A010716 -id:A010716 - cp's OEIS frontend

A101622 A Horadam-Jacobsthal sequence.

0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590

Offset: 0

Paul Barry, Dec 10 2004

Companion sequence to A084639.
This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 27 2017
Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [_Wolfdieter Lang_, Oct 18 2010]
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Stephen Wolfram, A New Kind of Science.
Wolfram Research, Wolfram Atlas of Simple Programs.
Index entries for sequences related to cellular automata.
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A131953.

[(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)

concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

a(n) = (2^(n+2) + (-1)^n - 5)/2.
G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).
a(n) = (A014551(n+2)-5)/2.
(1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007
From Paul Curtz, Jan 01 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) + 5.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = A000079(n+1) - A010693(n).
a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).
a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.
a(2n+1) - 2*a(2n) = 1.
a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017
a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

A023004 Number of partitions of n into parts of 5 kinds.

Original entry on oeis.org

1, 5, 20, 65, 190, 506, 1265, 2990, 6765, 14725, 31027, 63505, 126730, 247170, 472295, 885723, 1633000, 2963840, 5302075, 9358470, 16313440, 28107365, 47902010, 80803485, 134992865, 223474667, 366772720, 597049255, 964375855, 1546208695, 2461649861, 3892774130

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010716. - Alois P. Heinz, Oct 17 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2;
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. 5th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:=proc(n) option remember; `if`(n=0, 1, add(add(d*5, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^5,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

\ps100 for(n=0,100,print1((polcoeff(1/eta(x)^5,n,x)),","))

G.f.: Product_{m>=1} 1/(1-x^m)^5.
a(n) ~ 5^(3/2) * exp(Pi * sqrt(10*n/3)) / (32 * 3^(3/2) * n^2) * (1 - (3*sqrt(6/5) /Pi + 5*sqrt(5/6)*Pi/24) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.

Original entry on oeis.org

-1, 2, 9, 13, 31, 57, 119, 233, 471, 937, 1879, 3753, 7511, 15017, 30039, 60073, 120151, 240297, 480599, 961193, 1922391, 3844777, 7689559, 15379113, 30758231, 61516457, 123032919, 246065833, 492131671, 984263337, 1968526679, 3937053353, 7874106711

Offset: 1

Paul Curtz, Oct 10 2008, Oct 14 2008

			A020806 and its repeated differences in the next rows start as follows:
..1,..4,..2,..8,..5,..7,..1,..4,..2,..8, <- A020806
..3,.-2,..6,.-3,..2,.-6,..3,.-2,..6,.-3, <- A131969
.-5,..8,.-9,..5,.-8,..9,.-5,..8,.-9,..5,
.13,-17,.14,-13,.17,-14,.13,-17,.14,-13,
-30,.31,-27,.30,-31,.27,-30,.31,-27,.30,
.61,-58,.57,-61,.58,-57,.61,-58,.57,-61,
The diagonal is 1,-2,-9,-13,-31,... which yields a(n) after signs are flipped.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Join[{-1}, LinearRecurrence[{1, 2}, {2, 9}, 40]] (* Jean-François Alcover, Nov 06 2017 *)

Vec(-x*(1 - 3*x - 9*x^2) / ((1 + x)*(1 - 2*x)) + O(x^50)) \\ Colin Barker, Nov 06 2017

a(n+1) - 2*a(n) = (-1)^n*A010716(n), n>1, period 2.
G.f.: x*(1-3*x-9*x^2) / ((1+x)*(2*x-1)). - R. J. Mathar, Oct 24 2008
a(n) = 11*2^(n-2)/3 - 5*(-1)^n/3, n>1. - R. J. Mathar, Oct 24 2008
From Colin Barker, Nov 06 2017: (Start)
a(n) = (11*2^n - 20) / 12 for n>1 and even.
a(n) = (11*2^n + 20) / 12 for n>1 and odd.
a(n) = a(n-1) + 2*a(n-2) for n>3.
(End)

Edited and extended by R. J. Mathar, Oct 24 2008

A374359 a(1) = 2, a(n) = 5 for n > 1.

Original entry on oeis.org

2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Stefano Spezia, Jul 06 2024

Decimal expansion of 23/9, which is an approximation of the 5th root of 109 (A374357).

Simple continued fraction expansion of (1 + sqrt(29))/14 = (1 + A010484)/14.

			2.555555555555555555555555555555555555555...

Greg Martin and Winnie Miao, abc triples, arXiv:1409.2974 [math.NT], 2014. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A374357 (decimal expansion of the 5th root of 109), A374358 (continued fraction of the 5th root of 109).
Cf. A010484.
Essentially the same as A021022 and A010716.

