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Search: a045625 -id:a045625
Displaying 1-7 of 7 results found. page 1
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A284871 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) reversible strings of length n using a maximum of k different symbols. +10
8
1, 2, 0, 3, 1, 0, 4, 3, 4, 0, 5, 6, 15, 7, 0, 6, 10, 36, 39, 18, 0, 7, 15, 70, 126, 132, 29, 0, 8, 21, 120, 310, 540, 357, 70, 0, 9, 28, 189, 645, 1620, 2034, 1131, 126, 0, 10, 36, 280, 1197, 3990, 7790, 8316, 3276, 266, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A string and its reverse are considered to be equivalent.
REFERENCES
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]
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
T(n, k) = Sum_{d | n} mu(n/d) * (k^n + k^(ceiling(n/2))) / 2.
EXAMPLE
Table starts:
1 2 3 4 5 6 7 8 ...
0 1 3 6 10 15 21 28 ...
0 4 15 36 70 120 189 280 ...
0 7 39 126 310 645 1197 2044 ...
0 18 132 540 1620 3990 8568 16632 ...
0 29 357 2034 7790 23295 58779 131012 ...
0 70 1131 8316 39370 140610 412965 1050616 ...
0 126 3276 32760 195300 839790 2882376 8388576 ...
...
MATHEMATICA
b[n_, k_] := (k^n + k^Ceiling[n/2])/2;
a[n_, k_] := DivisorSum[n, MoebiusMu[n/#] b[#, k]&];
Table[a[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)
PROG
(PARI)
b(n, k) = (k^n + k^(ceil(n/2))) / 2;
a(n, k) = sumdiv(n, d, moebius(n/d) * b(d, k));
for(n=1, 10, for(k=1, 10, print1( a(n, k), ", "); ); print(); );
CROSSREFS
Columns 2-6 are A045625, A056314, A056315, A056316, A056317.
Cf. A277504, A143324.
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 04 2017
STATUS
approved
A056314 Number of primitive (aperiodic) reversible strings with n beads using a maximum of three different colors. +10
4
3, 3, 15, 39, 132, 357, 1131, 3276, 9945, 29508, 88935, 265668, 798252, 2391441, 7177584, 21523320, 64579920, 193709763, 581160255, 1743392040, 5230264026, 15690529437, 47071855131, 141214764600 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A string and its reverse are considered to be equivalent.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..24.
FORMULA
Sum mu(d)*A032120(n/d) where d|n.
CROSSREFS
Column 3 of A284871.
Cf. A045625.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
A056315 Number of primitive (aperiodic) reversible strings with n beads using a maximum of four different colors. +10
3
4, 6, 36, 126, 540, 2034, 8316, 32760, 131544, 524250, 2099196, 8388450, 33562620, 134217594, 536903100, 2147483520, 8590065660, 34359735816, 137439477756, 549755813250, 2199025344348, 8796093020154 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A string and its reverse are considered to be equivalent.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..22.
FORMULA
Sum mu(d)*A032121(n/d) where d|n.
CROSSREFS
Column 4 of A284871.
Cf. A045625, A056314.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
A056316 Number of primitive (aperiodic) reversible strings with n beads using a maximum of five different colors. +10
3
5, 10, 70, 310, 1620, 7790, 39370, 195300, 978050, 4882740, 24421870, 122069940, 610390620, 3051757490, 15258982680, 76293945000, 381470703120, 1907348623450, 9536748046870, 47683715818440 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A string and its reverse are considered to be equivalent.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..20.
FORMULA
Sum mu(d)*A032122(n/d) where d|n.
CROSSREFS
Column 5 of A284871.
Cf. A045625.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
A056317 Number of primitive (aperiodic) reversible strings with n beads using a maximum of six different colors. +10
3
6, 15, 120, 645, 3990, 23295, 140610, 839790, 5042610, 30232965, 181421850, 1088390415, 6530486970, 39182081385, 235093327980, 1410554953080, 8463334761210, 50779978307010, 304679900238330 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A string and its reverse are considered to be equivalent.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..19.
FORMULA
sum mu(d)*A056308(n/d) where d|n.
CROSSREFS
Column 6 of A284871.
Cf. A045625.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
A056331 Number of primitive (aperiodic) reversible string structures with n beads using a maximum of two different colors. +10
3
1, 1, 2, 4, 9, 16, 35, 66, 133, 261, 527, 1032, 2079, 4123, 8244, 16440, 32895, 65639, 131327, 262380, 524762, 1049071, 2098175, 4195230, 8390646, 16779231, 33558392, 67112892, 134225919, 268443306 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..30.
FORMULA
a(n) = sum mu(d)*A005418(n/d) where d divides n.
MATHEMATICA
a[n_] := DivisorSum[n, MoebiusMu[#] (2^(n/#-2) + 2^(Floor[n/#/2]-1))&];
Array[a, 30] (* Jean-François Alcover, Jun 29 2018 *)
CROSSREFS
Cf. A045625.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
A056318 Number of primitive (aperiodic) reversible strings with n beads using exactly two different colors. +10
1
0, 1, 4, 7, 18, 29, 70, 126, 266, 507, 1054, 2037, 4158, 8183, 16488, 32760, 65790, 131026, 262654, 524265, 1049524, 2097119, 4196350, 8388450, 16781292, 33554367, 67116784, 134217657, 268451838 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A string and its reverse are considered to be equivalent. Identical to A045625 for n>1.
REFERENCES
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]
LINKS
Table of n, a(n) for n=1..29.
FORMULA
Sum mu(d)*A056309(n/d) where d|n.
CROSSREFS
Cf. A045625.
KEYWORD
nonn
AUTHOR
Marks R. Nester
STATUS
approved
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Last modified August 30 04:38 EDT 2024. Contains 375526 sequences. (Running on oeis4.)