A007581 -id:A007581 - cp's OEIS frontend

A056478 Number of primitive (aperiodic) palindromic structures using a maximum of four different symbols.

1, 1, 0, 1, 1, 4, 3, 14, 13, 49, 46, 186, 181, 714, 700, 2789, 2780, 11050, 10997, 43946, 43895, 175259, 175088, 700074, 699875, 2798246, 2797536, 11188856, 11188191, 44747434, 44744591, 178973354, 178970560, 715860463, 715849600, 2863377048, 2863365834

Offset: 0

Marks R. Nester

nonn

Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A007581, A124303, A056460, A284826.

a(n) = Sum_{d|n} mu(d)*A124303(ceiling(n/(2*d))) for n > 0.

a(n) = Sum_{k=1..4} A284826(n, k) for n > 0. - Andrew Howroyd, Oct 02 2019

a(0)=1 prepended and terms a(32) and beyond from Andrew Howroyd, Oct 02 2019

A056505 Number of periodic palindromic structures using a maximum of four different symbols.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 13, 15, 40, 51, 133, 187, 479, 715, 1815, 2795, 7036, 11051, 27747, 43947, 110061, 175275, 438683, 700075, 1750959, 2798251, 6997643, 11188907, 27975351, 44747435, 111876715, 178973355, 447445692, 715860651, 1789684267, 2863377067, 7158491875

Offset: 0

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A007581, A285012.

a(n) = Sum_{k=1..4} A285012(n, k) for n > 0. - Andrew Howroyd, Oct 01 2019

Corrected by T. D. Noe, Oct 25 2006

a(0)=1 prepended and terms a(17) and beyond from Andrew Howroyd, Oct 01 2019

A362905 Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 8, 5, 3, 1, 1, 1, 16, 15, 11, 3, 1, 1, 1, 32, 51, 50, 14, 4, 1, 1, 1, 64, 187, 276, 99, 24, 4, 1, 1, 1, 128, 715, 1768, 969, 232, 30, 5, 1, 1, 1, 256, 2795, 12496, 11781, 3504, 429, 45, 5, 1, 1, 1, 512, 11051, 93600, 162877, 73440, 10659, 835, 55, 6, 1

Offset: 0

Andrew Howroyd, May 27 2023

Equivalently, T(n,k) is the number multisets with n elements drawn from {0..2^k-1} such that the bitwise-xor of all the elements gives zero.
T(n,k) is the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows.
T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and complementation of columns.

			Array begins:
=========================================
n/k| 0 1  2   3     4      5        6 ...
---+-------------------------------------
0  | 1 1  1   1     1      1        1 ...
1  | 1 1  1   1     1      1        1 ...
2  | 1 2  4   8    16     32       64 ...
3  | 1 2  5  15    51    187      715 ...
4  | 1 3 11  50   276   1768    12496 ...
5  | 1 3 14  99   969  11781   162877 ...
6  | 1 4 24 232  3504  73440  1878976 ...
7  | 1 4 30 429 10659 394383 18730855 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..4 are A000012, A004526(n+2), A053307, A362906, A363350.
Rows n=2..3 are A000079, A007581.
Main diagonal is A363351.
Cf. A054724, A340030, A340312, A363349.

A362905[n_,k_]:=(Binomial[2^k+n-1,n]+If[EvenQ[n],(2^k-1)Binomial[2^(k-1)+n/2-1,n/2],0])/2^k;Table[A362905[n-k,k],{n,0,15},{k,n,0,-1}] (* Paolo Xausa, Nov 19 2023 *)

T(n,k)={(binomial(2^k+n-1, n) + if(n%2==0, (2^k-1)*binomial(2^(k-1)+n/2-1,n/2)))/2^k}

T(n,k) = binomial(2^k+n-1, n)/2^k for odd n;
T(n,k) = (binomial(2^k+n-1, n) + (2^k-1)*binomial(2^(k-1)+n/2-1, n/2))/2^k for even n.
G.f. of column k: (1/(1-x)^(2^k) + (2^k-1)/(1-x^2)^(2^(k-1)))/2^k.

A273964 T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles right triangle, and new values introduced in 0..3 order.

