A084508 -id:A084508 - cp's OEIS frontend

A087289 a(n) = 2^(2*n+1) + 1.

3, 9, 33, 129, 513, 2049, 8193, 32769, 131073, 524289, 2097153, 8388609, 33554433, 134217729, 536870913, 2147483649, 8589934593, 34359738369, 137438953473, 549755813889, 2199023255553, 8796093022209, 35184372088833, 140737488355329, 562949953421313, 2251799813685249

Offset: 0

W. Edwin Clark, Aug 29 2003

Number of pairs of polynomials (f,g) in GF(2)[x] satisfying deg(f) <= n, deg(g) <= n and gcd(f,g) = 1.
An unpublished result due to Stephen Suen, David desJardins, and W. Edwin Clark. This is the case k = 2, q = 2 of their formula q^((n+1)*k) * (1 - 1/q^(k-1) + (q-1)/q^((n+1)*k)) for the number of ordered k-tuples (f_1, ..., f_k) of polynomials in GF(q)[x] such that deg(f_i) <= n for all i and gcd(f_1, ..., f_k) = 1.
Apparently the same as A084508 shifted left.
Terms in binary are palindromes of the form 1x1 where x is a string of 2*n zeros (A152577). - Brad Clardy, Sep 01 2011
For n > 0, a(n) is the number k such that the number of iterations of the map k -> (3k +1)/8 == 4 (mod 8) until reaching (3k +1)/8 <> 4 (mod 8) equals n. (see the Collatz problem: the start of the parity trajectory of a(n) is n times {100} = 100100100100...100abcd...). - Michel Lagneau, Jan 23 2012
An Engel expansion of 2 to the base 4 as defined in A181565, with the associated series expansion 2 = 4/3 + 4^2/(3*9) + 4^3/(3*9*33) + 4^4/(3*9*33*129) + .... Cf. A199561 and A207262. - Peter Bala, Oct 29 2013
For x = A083420(n), y = A000079(n+1), z = a(n) then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
A254046(n+1) is the 3-adic valuation of a(n). - Fred Daniel Kline, Jan 11 2017

			a(0) = 3 since there are three pairs, (0,1), (1,0) and (1,1) of polynomials (f,g) in GF(2)[x] of degree at most 0 such that gcd(f,g) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Morrison, Random polynomials over finite fields.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A087290, A087291, A087292, A099393.
Equals A004171 + 1.
Cf. also A007583, A181565, A199561, A207262, A254046, A283070.
Cf. A000079, A083420, A084508, A152577.

[2^(2*n+1) + 1: n in [0..30]]; // Vincenzo Librandi, May 16 2011

Table[2^(2 n + 1) + 1, {n, 0, 20}] (* or *) 3 NestList[4 # - 1 &, 1, 20]
(* or *) CoefficientList[Series[(3 - 6 x)/((1 - x) (1 - 4 x)), {x, 0, 20}], x] (* Michael De Vlieger, Mar 03 2017 *)

a(n)=2^(2*n+1)+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (3-6*x)/((1-x)*(1-4*x)).
a(n) = 3*A007583(n).
a(n) = 4*a(n-1) - 3. - Lekraj Beedassy, Apr 29 2005
a(n) = A099393(n+1) - 2*A099393(n). - Brad Clardy, Sep 01 2011
a(n) = 2^(2*n + 1)*a(-1-n) for all n in Z. - Michael Somos, Jan 11 2017
a(n) = A283070(n) - 1. - Peter M. Chema, Mar 02 2017
From Elmo R. Oliveira, Feb 22 2025: (Start)
E.g.f.: exp(x)*(2*exp(3*x) + 1).
a(n) = 5*a(n-1) - 4*a(n-2). (End)

A230184 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to at least one element with value 2-x(i,j).

Original entry on oeis.org

0, 3, 0, 3, 9, 0, 9, 27, 33, 0, 15, 159, 231, 129, 0, 33, 825, 3411, 1971, 513, 0, 63, 4395, 44487, 73857, 16815, 2049, 0, 129, 23307, 596973, 2432241, 1603431, 143451, 8193, 0, 255, 123729, 7957785, 82359177, 133265847, 34825803, 1223799, 32769, 0, 513

Offset: 1

R. H. Hardin, Oct 11 2013

Table starts
.0.....3........3...........9.............15................33
.0.....9.......27.........159............825..............4395
.0....33......231........3411..........44487............596973
.0...129.....1971.......73857........2432241..........82359177
.0...513....16815.....1603431......133265847.......11393567289
.0..2049...143451....34825803.....7303192425.....1576417829097
.0..8193..1223799...756450105...400233701367...218117038953009
.0.32769.10440387.16430979183.21933865129257.30179260908320577

			Some solutions for n=3 k=4
..2..0..1..1....0..2..1..1....2..0..0..1....0..2..2..0....1..1..2..2
..0..2..2..0....0..0..2..2....0..2..1..1....0..1..2..0....2..0..0..0
..0..2..1..1....1..1..0..2....1..1..0..2....1..2..0..2....0..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..238

