A056373 -id:A056373 - cp's OEIS frontend

A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 12, 18, 16, 5, 1, 12, 54, 40, 25, 6, 1, 40, 72, 160, 75, 36, 7, 1, 28, 405, 280, 375, 126, 49, 8, 1, 96, 390, 2176, 825, 756, 196, 64, 9, 1, 104, 1944, 2800, 8125, 2016, 1372, 288, 81, 10, 1, 280, 3411, 17920, 13175, 23976, 4312, 2304, 405

Offset: 1

N. J. A. Sloane, Dec 01 2007, based on email from Max Alekseyev, Nov 08 2007

From Andrew Howroyd, Apr 22 2017: (Start)
Number of step shifted (decimated) sequences of length n using a maximum of m different symbols. See A056371 for an explanation of step shifts. -
Number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence.  Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {0..n-1}. (End)

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, ...
3, 9, 18, 54, 72, 405, 390, 1944, 3411, 14985, 17802, 139968, 133104, 798525, 1804518, 5454378, 8072532, 64599849, 64573626, 437732424, ...
4, 16, 40, 160, 280, 2176, 2800, 17920, 44224, 263296, 419872, 4280320, 5594000, 44751616, 134391040, 539054080, 1073758360, 11453771776, 15271054960, 137575813120, ...
5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1830
Max Alekseyev, First 20 rows and columns of table
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row m=2 is A056371
Row m=3 is A056372
Row m=4 is A056373
Row m=5 is A056374
Row m=6 is A056375
Column n=2 is A000290
Column n=3 is A002411
Column n=4 is A019582
Cf. A285522, A285548.

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n]==1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; Table[a[m-n+1, n], {m, 1, 15}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Dec 01 2015 *)

for(i=1,15,for(m=1,i,n=i-m+1; print1(sum(k=1, n, if(gcd(k,n)==1, m^sumdiv(n,d,eulerphi(d)/znorder(Mod(k,d))),0))/eulerphi(n)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

Offset corrected by Andrew Howroyd, Apr 20 2017

A056393 Number of step shifted (decimated) sequence structures using a maximum of four different symbols.

Original entry on oeis.org

1, 2, 4, 11, 17, 107, 131, 811, 1893, 11107, 17599, 179371, 233449, 1866057, 5603787, 22469291, 44744047, 477262537, 636308685, 5732457131, 15272176697, 73301054891, 133274359129, 1466413263531

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. 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]

Cf. A056373, A288620.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

a(n) = Sum_{k=1..4} A288620(n, k). - Andrew Howroyd, Jun 13 2017

A056378 Number of step shifted (decimated) sequences using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 12, 60, 792, 1404, 10716, 31200, 205032, 349956, 3727932, 5065804, 41574312, 127199028, 517290132, 1041517620, 11195637720, 15012935676, 135825699612, 363040469732, 1746670165416, 3181465294092

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

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. A056373.

A056373(n)-4*A056372(n)+6*A056371(n)-4.

A056379 Number of step shifted (decimated) sequences using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 30, 900, 2800, 32010, 139080, 1276200, 2960940, 41626230, 75086430, 801522300, 3162262170, 16463793480, 42395689530, 579164463000, 983928850100, 11241277288950, 37913042835300

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

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. A056374.

A056374(n)-5*A056373(n)+10*A056372(n)-10*A056371(n)+5.

A056380 Number of step shifted (decimated) sequences using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 2520, 48060, 317520, 4109040, 12923136, 238785300, 559279980, 7612396920, 37864711260, 246263046840, 787758864480, 13282478342640, 27723264985920, 387585098313300, 1595144664456720

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

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. A056375.

A056375(n)-6*A056374(n)+15*A056373(n)-20*A056372(n)+15*A056371(n)-6.

A056383 Number of primitive (aperiodic) step shifted (decimated) sequences using a maximum of four different symbols.

Original entry on oeis.org

4, 12, 36, 144, 276, 2124, 2796, 17760, 44184, 263004, 419868, 4278000, 5593996, 44748804, 134390724, 539036160, 1073758356, 11453725416, 15271054956, 137575549680, 366528035564, 1759219864020, 3198580043436

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

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. A056371, A056373.

a(n) = Sum_{d|n} mu(d)*A056373(n/d).

cp's OEIS Frontend

A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

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A056393 Number of step shifted (decimated) sequence structures using a maximum of four different symbols.

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A056378 Number of step shifted (decimated) sequences using exactly four different symbols.

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A056379 Number of step shifted (decimated) sequences using exactly five different symbols.

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A056380 Number of step shifted (decimated) sequences using exactly six different symbols.

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A056383 Number of primitive (aperiodic) step shifted (decimated) sequences using a maximum of four different symbols.

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