Original entry on oeis.org

1, 2, 2, 5, 10, 5, 15, 105, 105, 15, 51, 1264, 2766, 1264, 51, 187, 15418, 49702, 49702, 15418, 187, 715, 188465, 784878, 543906, 784878, 188465, 715, 2795, 2304141, 13322548, 4010258, 4010258, 13322548, 2304141, 2795, 11051, 28171070, 232027264

Offset: 1

R. H. Hardin, Jun 05 2016

Table starts
.........1.................2...................5............15......51......187
.........2................10.................105..........1264...15418...188465
.........5...............105................2766.........49702..784878.13322548
........15..............1264...............49702........543906.4010258.16365681
........51.............15418..............784878.......4010258.5029904...725420
.......187............188465............13322548......16365681..725420.....2890
.......715...........2304141...........232027264......55643290...30060.......12
......2795..........28171070..........4056338728.....210405148....1337........2
.....11051.........344428103.........70402715640.....904155053......54........0
.....43947........4211086174.......1221782263634....4384022592......11........0
....175275.......51486063258......21228810396872...21350702064.......0........0
....700075......629484802987.....368923543916728..105038486755.......0........0
...2798251.....7696279187803....6410003754557834..514302715276.......0
..11188907....94097130028610..111371494441358332.2558243582787
..44747435..1150461107685501.1935130749075453956
.178973355.14065899352150196
.715860651

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

R. H. Hardin, Table of n, a(n) for n = 1..153

Column 1 is A007581(n-1).

Empirical for column k:
k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)
k=2: [order 7] for n>8

A273968 T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles triangle, and new values introduced in 0..3 order.

Original entry on oeis.org

1, 2, 2, 5, 10, 5, 15, 105, 105, 15, 51, 1264, 1946, 1264, 51, 187, 13045, 24587, 24587, 13045, 187, 715, 136063, 211121, 49534, 211121, 136063, 715, 2795, 1273961, 2116450, 36817, 36817, 2116450, 1273961, 2795, 11051, 11936399, 15084593, 40569

Offset: 1

R. H. Hardin, Jun 05 2016

Table starts
......1..........2..........5....15.....51.....187......715......2795.....11051
......2.........10........105..1264..13045..136063..1273961..11936399..95697853
......5........105.......1946.24587.211121.2116450.15084593.112714143.511843593
.....15.......1264......24587.49534..36817...40569....36570.....90627.....89429
.....51......13045.....211121.36817....356......10........0.........0
....187.....136063....2116450.40569.....10.......0........0
....715....1273961...15084593.36570......0.......0
...2795...11936399..112714143.90627......0
..11051...95697853..511843593.89429
..43947..800561388.2841058530
.175275.5824237329
.700075

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

R. H. Hardin, Table of n, a(n) for n = 1..82

Column 1 is A007581(n-1).

Empirical for column k:

k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)

A110613 a(n+3) = 5a(n+2) - 2a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.

Original entry on oeis.org

1, 0, 3, 7, 29, 107, 421, 1659, 6597, 26299, 105029, 419771, 1678405, 6712251, 26846277, 107379643, 429507653, 1718008763, 6871991365, 27487878075, 109951337541, 439805000635, 1759219303493, 7036875815867, 28147500467269

Offset: 0

Creighton Dement, Jul 31 2005

A Jacobsthal related sequence (A001045). This sequence was calculated using the same rules given for A108618; the "initial seed" is the floretion given in the program code, below.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-2,-8).

Cf. A001045, A007582, A007581, A049775, A087440, A108618, A110614.

seriestolist(series((1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2tessumseq[(.5'i - .5'k - .5i' + .5k' - .5'ij' - .5'ji' - .5'jk' - .5'kj')('i + j' + 'ij' + 'ji')] Sumtype is set to:sum[Y[15]] = sum(*) (from 3rd term, disregarding signs)

LinearRecurrence[{5, -2, -8}, {1, 0, 3}, 50] (* G. C. Greubel, Sep 01 2017 *)

x='x+O('x^50); Vec((1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1))) \\ G. C. Greubel, Sep 01 2017

G.f.: (1-5*x+5*x^2)/((4*x-1)*(2*x-1)*(x+1)).
Program "Superseeker" finds:
a(n) + a(n+1) = A007582(n) = A007581(n+1) - A007581(n).
a(n+2) - a(n) = A049775(n).
a(n) + 2*a(n+1) + a(n+2) = A087440(n+1).

A092055 a(n) = binomial(2 + 2^n,3).

Original entry on oeis.org

1, 4, 20, 120, 816, 5984, 45760, 357760, 2829056, 22500864, 179481600, 1433753600, 11461636096, 91659526144, 733141975040, 5864598896640, 46914643623936, 375308558925824, 3002434111406080, 24019335451770880, 192154133857304576, 1537230871833083904, 12297838178567454720

Offset: 0

Henry Bottomley, Feb 19 2004

a(n) = Sum_{i=1...(2^n)} i*(i+1)/2, this sequence is thus similar to A016131 as it is a sum of triangular numbers on the interval <1,2^n>, A016131 is a sum of triangular numbers on the interval <1,2^n - 1>. - Ctibor O. Zizka, Mar 03 2009

a(n) is the number of unordered (not necessarily distinct) triples of subsets taken from the power set of {1,2,...,n}. Cf. A007582 (pairs of such subsets). - Geoffrey Critzer, Jul 10 2013

			a(5) = C(2+2^5,3) = C(34,3) = 5984.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Cf. A000079, A007581, A092056, A016131.

