A084509 -id:A084509 - cp's OEIS frontend

A084508 Partial sums of A084509. Positions of ones in the first differences of A084506.

0, 1, 3, 9, 33, 129, 513, 2049, 8193, 32769, 131073, 524289, 2097153, 8388609, 33554433, 134217729, 536870913, 2147483649, 8589934593, 34359738369, 137438953473, 549755813889, 2199023255553, 8796093022209, 35184372088833

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A084506, A084509, A087289.

LinearRecurrence[{5,-4},{0,1,3,9},30] (* Harvey P. Dale, May 21 2021 *)

a(n) = n for n < 2, a(n) = 2^(2*n - 3) + 1 = A087289(n-2) for n >= 2. - Antti Karttunen, Oct 24 2012 [Corrected by Petros Hadjicostas, Aug 02 2020]
From Chai Wah Wu, Jan 28 2021: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n > 3.
G.f.: x*(-2*x^2 - 2*x + 1)/((x - 1)*(4*x - 1)). (End)

A145463 Eigentriangle, row sums = A084509.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 13, 3, 2, 6, 47, 13, 6, 6, 24, 173, 47, 26, 18, 24, 96, 639, 173, 94, 78, 72, 96, 384, 2357, 639, 346, 282, 312, 288, 384, 1536, 8695, 2357, 1278, 1038, 1128, 1248, 1152, 1536, 6144, 32077, 8695, 4714, 3834, 4152, 4512, 4992, 4608, 6144, 24576

Offset: 1

Gary W. Adamson, Oct 11 2008

Row sums = A084509: (1, 2, 6, 24, 96, 384, 1536,...).
Right border = A084509 shifted: (1, 1, 2, 6, 24,...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
3, 1, 2;
13, 3, 2, 6;
47, 13, 6, 6, 24;
173, 47, 26, 18, 24, 96;
639, 173, 94, 78, 72, 96, 384;
2357, 639, 346, 282, 312, 288, 384, 1536;
...
Row 4 = (13, 3, 2, 6) = termwise products of (13, 3, 1, 1) and (1, 1, 2, 6).

A084509, Cf. A084519

Triangle read by rows, M * (A084509 * 0^(n-k)). M = an infinite lower triangular matrix with A084519: (1, 1, 3, 13, 47, 173,...) in every column; and (A084509 * 0^(n-k)) = an infinite lower triangular matrix with A084509 (1, 2, 6, 24, 96,...) shifted: (1, 1, 2, 6, 24, 96, 384,...) as the right diagonal and the rest zeros.

A164346 a(n) = 3 * 4^n.

Original entry on oeis.org

3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968

Offset: 0

Klaus Brockhaus, Aug 13 2009

Binomial transform of A000244 without initial 1.
Second binomial transform of A007283.
Third binomial transform of A010701.
Inverse binomial transform of A005053 without initial 1.
First differences of A024036. - Omar E. Pol, Feb 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 (powers of 4), A000244 (powers of 3), A007283 (3*2^n), A010701 (all 3's), A005053, A002001, A096045, A140660 (3*4^n+1), A002023 (6*4^n), A002063(9*4^n), A056120, A084509.

Magma
```
[ 3*4^n: n in [0..22] ];
```

3 4^Range[0,30]  (* Harvey P. Dale, Mar 11 2011 *)

A164346(n)=3*4^n \\ M. F. Hasler, Jul 28 2015

def A164346(n): return 3<<(n<<1) # Chai Wah Wu, Aug 30 2024

a(n) = 4*a(n-1) for n > 1; a(0) = 3.
G.f.: 3/(1-4*x).
a(n) = A002001(n+1). a(n) = A096045(n)+2. a(n) = A140660(n)-1.
a(n) = A002023(n)/2. a(n) = A002063(n)/3. a(n) = A056120(n+3)/9.
Apparently a(n) = A084509(n+3)/2.
a(n) = A110594(n+1), n>1. - R. J. Mathar, Aug 17 2009
a(n) = 3*A000302(n). - Omar E. Pol, Feb 18 2013
a(n) = A000079(2*n) + A000079(2*n+1). - M. F. Hasler, Jul 28 2015
E.g.f.: 3*exp(4*x). - G. C. Greubel, Sep 15 2017

A084501 An infinite juggling sequence of three balls: successively larger ground-state 3-ball site swaps listed in lexicographic order.

