n) = A104534(n) + 2.

cp's OEIS Frontend

A251710 7-step Fibonacci sequence starting with (0,0,0,0,0,1,0).

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A251711 7-step Fibonacci sequence starting with (0,0,0,0,1,0,0).

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A251712 7-step Fibonacci sequence starting with (0,0,0,1,0,0,0).

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A251713 7-step Fibonacci sequence starting with (0,0,1,0,0,0,0).

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A251714 7-step Fibonacci sequence starting with (0,1,0,0,0,0,0).

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

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A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

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A111431 a(n) = Fibonacci(tribonacci(n)).

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