Mathematica
```
LinearRecurrence[{1},{2,5},100]
```

G.f.: x*(2 + 3*x)/(1 - x).
a(n) = a(n-1) for n > 2.
E.g.f.: 5*exp(x) - 3*x - 5.

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

Original entry on oeis.org

2, 1, 5, 4, 8, 7, 11, 10, 14, 13, 17, 16, 20, 19, 23, 22, 26, 25, 29, 28, 32, 31, 35, 34, 38, 37, 41, 40, 44, 43, 47, 46, 50, 49, 53, 52, 56, 55, 59, 58, 62, 61, 65, 64, 68, 67, 71, 70, 74, 73, 77, 76, 80, 79, 83, 82, 86, 85, 89, 88, 92, 91, 95, 94, 98, 97, 101, 100, 104, 103, 107

Offset: 0

Vincenzo Librandi, Oct 16 2009

A sequence defined by a(1)=1, a(n)=k*n-a(n-1), k a constant parameter, has recurrence a(n)= 3*a(n-1) -3*a(n-2) +a(n-3). Its generating function is x*(1+2*(k-1)*x+(1-k)*x^2)/((1+x)*(1-x)^2). The closed form is a(n) = k*n/2+k/4+(-1)^n*(3*k/4-1). This applies with k=3 to this sequence here, and for example to sequences A165033, and A166519-A166525. - R. J. Mathar, Oct 17 2009
From Paul Curtz, Feb 20 2010: (Start)
Also: A001651, terms swapped by pairs.
a(n) mod 9 defines a period-6 sequence which is a permutation of A141425. (End)

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

Cf. A001651, A010685, A010716, A016777, A016945, A073010.

[(3 +5*(-1)^n+6*n)/4: n in [0..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(2 x^2 - x + 2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Harvey P. Dale, Mar 25 2011 *)
Table[(3 + 5 (-1)^n + 6 n) / 4, {n, 0, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

a(n) = 3*n - a(n-1).
From Paul Curtz, Feb 20 2010: (Start)
a(n+1)-a(n) = (-1)^(n+1)*A010685(n).
Second differences: |a(n+2)-2*a(n+1)+a(n)| = A010716(n).
a(2*n) + a(2*n+1) = A016945(n) = 6*n+3.
a(2*n) = A016945(n).
a(2*n+1) = A016777(n). (End)
G.f. ( 2-x+2*x^2 ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011
E.g.f.: (1/4)*exp(-x)*(5 + 3*exp(2*x) + 6*x*exp(2*x)). - G. C. Greubel, May 15 2016
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 24 2023

a(0)=2 added by Paul Curtz, Feb 20 2010

A190357 Decimal expansion of 1/4 - 2/Pi^2.

Original entry on oeis.org

0, 4, 7, 3, 5, 7, 6, 3, 2, 7, 1, 5, 3, 2, 4, 4, 5, 7, 1, 1, 2, 2, 4, 1, 0, 7, 3, 5, 8, 0, 5, 4, 4, 7, 2, 2, 1, 9, 1, 2, 8, 2, 4, 5, 0, 6, 5, 5, 5, 0, 6, 9, 0, 2, 3, 0, 8, 7, 8, 1, 9, 3, 6, 2, 1, 1, 6, 5, 3, 7, 5, 8, 0, 7, 5, 5, 2, 9, 1, 1, 7, 6, 1, 7, 5, 8, 1, 4, 5, 2, 1, 4, 8, 7, 5, 6, 3, 2, 4, 7, 7

Offset: 0

Jonathan Vos Post, May 09 2011

Constant given on p. 1 of Schlage-Puchta.

			0.047357632715324457112241...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jan-Christoph Schlage-Puchta, Sumsets avoiding squarefree integers, arXiv:1105.1305 [math.NT], May 06 2011.
Index entries for transcendental numbers

Cf. A005117, A010716 (1/18), A059956 (6/Pi^2).

R:=RealField(100); [(1/4) - (2/Pi(R)^2)]; // G. C. Greubel, Apr 05 2018

Join[{0}, RealDigits[(1/4) - (2/Pi^2), 10, 100][[1]]] (* G. C. Greubel, Apr 05 2018 *)

(1/4) - (2/Pi^2) \\ G. C. Greubel, Apr 05 2018

Equals (1/4) - (2/Pi^2).

A084962 Iterations of the Fibonacci sequence starting at 6.