Original entry on oeis.org

3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 2, 4, 2, 3, 4, 4, 1, 5, 2, 2, 5, 3, 1, 3, 3, 3, 3, 3, 3, 4, 2, 3, 4, 2, 3, 3, 4, 4, 1, 3, 5, 2, 2, 3, 5, 3, 1, 4, 2, 3, 3, 4, 2, 4, 2, 4, 4, 1, 3, 4, 4, 4, 0, 4, 5, 1, 2, 4, 5, 3, 0, 5, 2, 2, 3, 5, 2, 4, 1, 5, 3, 1, 3, 5, 3, 4, 0, 5, 5, 1, 1, 5, 5, 2, 0, 6, 2, 2, 2, 6, 2, 3, 1, 6, 3

Offset: 1

Antti Karttunen, Jun 02 2003

Every possible 3-ball asynchronic site swap of finite period occurs as a subsequence of this sequence. E.g., "51" (three-ball shower) occurs first time at a(65)=5, a(66)=1.
We obtain the sequence by traversing each possible loop of successively larger lengths in 3-ball state graph as depicted in Polster's book, or section 7 of Knutson's Siteswap FAQ (but not limited by throw height), starting from and ending to the ground state 7 (xxx) and by concatenating those sequences in lexicographic order.
One can take any subsequence A084501[i..j] such that A084503(i-1) = A084503(j) = 7 and try to juggle it periodically or give it to one of the Siteswap animators available at J.I.S., e.g., by taking the first 39 terms, one gets a site swap pattern "333423333424234415225313333334234233441".

			The successive site swaps are: 3; 3,3; 4,2; 3,3,3; 3,4,2; 4,2,3; 4,4,1; 5,2,2; 5,3,1; 3,3,3,3; ... See A084502.

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 45.

A. Karttunen, Table of n, a(n) for n = 1..283991
A. Karttunen, Scheme-program for computing this sequence
A. Knutson, Siteswap FAQ, Section 7, A deeper level: landing schedules, or juggling states
Juggling Information Service, Site Swap animators and other juggling software
Index entries for sequences related to juggling

Subsets: A084511, A084521.
The number of such site swaps of length n is given by A084509.
First position where n appears: A084507.

A164908 a(n) = (3*4^n - 0^n)/2.

Original entry on oeis.org

1, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984, 1688849860263936

Offset: 0

Klaus Brockhaus, Aug 31 2009

Binomial transform of A164907. Inverse binomial transform of A057651.
Partial sums are in A083420.
Decimal representations of the n-th iterations of elementary cellular automata rules 14, 46, 142 and 174 generate this sequence (see A266298 and A266299). - Karl V. Keller, Jr., Aug 31 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (4).

Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.

Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.

Magma
```
[ (3*4^n-0^n)/2: n in [0..22] ];
```

a[n_]:=(MatrixPower[{{2,2},{2,2}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* Harvey P. Dale, Feb 14 2012 *)

a(n)=3*4^n\2 \\ Charles R Greathouse IV, Oct 12 2015

print([int(6*4**(n-1)) for n in range(50)]) # Karl V. Keller, Jr., Aug 30 2021

a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1+2*x)/(1-4*x).
a(n) = floor(6*4^(n-1)). - Karl V. Keller, Jr., Aug 30 2021
E.g.f.: (3*exp(4*x) - 1)/2. - Elmo R. Oliveira, Mar 31 2025

A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 3, 13, 47, 173, 639, 2357, 8695, 32077, 118335, 436549, 1610471, 5941181, 21917583, 80856053, 298285687, 1100404333, 4059496479, 14975869477, 55247410055, 203812962077, 751885445295, 2773777080149, 10232728055191

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This sequence counts the length n asynchronic site swaps given in A084511/A084512.
First differences of A084518. INVERTi transform of A084509. Cf. also A084529, A003319.
Equals left border of triangle A145463. - Gary W. Adamson, Oct 11 2008

Carsten Elsner, Dominic Klyve and Erik R. Tou, A zeta function for juggling sequences, Journal of Combinatorics and Number Theory, Volume 4, Issue 1, 2012, pp. 1-13; ISSN 1942-5600

Fan Chung, R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194
Index entries for sequences related to juggling
Index entries for linear recurrences with constant coefficients, signature (3, 2, 2).

