A168084 -id:A168084 - cp's OEIS frontend

A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521472, 1042432, 2083841, 4165637, 8327186, 16646200, 33276064, 66519472, 132973664, 265816832, 531372800, 1062224128

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 18 2006

nonn

The (1,10)-entry of the matrix M^n, where M is the 10 X 10 matrix {{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}}.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
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).

Cf. A000322, A001591, A001592, A079262.
Cf. A001591, A001592, A122189, A079262, A104144, A168082, A168083, A168084. - Richard Choulet, Feb 22 2010
Cf. A257227, A257228 for primes in this sequence.

with(linalg): p:=-1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9+x^10: M[1]:=transpose(companion(p,x)): for n from 2 to 40 do M[n]:=multiply(M[n-1],M[1]) od: seq(M[n][1,10],n=1..40);
k:=10: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);k:=10: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); # Richard Choulet, Feb 22 2010

M = {{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}}; v[1] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 50}]
a={1,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]  (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=10},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

a(n) = Sum_{j=1..10} a(n-j) for n>=10; a(n) = 0 for 0<=n<=8, a(9) = 1 (follows from the minimal polynomial of M; a Maple program based on this recurrence relation is much slower than the given Maple program, based on the definition).
G.f.: -x^9/(-1+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
Another form of the g.f. f: f(z)=(z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=10. 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=10 and sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

Edited by N. J. A. Sloane, Oct 29 2006 and Mar 05 2011

A220493 Fibonacci 15-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-15).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32767, 65533, 131064, 262124, 524240, 1048464, 2096896, 4193728, 8387328, 16774400, 33548288, 67095552, 134189056, 268374016, 536739840, 1073463296, 2146893825, 4293722117, 8587313170

Offset: 1

Ruskin Harding, Feb 20 2013

Also called Pentadecanacci numbers. In previous similar sequences, a(1), ..., a(n-1) have been set equal to zero and a(n)=1. For example, A168084 (Fibonacci 13-step numbers) has 12 0's as the first 12 terms and a(13)=1.

Robert Israel, Table of n, a(n) for n = 1..3320
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
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.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Cf. A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), A001591 (pentanacci).

f:= gfun:-rectoproc({a(n) = add(a(n-i),i=1..15), seq(a(n)=0,n=-14..0),a(1)=1},a(n),remember):
map(f, [$1..100]); # Robert Israel, Feb 19 2019

FibonacciSequence[n_, kMax_] := Module[{a, s}, a = Join[{1}, Table[0, {n - 1}]]; lst = {}; Table[s = Plus @@ a; a = RotateLeft[a]; a[[n]] = s, {k, 1, kMax}]]; FibonacciSequence[15, 50] (* T. D. Noe, Feb 20 2013 *)

G.f.: x/(1-Sum_{k=1..15} x^k). - Robert Israel, Feb 19 2019

A220469 Fibonacci 14-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-14).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16383, 32765, 65528, 131052, 262096, 524176, 1048320, 2096576, 4193024, 8385792, 16771072, 33541120, 67080192, 134156288, 268304384, 536592385, 1073152005, 2146238482, 4292345912, 8584429728

Offset: 1

Ruskin Harding, Feb 20 2013

nonn

Also called tetradecanacci numbers. In previous similar sequences, a(1),...,a(n-1) have been set equal to zero and a(n)=1. For example, A168084 (Fibonacci 13-step numbers) has 12 0's as the first 12 terms and a(13)=1.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7
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
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).

Cf. A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), A001591 (pentanacci).

FibonacciSequence[n_, kMax_] := Module[{a, s}, a = Join[{1}, Table[0, {n - 1}]]; lst = {}; Table[s = Plus @@ a; a = RotateLeft[a]; a[[n]] = s, {k, 1, kMax}]]; FibonacciSequence[14, 50] (* T. D. Noe, Feb 20 2013 *)
Drop[LinearRecurrence[PadRight[{},14,1],Join[PadRight[{},13,0],{1}],50],13] (* Harvey P. Dale, Feb 25 2013 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},{1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096},35] (* Ray Chandler, Aug 03 2015 *)

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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A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

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A220493 Fibonacci 15-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-15).

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A220469 Fibonacci 14-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-14).

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

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