Original entry on oeis.org

6, 8, 21, 10946

Offset: 0

Hollie L. Buchanan II, Jun 14 2003

nonn

The next term, a(4) = 1.695... * 10^2287, has 2288 digits and is too large to display.

This sequence is of interest because the sequences with this recurrence and a(0) in {0, 1, 2, 3, 4} all converge to 1 and the sequence with a(0) = 5 is constant.

			a(3) = Fibonacci(a(2)) = Fibonacci(21) = 10946.

Alonso del Arte, Table of a(n) for n=0..4, with digits of a(4) grouped in hundreds

Cf. A000045, A007097, A010716 (starting with 5), A084963 (starting with 7).

a:= proc(n) option remember; `if`(n=0, 6,
      (<<0|1>, <1|1>>^a(n-1))[1,2])
    end:
seq(a(n), n=0..4);  # Alois P. Heinz, May 09 2020

fibonaccieth6[m_] := Module[{ex = 6}, Do[ex = Fibonacci[ex], {m}]; ex] Table[fibonaccieth6[m], {m, 0, 4}]
NestList[Fibonacci[#] &, 6, 4] (* Alonso del Arte, Apr 30 2020 *)

val fiboLimited: LazyList[Int] = 0 #:: 1 #:: fiboLimited.zip(fiboLimited.tail).map { n => n._1 + n._2 }
def fibonaccieth(start: Int): LazyList[Int] = LazyList.iterate(start)(fiboLimited)
fibonaccieth(6).takeWhile( > 0).toList // _Alonso del Arte, Apr 30 2020

a(0) = 6, a(n) = Fibonacci(a(n-1)) for n>0.

A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

Original entry on oeis.org

0, -1, -1, -4, -5, -2, -7, -8, -3, -10, -11, -4, -13, -14, -5, -16, -17, -6, -19, -20, -7, -22, -23, -8, -25, -26, -9, -28, -29, -10, -31, -32, -11, -34, -35, -12, -37, -38, -13, -40, -41, -14, -43, -44, -15, -46, -47, -16, -49, -50, -17, -52, -53, -18, -55, -56, -19, -58, -59, -20

Offset: 0

Paul Curtz, Sep 07 2011

0,  -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5,    = a(n)/b(n),
1,    0,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
1,   -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
3,   -2,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
5,   -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
11, -10,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .
a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).
b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).
a(n+5)-a(n+2)=b(n+5)  (=-1,-3,-3,=-A169609(n)).

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Sep 18 2012 *)

a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.
From Chai Wah Wu, May 07 2024: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 7.
G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

A176260 Periodic sequence: Repeat 5, 1.

Original entry on oeis.org

5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5

Offset: 0

Klaus Brockhaus, Apr 13 2010

Interleaving of A010716 and A000012.
Also continued fraction expansion of (5+3*sqrt(5))/2.
Also decimal expansion of 17/33.
Essentially first differences of A047264.
Binomial transform of 5 followed by -A122803 without initial terms 1, -2.
Inverse binomial transform of 5 followed by A007283 without initial term 3.
Second inverse binomial transform of A168607 without initial term 3.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 6*x^5 + ... is the o.g.f. for A008805. - Peter Bala, Mar 13 2015

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010716 (all 5's sequence), A000012 (all 1's sequence), A090550 (decimal expansion of (5+3*sqrt(5))/2), A010686 (repeat 1, 5), A047264 (congruent to 0 or 5 mod 6), A122803 (powers of -2), A007283 (3*2^n), A168607 (3^n+2), A008805.

&cat[ [5, 1]: n in [0..52] ];
[ 3+2*(-1)^n: n in [0..104] ];

a(n) = 3+2*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 5, a(1) = 1.
a(n) = -a(n-1)+6 for n > 0; a(0) = 5.
a(n) = 5*((n+1) mod 2)+(n mod 2).
a(n) = A010686(n+1).
G.f.: (5+x)/(1-x^2).
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 5, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+2^(2-s)). (End)
E.g.f.: 5*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025

cp's OEIS Frontend

A101622 A Horadam-Jacobsthal sequence.

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A023004 Number of partitions of n into parts of 5 kinds.

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A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.

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A374359 a(1) = 2, a(n) = 5 for n > 1.

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A166517 a(n) = (3 + 5*(-1)^n + 6*n)/4.

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Magma

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A190357 Decimal expansion of 1/4 - 2/Pi^2.

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Magma

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PARI

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A084962 Iterations of the Fibonacci sequence starting at 6.

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A166517 a(n) = (3 + 5(-1)^n + 6n)/4.