A079262 -id:A079262 - cp's OEIS frontend

A251740 8-step Fibonacci sequence starting with 0,0,0,0,0,1,0,0.

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 16, 32, 63, 126, 252, 503, 1004, 2004, 4000, 7984, 15936, 31809, 63492, 126732, 252961, 504918, 1007832, 2011664, 4015344, 8014752, 15997695, 31931898, 63737064, 127221167, 253937416, 506867000, 1011722336, 2019429328

Offset: 0

Arie Bos, Dec 07 2014

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, A251672, A251741, A251742, A251744, A251745.

LinearRecurrence[Table[1, {8}], {0, 0, 0, 0, 0, 1, 0, 0}, 43] (* Michael De Vlieger, Dec 08 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^5*(-1+x+x^2)/(-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+2)-A079262(n+1)-A079262(n). - R. J. Mathar, Mar 28 2025

A251741 8-step Fibonacci sequence starting with 0,0,0,0,1,0,0,0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 4, 8, 16, 31, 62, 124, 248, 495, 988, 1972, 3936, 7856, 15681, 31300, 62476, 124704, 248913, 496838, 991704, 1979472, 3951088, 7886495, 15741690, 31420904, 62717104, 125185295, 249873752, 498755800, 995532128, 1987113168

Offset: 0

Arie Bos, Dec 07 2014

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, A251672, A251740, A251742, A251744, A251745.

LinearRecurrence[Table[1, {8}], {0, 0, 0, 0, 1, 0, 0, 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^4*(-1+x+x^2+x^3)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8) . - R. J. Mathar, Mar 28 2025

A251742 8-step Fibonacci sequence starting with 0,0,0,1,0,0,0,0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 4, 8, 15, 30, 60, 120, 240, 479, 956, 1908, 3808, 7601, 15172, 30284, 60448, 120656, 240833, 480710, 959512, 1915216, 3822831, 7630490, 15230696, 30400944, 60681232, 121121631, 241762552, 482565592, 963215968, 1922609105

Offset: 0

Arie Bos, Dec 07 2014

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, A251672, A251740, A251741, A251744, A251745.

LinearRecurrence[Table[1, {8}], {0, 0, 0, 1, 0, 0, 0, 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^3*(-1+x+x^2+x^3+x^4)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8) . - R. J. Mathar, Mar 28 2025
a(n) = A172318(n-3)-2*A172318(n-4)+A172318(n-8) . - R. J. Mathar, Mar 28 2025

A251744 8-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 112, 224, 447, 892, 1780, 3553, 7092, 14156, 28256, 56400, 112576, 224705, 448518, 895256, 1786959, 3566826, 7119496, 14210736, 28365072, 56617568, 113010431, 225572344, 450249432, 898711905, 1793856984

Offset: 0

Arie Bos, Dec 07 2014

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, A251672, A251740, A251741, A251742, A251745.

LinearRecurrence[Table[1, {8}], {0, 0, 1, 0, 0, 0, 0, 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^2*(-1+x+x^2+x^3+x^4+x^5)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8) . - R. J. Mathar, Mar 28 2025

A251745 8-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 96, 192, 383, 764, 1525, 3044, 6076, 12128, 24208, 48320, 96448, 192513, 384262, 766999, 1530954, 3055832, 6099536, 12174864, 24301408, 48506368, 96820223, 193256184, 385745369, 769959784, 1536863736, 3067627936

Offset: 0

Arie Bos, Dec 07 2014

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, A251672, A251740, A251741, A251742, A251744.

LinearRecurrence[Table[1, {8}], {0, 1, 0, 0, 0, 0, 0, 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*(-1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8) . - R. J. Mathar, Mar 28 2025

A168084 Fibonacci 13-step numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16381, 32760, 65516, 131024, 262032, 524032, 1048000, 2095872, 4191488, 8382464, 16763904, 33525760, 67047424, 134086657, 268156933, 536281106, 1072496696

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).

Cf. A000078, A001591, A001592, A122189, A079262, A168083.

k:=13: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); k:=13: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); # Richard Choulet, Feb 22 2010

a={1,0,0,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]
With[{nn=13},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

Another form of the g.f. f: f(z)=(z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=13. 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=13 and convention sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

A172318 9th column of the array A172119.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64512, 128768, 257025, 513030, 1024024, 2043984, 4079856, 8143520, 16254720, 32444928, 64761088, 129265151, 258017272, 515010520, 1027977056

Offset: 0

Richard Choulet, Jan 31 2010

			a(7)=C(7,7)*2^7=128. a(10)=C(10,10)*2^10-C(2,1)*2^1=1020.

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,0,0,-1).

Cf. A172317, A172316, A172119, A001949, A107066.

Partial sums of A079262.

for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od; k:=8:taylor(1/(1-2*z+z^(k+1)),z=0,30);

G.f.: 1/(1-2*z+z^9).
a(n) = sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))) with k=8.
Recurrence relation: a(n+9) = 2*a(8) - a(n).

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

A254412 Indices of primes in the 8th-order Fibonacci number sequence, A123526.

Original entry on oeis.org

11, 13, 15, 24, 30, 33, 57, 104, 121, 132, 149, 158, 178, 220, 295, 389, 1070, 1101, 1373, 1761, 1778, 2333, 2731, 4541, 5189, 5237, 5738, 8025, 8787, 10561, 11783, 13435, 14638, 15337, 20985, 21722, 24770, 31009, 57367, 65877, 129773, 134630, 167020

Offset: 1

Robert Price, Jan 30 2015

a(44) > 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, A123526, A254413.

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

cp's OEIS Frontend

A251740 8-step Fibonacci sequence starting with 0,0,0,0,0,1,0,0.

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A251741 8-step Fibonacci sequence starting with 0,0,0,0,1,0,0,0.

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A251742 8-step Fibonacci sequence starting with 0,0,0,1,0,0,0,0.

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A251744 8-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0.

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A251745 8-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0.

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A168084 Fibonacci 13-step numbers.

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A172318 9th column of the array A172119.

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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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