A123526 -id:A123526 - cp's OEIS frontend

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

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

A254413 Primes in the 8th-order Fibonacci numbers A123526.

Original entry on oeis.org

29, 113, 449, 226241, 14307889, 113783041, 1820091580429249, 233322881089059894782836851617, 29566627412209231076314948970028097, 59243719929958343565697204780596496129, 7507351981539044730893385057192143660843521

Offset: 1

Robert Price, Jan 30 2015

nonn

a(12) is too large to display here. It has 46 digits and is the 158th term in A123526.

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192, A060455, A253318, A079262, A253705, A123526, A254412.

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,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Select[With[{lr=PadRight[{},8,1]},LinearRecurrence[lr,lr,200]],PrimeQ] (* Harvey P. Dale, Dec 03 2022 *)

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

A127193 A 9th-order Fibonacci sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8185, 16353, 32673, 65281, 130433, 260609, 520705, 1040385, 2078721, 4153345, 8298505, 16580657, 33128641, 66192001, 132253569, 264246529, 527972353, 1054904321

Offset: 1

Luis A Restrepo (luisiii(AT)mac.com), Jan 07 2007

9-Bonacci constant = 1.99802947...

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).

Cf. Fibonacci numbers A000045, tribonacci numbers A000213, tetranacci numbers A000288, pentanacci numbers A000322, hexanacci numbers A000383, 7th-order Fibonacci numbers A060455, octanacci numbers, A123526.

LinearRecurrence[{1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1},40] (* Ray Chandler, Aug 01 2015 *)
With[{c=Table[1,{9}]},LinearRecurrence[c,c,40]] (* Harvey P. Dale, Apr 08 2016 *)

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

For a(1)=...=a(9)=1, a(10)=9, a(n)= 2*a(n-1) - a(n-10). - Vincenzo Librandi, Dec 20 2010

G.f.: x*(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8+7*x^9)/(1-2*x+x^10). - G. C. Greubel, Jul 28 2017

A127194 A 10th-order Fibonacci sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 19, 37, 73, 145, 289, 577, 1153, 2305, 4609, 9217, 18424, 36829, 73621, 147169, 294193, 588097, 1175617, 2350081, 4697857, 9391105, 18772993, 37527562, 75018295, 149962969, 299778769, 599263345, 1197938593

Offset: 1

Luis A Restrepo (luisiii(AT)hotmail.com), Jan 11 2007

10th-order Fibonacci constant = 1.999018633...

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).

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.

Cf. A257073, A257074.

With[{t=Table[1,{10}]},LinearRecurrence[t,t,40]] (* Harvey P. Dale, Nov 12 2013 *)

a(n)=([0,1,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1,1]^(n-1)*[1;1;1;1;1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

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) / (-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10). - R. J. Mathar, Nov 23 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

A123525 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

Original entry on oeis.org

2, 14, 102, 836, 7730, 79962, 916454, 11533832, 158149026, 2346622310, 37458934502, 640013453004, 11652216012242, 225169809833906, 4602407562991590, 99194703240441872

Offset: 1

Karol A. Penson, Oct 02 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf.: A002720, A052852, A123510, A123511, A123512, A123526, A123527.

Rest[With[{nmax = 50}, CoefficientList[Series[(1/(1 - x)^2)*Exp[x/(1 - x)]*LaguerreL[1, 1/(x - 1)]*x, {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Oct 14 2017 *)

E.g.f.: (1/(1-x)^2)*exp(x/(1-x))*LaguerreL(1,1/(x-1))*x.
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: (n-2)*(n-1)*a(n) = 2*(n-2)*n^2*a(n-1) - (n-1)^3*n*a(n-2).
a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 5/4) / sqrt(2) * (1 + 31/(48*sqrt(n))).
(End)

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A254413 Primes in the 8th-order Fibonacci numbers A123526.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A127193 A 9th-order Fibonacci sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A127194 A 10th-order Fibonacci sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A123525 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

A123525 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.