[Binomial(2^n+2, 3): n in [0..30]]; // G. C. Greubel, Dec 27 2024

seq(binomial(2+2^n, 3), n=0..25); # Zerinvary Lajos, Feb 22 2008

nn=20;Table[Coefficient[Series[1/(1-x)^(2^n),{x,0,nn}],x^3],{n,0,nn}] (* Geoffrey Critzer, Jul 10 2013 *)
Binomial[2+2^Range[0,30], 3] (* G. C. Greubel, Dec 27 2024 *)

Vec((1-10*x+20*x^2)/((1-2*x)*(1-4*x)*(1-8*x)) + O(x^100)) \\ Colin Barker, Sep 13 2014

def A092055(n): return binomial(pow(2,n)+2,3)
print([A092055(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(n) = (2^(3*n-1) +3*2^(2*n-1) + 2^n)/3 = A092056(3, n).
a(n) = A000079(n)*A007581(n).
a(n) = 2*a(n-1) + 4^(n-1) + 8^(n-1).
a(n) = [x^3] 1/(1-x)^(2^n). - Geoffrey Critzer, Jul 11 2013
a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3). - Colin Barker, Sep 13 2014
G.f.: (1-10*x+20*x^2) / ((1-2*x)*(1-4*x)*(1-8*x)). - Colin Barker, Sep 13 2014
E.g.f.: (1/6)*(2*exp(2*x) + 3*exp(4*x) + exp(8*x)). - G. C. Greubel, Dec 27 2024

A056275 Number of primitive (aperiodic) word structures of length n using a 4-ary alphabet.

Original entry on oeis.org

1, 1, 4, 13, 50, 181, 714, 2780, 11046, 43895, 175274, 699875, 2798250, 11188191, 44747380, 178970560, 715860650, 2863365834, 11453377194, 45813202675, 183252461532, 733008625151, 2932033104554, 11728127521060, 46912504507000, 187649998452735, 750599971438464

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A008683, A054719.

a(n) = Sum_{d|n} mu(d)*A007581(n/d-1).

A056473 Number of palindromic structures using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 10, 10, 65, 65, 350, 350, 1701, 1701, 7770, 7770, 34105, 34105, 145750, 145750, 611501, 611501, 2532530, 2532530, 10391745, 10391745, 42355950, 42355950, 171798901

Offset: 1

Marks R. Nester

Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-26,26,24,-24).

Cf. A000453, A007581.

StirlingS2[Floor[(Range[40]+1)/2],4] (* or *) LinearRecurrence[ {1,9,-9,-26,26,24,-24},{0,0,0,0,0,0,1},40] (* Harvey P. Dale, Mar 08 2013 *)

stirling2( [(n+1)/2], 4).
G.f.: x^7/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)). [Colin Barker, Jul 24 2012]
a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=0, a(7)=1, a(n)=a(n-1)+9*a(n-2)-9*a(n-3)- 26*a(n-4)+ 26*a(n-5)+24*a(n-6)-24*a (n-7). - Harvey P. Dale, Mar 08 2013

A086444 Sequence associated with palindromic structures.

Original entry on oeis.org

1, 3, 10, 37, 150, 653, 2990, 14157, 68470, 335533, 1657470, 8227277, 40957190, 204250413, 1019649550, 5093448397, 25452860310, 127221189293, 635976675230, 3179495693517, 15896315681830, 79478090576173, 397379990430510

Offset: 0

Paul Barry, Jul 20 2003

Binomial transform of A007581. Second binomial transform of A007051 (with extra leading 1).

Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

LinearRecurrence[{10,-31,30},{1,3,10},30] (* Harvey P. Dale, Sep 14 2019 *)

G.f.: (1-7x+11x^2)/((1-2x)(1-3x)(1-5x));

a(n) = 2^n/3 + 3^n/2 + 5^n/6.

cp's OEIS Frontend

A056478 Number of primitive (aperiodic) palindromic structures using a maximum of four different symbols.

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A056505 Number of periodic palindromic structures using a maximum of four different symbols.

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A362905 Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.

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A273964 T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles right triangle, and new values introduced in 0..3 order.

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A273968 T(n,k)=Number of nXk 0..3 arrays with no three equal values forming an isosceles triangle, and new values introduced in 0..3 order.

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A110613 a(n+3) = 5*a(n+2) - 2*a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.

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A092055 a(n) = binomial(2 + 2^n,3).

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A056275 Number of primitive (aperiodic) word structures of length n using a 4-ary alphabet.

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A110613 a(n+3) = 5a(n+2) - 2a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.