A127194 -id:A127194 - cp's OEIS frontend

A257073 Indices of primes in the 10th-order Fibonacci number sequence, A127194.

12, 13, 14, 17, 18, 27, 28, 29, 32, 50, 73, 75, 76, 81, 105, 184, 213, 813, 896, 1553, 4132, 5661, 5787, 7896, 8726, 9225, 9345, 11332, 11391, 12627, 13163, 14173, 15900, 17772, 17827, 20847, 21034, 25607, 94510, 98297, 101251, 127488, 138108, 188706

Offset: 1

Robert Price, Apr 15 2015

nonn

a(45) > 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.
OEIS Wiki, Index of Prime Fibonacci Numbers and Variants
OEIS Wiki, Index to OEIS Section Fi

Cf. A127194, A257074.

a={1,1,1,1,1,1,1,1,1,1}; step=10; lst={}; For[n=step+1,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Position[With[{c=Table[1,{10}]},LinearRecurrence[c,c,1000]],?PrimeQ]//Flatten (* The program generates the first 27 terms of the sequence. *) (* _Harvey P. Dale, Nov 11 2024 *)

A257074 Primes in the 10th-order Fibonacci numbers A127194.

Original entry on oeis.org

19, 37, 73, 577, 1153, 588097, 1175617, 2350081, 18772993, 4877942342401, 40459828019985257473, 161680527238968327169, 323202386497090728961, 10317126837306838775329, 171065716870341976441855042753

Offset: 1

Robert Price, Apr 15 2015

nonn

a(16) is too large to display here. It has 53 digits and is the 184th term in A127194.

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.
OEIS Wiki, Index of Prime Fibonacci Numbers and Variants
OEIS Wiki, Index to OEIS Section Fi

Cf. A127194, A257073.

a={1,1,1,1,1,1,1,1,1,1}; step=10; offset=1; lst={}; For[n=step+offset,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Select[With[{t=Table[1,{10}]},LinearRecurrence[t,t,400]],PrimeQ] (* Harvey P. Dale, Jan 17 2024 *)

A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Robert G. Wilson v, Oct 13 2009

Essentially a duplicate of A000079. - N. J. A. Sloane, Oct 15 2009
a(n) is the number of compositions of n into an odd number of parts.
Also 0 together with A011782. - Omar E. Pol, Oct 28 2013
Inverse INVERT transform of A001519. - R. J. Mathar, Dec 08 2022

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 16*x^6 + 32*x^7 + 64*x^8 + 128*x^9 + ...

Indranil Ghosh, Table of n, a(n) for n = 0..3317
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000045, A000079, A000213, A000288, A000322, A000383, A001519.
Cf. A011782, A034008, A060455, A123526, A127193, A127194, A127624.
Cf. A131577, A163551.

[n le 1 select n else 2^(n-2): n in [0..40]]; // G. C. Greubel, Jul 27 2024

a:= n-> `if`(n<2, n, 2^(n-2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 02 2021

a[0] = 0; a[1] = 1; a[n_] := a[n] = Plus @@ Array[a, n - 1]; Array[a, 35, 0]

[(2^n +2*int(n==1) -int(n==0))/4 for n in range(41)] # G. C. Greubel, Jul 27 2024

a(n) = A000079(n-1) for n > 0.
O.g.f.: x*(1 - x) / (1 - 2*x) = x / (1 - x / (1 - x)).
a(n) = (1-n) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (exp(2*x) + 2*x - 1)/4. - Stefano Spezia, Aug 07 2022

A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 22, 25, 30, 35, 41, 49, 57, 67, 79, 93, 109, 129, 151, 178, 209, 246, 290, 340, 401, 471, 554, 652, 767, 902, 1061, 1248, 1468, 1727, 2031, 2390, 2810, 3306, 3889, 4574, 5381, 6329

Offset: 0

Luis A Restrepo (luisiii(AT)mac.com), Feb 04 2007

a(n) = O(n^c), where c is the larger real root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 1.176280818..., the smallest known Salem constant.

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

E. Ghate and E. Hironaka, The Arithmetic And Geometry Of Salem Numbers, Bull. Amer. Math. Soc. 38 (2001), 293-314.
Eric Weisstein's World of Mathematics, MathWorld: Salem Constants
Eric Weisstein's World of Mathematics, MathWorld: Substitution System
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,1,1,1,0,-1,-1). [From _R. J. Mathar_, Jun 30 2010]

Cf. A070178, A029826, A107480, A127193, A127194, A127624.

LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1},{1,1,1,1,1,1,1,1,1,1},70] (* Harvey P. Dale, May 31 2013 *)

G.f.: ( 1+2*x+2*x^2+x^3-x^5-2*x^6-3*x^7-3*x^8-2*x^9 ) / ( 1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10 ). [R. J. Mathar, Jun 30 2010]

Edited by Don Reble, Mar 09 2007

A127624 An 11th-order Fibonacci sequence: a(n) = a(n-1) + ... + a(n-11).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40951, 81881, 163721, 327361, 654561, 1308801, 2616961, 5232641, 10462721, 20920321, 41830401, 83640321, 167239691, 334397501, 668631281

Offset: 1

Luis A Restrepo (Luisiii(AT)mac.com), Jan 19 2007

The ratio a(n+1)/a(n) approaches the unique real root of r^11 = r^10 + ... + r + 1; r is about 1.99951040197828549144.

All terms have last digit 1.

Robert Price, Table of n, a(n) for n = 1..1000
E. S. Croot, Notes on Linear Recurrence Sequences
M. A. Lerma, Recurrence Relations
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1).

Cf. Fibonacci numbers A000045, tribonacci numbers A000213, tetranacci numbers A000288, pentanacci numbers A000322, hexanacci numbers A000383, heptanacci numbers A060455, octanacci numbers A123526, 9th-order Fibonacci sequence A127193, 10th-order Fibonacci sequence A127194.

Cf. A257966 (indices of primes in a), A257967 (primes in a).

Module[{nn=11,lr},lr=PadRight[{},nn,1];LinearRecurrence[lr,lr,20]] (* Harvey P. Dale, Feb 04 2015 *)

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

O.g.f: x*(-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) / (-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11). - R. J. Mathar, Dec 02 2007

Edited by Dean Hickerson, Mar 09 2007

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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A257073 Indices of primes in the 10th-order Fibonacci number sequence, A127194.

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A257074 Primes in the 10th-order Fibonacci numbers A127194.

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A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

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A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

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A127624 An 11th-order Fibonacci sequence: a(n) = a(n-1) + ... + a(n-11).

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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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