A127193 -id:A127193 - cp's OEIS frontend

A251751 9-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0,0.

0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 112, 224, 448, 895, 1788, 3572, 7137, 14260, 28492, 56928, 113744, 227264, 454080, 907265, 1812742, 3621912, 7236687, 14459114, 28889736, 57722544, 115331344, 230435424, 460416768, 919926271, 1838039800

Offset: 0

Arie Bos, Dec 07 2014

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251747, A251748, A251749, A251750, A251752.

Cf. A255534 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 1, 0, 0, 0, 0, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).
G.f.: x^2*(-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+x^9) . - R. J. Mathar, Mar 28 2025
a(n) = A172319(n-2)-2*A172319(n-3)+A172319(n-9). - R. J. Mathar, Mar 28 2025

A251752 9-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0,0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 96, 192, 384, 767, 1532, 3061, 6116, 12220, 24416, 48784, 97472, 194752, 389120, 777473, 1553414, 3103767, 6201418, 12390616, 24756816, 49464848, 98832224, 197469696, 394550272, 788323071, 1575092728

Offset: 0

Arie Bos, Dec 07 2014

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251747, A251748, A251749, A251750, A251751.

Cf. A255536 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 1, 0, 0, 0, 0, 0, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).
G.f.: x*(-1+x+x^2+x^3+x^4+x^5+x^6+x^7)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025
a(n) = A172319(n-1)-2*A172319(n-2)+A172319(n-9). - R. J. Mathar, Mar 28 2025

A251748 9-step Fibonacci sequence starting with 0,0,0,0,0,1,0,0,0.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 4, 8, 16, 32, 63, 126, 252, 504, 1007, 2012, 4020, 8032, 16048, 32064, 64065, 128004, 255756, 511008, 1021009, 2040006, 4075992, 8143952, 16271856, 32511648, 64959231, 129790458, 259325160, 518139312, 1035257615, 2068475224

Offset: 0

Arie Bos, Dec 07 2014

The only primes in this sequence whose indices are less than 2*10^5 are 2 and 65865769729, which correspond to indices of 10 and 45. - Robert Price, Feb 24 2015

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251747, A251749, A251750, A251751, A251752.

LinearRecurrence[Table[1, {9}], {0, 0, 0, 0, 0, 1, 0, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).

G.f.: x^5*(-1+x+x^2+x^3)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025

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

A163551 13th-order Fibonacci numbers: a(n) = a(n-1) + ... + a(n-13) with a(1)=...=a(13)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196597, 393169, 786289, 1572481, 3144769, 6289153, 12577537, 25153537, 50304001, 100601857, 201191425, 402358273, 804667393

Offset: 1

Jainit Purohit (mjainit(AT)gmail.com), Jul 30 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1).

Cf. A000045 (Fibonacci numbers), A000213 (tribonacci), A000288 (tetranacci), A000322 (pentanacci), A000383 (hexanacci), A060455 (heptanacci), A123526 (octanacci), A127193 (nonanacci), A127194 (decanacci), A127624 (undecanacci), A207539 (dodecanacci).

With[{c=Table[1,{13}]},LinearRecurrence[c,c,40]] (* Harvey P. Dale, Aug 09 2013 *)

x='x+O('x^50); Vec((1-x^2 -2*x^3-3*x^4 -4*x^5-5*x^6 -6*x^7-7*x^8 -8*x^9 -9*x^10 -10*x^11 -11*x^12) / (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11-x^12-x^13)) \\ G. C. Greubel, Jul 28 2017

a(n) = a(n-1)+a(n-2)+...+a(n-13) for n > 12, a(0)=a(1)=...=a(12)=1.

G.f.: (-1)*(-1+x^2+2*x^3+3*x^4+4*x^5+5*x^6+6*x^7+7*x^8+8*x^9+9*x^10 +10*x^11 +11*x^12) / (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11-x^12-x^13). - Michael Burkhart, Feb 18 2012

Values adapted to the definition by R. J. Mathar, Aug 01 2009

A249169 Fibonacci 16-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-16).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65535, 131069, 262136, 524268, 1048528, 2097040, 4194048, 8388032, 16775936, 33551616, 67102720, 134204416, 268406784, 536809472, 1073610752, 2147205120, 4294377472, 8588689409

Offset: 15

Alan N. Inglis, Oct 22 2014

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

Other k-step Fibonacci sequences: Cf. A000045, A000213, A000288, A000322, A000383, A060455, A123526, A127193, A127194, A168083, A207539, A168084, A220469, A220493.

a:= proc(n) option remember; `if`(n<15, 0,
      `if`(n=15, 1, add(a(n-j), j=1..16)))
    end:
seq(a(n), n=15..50);  # Alois P. Heinz, Oct 23 2014

CoefficientList[Series[-1 /(x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

a(n) = a(n-1) + a(n-2) + ... + a(n-16).

G.f.: -x^15 / (x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5 +x^4+x^3+x^2+x-1). - Alois P. Heinz, Oct 23 2014

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A251751 9-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0,0.

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A251752 9-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0,0.

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A251748 9-step Fibonacci sequence starting with 0,0,0,0,0,1,0,0,0.

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

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A163551 13th-order Fibonacci numbers: a(n) = a(n-1) + ... + a(n-13) with a(1)=...=a(13)=1.

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A249169 Fibonacci 16-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-16).

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