A000322 -id:A000322 - 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

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

A112959 a(1) = a(2) = a(3) = a(4) = a(5) = 1; for n>1: a(n+5) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608845

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic pentanacci sequence.

This is to A000283 as a pentanacci (A000322) is to Fibonacci. Primes in this begin a(6) = 5 and a(7) = 29. a(8), a(9), a(10) and a(11) are semiprime.

			5^2 + 29^2 + 869^2 + 756029^2 + 571580604869^2 = 326704387862983487112029.

Seiichi Manyama, Table of n, a(n) for n = 1..16

Cf. A000283, A000288, A112957, A112958, A112960.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == a[5] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2 + a[n-5]^2}, a, {n, 16}] (* Vincenzo Librandi, Aug 21 2016 *)

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

A214826 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 4.

Original entry on oeis.org

1, 4, 4, 9, 17, 30, 56, 103, 189, 348, 640, 1177, 2165, 3982, 7324, 13471, 24777, 45572, 83820, 154169, 283561, 521550, 959280, 1764391, 3245221, 5968892, 10978504, 20192617, 37140013, 68311134, 125643764, 231094911, 425049809

Offset: 0

Abel Amene, Jul 29 2012

See Comments in A214727.

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

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,4,4];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,4,4},33] (* Ray Chandler, Dec 08 2013 *)

my(x='x+O('x^40)); Vec((1+3*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+3*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

G.f.: (1+3*x-x^2)/(1-x-x^2-x^3).

a(n) = K(n) - 2*T(n+1) + 5*T(n), where K(n) = A001644(n) and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

Original entry on oeis.org

1, 8, 8, 17, 33, 58, 108, 199, 365, 672, 1236, 2273, 4181, 7690, 14144, 26015, 47849, 88008, 161872, 297729, 547609, 1007210, 1852548, 3407367, 6267125, 11527040, 21201532, 38995697, 71724269, 131921498, 242641464, 446287231, 820850193, 1509778888

Offset: 0

Abel Amene, Aug 07 2012

See comments in A214727.

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

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,8,8];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+7*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

CoefficientList[Series[(x^2-7*x-1)/(x^3+x^2+x-1), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jun 18 2014 *)
LinearRecurrence[{1,1,1}, {1,8,8}, 40] (* G. C. Greubel, Apr 24 2019 *)

my(x='x+O('x^40)); Vec((1+7*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019

((1+7*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

G.f.: (1+7*x-x^2)/(1-x-x^2-x^3).

a(n) = -A000073(n) + 7*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A247192 Indices of primes in the hexanacci numbers sequence A000383.

Original entry on oeis.org

7, 9, 30, 31, 33, 46, 52, 54, 82, 102, 109, 124, 210, 301, 351, 365, 369, 1045, 2044, 2125, 2143, 2815, 4377, 4754, 4893, 7310, 11558, 17602, 17929, 28389, 32100, 44298, 106725, 151678, 197953

Offset: 1

Robert Price, Dec 03 2014

a(36) > 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.

a={1,1,1,1,1}; For[n=5, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[5]]=sum]

A253318 Indices of primes in the 7th-order Fibonacci number sequence, A060455.

Original entry on oeis.org

7, 8, 11, 12, 14, 15, 16, 17, 18, 19, 21, 23, 26, 32, 33, 36, 42, 44, 71, 72, 137, 180, 193, 285, 679, 955, 1018, 1155, 1176, 1191, 2149, 2590, 2757, 3364, 4233, 6243, 6364, 7443, 10194, 11254, 13318, 18995, 20478, 22647, 29711, 34769, 61815, 71993, 107494, 135942, 148831

Offset: 1

Robert Price, Dec 30 2014

nonn

a(52) > 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.

a={1,1,1,1,1,1,1}; step=7; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[7]]=sum]; lst

A061451 Array T(n,k) of k-th order Fibonacci numbers read by antidiagonals in up-direction.

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 5, 7, 9, 8, 6, 9, 13, 17, 13, 7, 11, 17, 25, 31, 21, 8, 13, 21, 33, 49, 57, 34, 9, 15, 25, 41, 65, 94, 105, 55, 10, 17, 29, 49, 81, 129, 181, 193, 89, 11, 19, 33, 57, 97, 161, 253, 349, 355, 144, 12, 21, 37, 65, 113, 193, 321, 497, 673, 653, 233

Offset: 1

Frank Ellermann, Jun 11 2001

			2, 3, 5, 8 ... first order, a(1)=2, a(3)=3, a(6)=5, a(10)=8, ...
3, 5, 9,17 ... 2nd order, a(2)=3, a(5)=5, a(9)=9, ...
4, 7,13,25 ... 3rd order, a(4)=4, a(8)=7, ...
5, 9,17,33 ... 4th order, a(7)=5, ...

N. Wirth, Algorithmen und Datenstrukturen, 1975, table 2.15 (ch. 2.3.4)

T. D. Noe, Rows n=1..50 of triangle, flattened

Rows 1..6: A000045, A000213, A000288, A000322, A000383, A060455.

max = 12; Clear[f]; f[k_, n_] /; n > k := f[k, n] = Sum[f[k, n - j], {j, 1, k + 1}]; f[k_, n_] = 1; t = Table[ Table[ f[k, n], {n, k + 1, max}], {k, 1, max}]; Table[ t[[k - n + 1, n]], {k, 1, max - 1}, {n, 1, k}] // Flatten (* Jean-François Alcover, Apr 10 2013 *)

A251653 5-step Fibonacci sequence starting with 0,0,1,0,0.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 4, 7, 14, 28, 55, 108, 212, 417, 820, 1612, 3169, 6230, 12248, 24079, 47338, 93064, 182959, 359688, 707128, 1390177, 2733016, 5372968, 10562977, 20766266, 40825404, 80260631, 157788246, 310203524, 609844071, 1198921876, 2357018348, 4633776065, 9109763884, 17909324244, 35208804417

Offset: 0

Arie Bos, Dec 06 2014

Doubling the entries > 1 as 1, 2, 2, 4, 4, 7, 7, 14, 14, 28, 28, 55, 55,... (offset 0) gives Nyblom's palindromic binary strings having no 5-runs of 1's. - R. J. Mathar, Mar 28 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_5(n)
H. Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, even length, r=4.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Other 5-step sequences are A000322, A001591, A023424, A074048, A124312, A124313, A122997, A135055, A135056, A145029

(see www.jsoftware.com) First construct the generating matrix
1  1  1  1  1
1  2  2  2  2
2  3  4  4  4
4  6  7  8  8
8 12 14 15 16
Given that matrix one can produce the first 5*200 numbers by
, M(+/ . *)^:(i.250) 0 0 1 0 0x

LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 1, 0, 0}, 100] (* G. C. Greubel, May 27 2016 *)

a(n+5) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4).

G.f.: x^2*(x^2 + x - 1)/(x^5 + x^4 + x^3 + x^2 + x - 1). - Chai Wah Wu, May 27 2016

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.

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

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A112959 a(1) = a(2) = a(3) = a(4) = a(5) = 1; for n>1: a(n+5) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2.

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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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A214826 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 4.

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A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

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