Cf. A145463. - Gary W. Adamson, Oct 11 2008

INVERTi([seq(A084509(n),n=1..80)]);
with(combinat); A084519 := proc(n) option remember; local c,i,k; A084509(n)-add(add(mul(A084519(i),i=c),c=composition(n,k)),k=2..n); end;

LinearRecurrence[{3,2,2},{1,1,3},30] (* Harvey P. Dale, Jul 20 2013 *)

a(n) seems to satisfy the recurrence: a(1) = a(2) = 1, a(3) = 3 and a(n) = 3*a(n-1)+2*a(n-2)+2*a(n-3). If so, a(n) = floor(A*B^n+1/2) where B = 3.6890953... is the real positive root of x^3-3x^2-2x-2 = 0 and A = 0.0687059... is the real positive root of 118*x^3+118*x^2+35*x-3 = 0. - Benoit Cloitre, Jun 14 2003 [This conjecture is established in the Chung-Graham paper.]

G.f.: x*(1-2*x-2*x^2)/(1-3*x-2*x^2-2*x^3). - Colin Barker, Jan 14 2012

A084502 Successively larger 3-ball ground-state site swaps of A084501 in concatenated decimal notation.

Original entry on oeis.org

3, 33, 42, 333, 342, 423, 441, 522, 531, 3333, 3342, 3423, 3441, 3522, 3531, 4233, 4242, 4413, 4440, 4512, 4530, 5223, 5241, 5313, 5340, 5511, 5520, 6222, 6231, 6312, 6330, 6411, 6420, 33333, 33342, 33423, 33441, 33522, 33531, 34233, 34242, 34413

Offset: 1

Antti Karttunen, Jun 02 2003

Note that this decimal representation works only up to the A084500(A084507(10))-1 = 7707th term which is 99600000, after which follows the 7708th solution 10,2,2,2,2,2,2,2 which would be usually represented as "A2222222".

A. Karttunen, Scheme-program for computing this sequence
Index entries for sequences related to juggling

The number of 'digits' in term a(n) is given by A084506.
The number of terms of length n is given by A084509.
Cf. A084512, A084522.

A084529 Number of 'prime' ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 3, 12, 42, 142, 502, 1702, 5878

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This sequence counts the length n asynchronic site swaps given in A084521/A084522.

A juggling sequence is defined as 'prime', if it does not visit any state more than once. This means that in A084523 no integer occurs twice between consecutive sevens.

B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, pp. 50-51.

A. Karttunen, Scheme-program for counting this sequence
Index entries for sequences related to juggling

First differences of A084528. Cf. A084509, A084519.

A255253 Complete list of siteswaps (indecomposable ground-state in concatenated decimal notation organized first by sum of digits and then by magnitude).

Original entry on oeis.org

0, 1, 2, 3, 4, 31, 40, 5, 6, 42, 51, 60, 312, 330, 411, 420, 501, 600, 7, 8, 53, 62, 71, 3122, 3302, 4013, 4112, 4130, 4202, 4400, 5111, 5120, 5201, 5300, 6011, 6020, 7001, 8000, 9, 423, 441, 450, 522, 531, 603, 612, 630

Offset: 1

Gordon Hamilton, Feb 18 2015

Siteswaping is worthy of exploration in the elementary school classroom. In my experience (Gordon Hamilton) students across a full spectrum of ability find the subject matter intriguing and the mathematics engaging.
By "indecomposable" we mean that the juggling state sequence associated to each loop should not return to the ground state 7 (xxx) until after the last throw.
By "ground state" we mean that the permutation is chosen that is as large as possible. Example: 3302 is the same as 3023 and 0233 and 2330. Only the 3302 is in the list because it is the largest number.
The list breaks down at term 57, which requires a digit for "10." In the classroom this can be solved by writing "10" vertically or using commas.

			There are 13 siteswap sequences that have a digit-sum of 9. In order, these are 9, 423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900.

Cf. A065178, A084509, A065180.

cp's OEIS Frontend

A084508 Partial sums of A084509. Positions of ones in the first differences of A084506.

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A145463 Eigentriangle, row sums = A084509.

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A164346 a(n) = 3 * 4^n.

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A084501 An infinite juggling sequence of three balls: successively larger ground-state 3-ball site swaps listed in lexicographic order.

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A164908 a(n) = (3*4^n - 0^n)/2.

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A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n.

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A084502 Successively larger 3-ball ground-state site swaps of A084501 in concatenated decimal notation.

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A084529 Number of 'prime' ground-state 3-ball juggling sequences of period n.

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