Column 2 is A084508(n+1)

Row 1 is A062510(n-1)

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 5*a(n-1) -4*a(n-2)
k=3: a(n) = 9*a(n-1) -4*a(n-2)
k=4: a(n) = 26*a(n-1) -97*a(n-2) +89*a(n-3) -18*a(n-4) +a(n-5)
k=5: [order 6] for n>8
k=6: [order 23] for n>24
k=7: [order 34] for n>37
Empirical for row n:
n=1: a(n) = a(n-1) +2*a(n-2)
n=2: a(n) = 4*a(n-1) +7*a(n-2) +a(n-3) -6*a(n-4) -5*a(n-5) for n>6
n=3: [order 21] for n>23
n=4: [order 93] for n>96

A084509 Number of ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 2, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984

Offset: 0

Antti Karttunen, Jun 02 2003

This sequence counts the length n asynchronic site swaps given in A084501/A084502.
Equals row sums of triangle A145463. - Gary W. Adamson, Oct 11 2008
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4, 1>5} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the first element is the largest. - Sergey Kitaev, Dec 11 2020
a(n) is the number of permutations p[1]..p[n] of {1,...,n} with p[j+1] < p[j]+4 for 0 < j < n. - Don Knuth, Apr 25 2022

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 48.

Michael De Vlieger, Table of n, a(n) for n = 0..1662
Fan Chung and R. L. Graham, Primitive juggling sequences, Amer. Math. Monthly 115(3) (2008), 185-19.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for sequences related to juggling
Index entries for linear recurrences with constant coefficients, signature (4).

First differences of A084508.
INVERT transform of A084519.
Cf. A002023, A084501, A084502, A084529, A145463.

A084509 := n -> `if`((n<4),n!,6*(4^(n-3)));
INVERT([seq(A084519(n),n=1..12)]);

LinearRecurrence[{4},{1,2,6},30] (* Harvey P. Dale, Aug 23 2018 *)

a(n) = n! for n <= 4, a(n) = 6*4^(n-3) = A002023(n-3) for n >= 3.

G.f.: 1 + x*(1 - 2*x - 2*x^2)/(1 - 4*x). - Philippe Deléham, Aug 16 2005

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020

A084506 The length of each successively larger 3-ball ground-state site swap given in A084501, i.e., the number of digits in each term of A084502.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 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

Antti Karttunen, Jun 02 2003

nonn

A. Karttunen, Table of n, a(n) for n = 1..32770
A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

Partial sums: A084505.
Differs from A084556 first time at the 130th term, where A084506(130) = 6, while A084556(130) = 5.
Cf. A084516, A084526, A084500, A084508.

A152577 a(n) = 10^(2*n - 1) + 1.

Original entry on oeis.org

11, 1001, 100001, 10000001, 1000000001, 100000000001, 10000000000001, 1000000000000001, 100000000000000001, 10000000000000000001, 1000000000000000000001, 100000000000000000000001

Offset: 1

Cino Hilliard, Dec 08 2008

These numbers are all divisible by 11. This follows from the identity a^n - b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd values of n. In this example a=10 and b=1 so a+b = 11. The sum of digits rule for divisibility by 11 also applies.

Bisection of A000533. Also, bisection of A062397. a(n) is also A084508(n+1) written in base 2. a(n) is also A087289(n-1) written in base 2. a(n) is also the concatenation of "1", 2(n-1) digits "0" and "1". - Omar E. Pol, Dec 13 2008

			From _Omar E. Pol_, Dec 14 2008: (Start)
n ....... a(n)
1 ....... 11
2 ...... 1001
3 ..... 100001
4 .... 10000001
5 ... 1000000001
(End)

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A000533, A062397, A084508, A087289. - Omar E. Pol, Dec 13 2008

LinearRecurrence[{101,-100},{11,1001},20] (* Harvey P. Dale, Nov 05 2015 *)

g(n)=forstep(x=1,n,2,y=(10^x+1);print1(y","))

a(n) = 100*a(n-1) - 99 (with a(1)=11). - Vincenzo Librandi, Dec 14 2010
G.f.: -11*x*(-1+10*x) / ( (100*x-1)*(x-1) ). - R. J. Mathar, Sep 01 2011
a(n) = 11*A095372(n-1). - R. J. Mathar, Sep 01 2011
a(n) = 101*a(n-1)-100*a(n-2). - Wesley Ivan Hurt, Apr 24 2021
E.g.f.: (exp(100*x) + 10*exp(x) - 11)/10. - Stefano Spezia, Mar 13 2025

cp's OEIS Frontend

A087289 a(n) = 2^(2*n+1) + 1.

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A230184 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to at least one element with value 2-x(i,j).

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A084509 Number of ground-state 3-ball juggling sequences of period n.

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A084506 The length of each successively larger 3-ball ground-state site swap given in A084501, i.e., the number of digits in each term of A084502.

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A152577 a(n) = 10^(2*n - 1) + 1.

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