A327396
Triangle read by rows: T(n,k) is the number of n-bead necklace structures with beads of exactly k colors and no adjacent beads having the same color.
Original entry on oeis.org
0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 3, 5, 2, 1, 0, 0, 3, 10, 8, 2, 1, 0, 1, 7, 33, 40, 18, 3, 1, 0, 0, 11, 83, 157, 104, 28, 3, 1, 0, 1, 19, 237, 650, 615, 246, 46, 4, 1, 0, 0, 31, 640, 2522, 3318, 1857, 495, 65, 4, 1, 0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1
Offset: 1
Triangle begins:
0;
0, 1;
0, 0, 1;
0, 1, 1, 1;
0, 0, 1, 1, 1;
0, 1, 3, 5, 2, 1;
0, 0, 3, 10, 8, 2, 1;
0, 1, 7, 33, 40, 18, 3, 1;
0, 0, 11, 83, 157, 104, 28, 3, 1;
0, 1, 19, 237, 650, 615, 246, 46, 4, 1;
0, 0, 31, 640, 2522, 3318, 1857, 495, 65, 4, 1;
0, 1, 63, 1817, 9888, 17594, 13311, 4911, 944, 97, 5, 1;
...
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R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace((y-1)*exp(-x + O(x*x^(n\m))) - y + exp(-x + sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d)) ), x, x^m))/x), -n)]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 09 2019
A306896
a(n) = Sum_{d|n} (2^d + 2*(-1)^d)*phi(n/d).
Original entry on oeis.org
0, 6, 6, 24, 30, 84, 126, 288, 522, 1080, 2046, 4224, 8190, 16548, 32850, 65856, 131070, 262836, 524286, 1049760, 2097438, 4196412, 8388606, 16782048, 33554550, 67117128, 134218782, 268452240, 536870910, 1073777040, 2147483646, 4295033472, 8589938742, 17180000352, 34359739050
Offset: 1
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See A306888.
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Table[DivisorSum[n, (2^# + 2 (-1)^#) EulerPhi[n/#] &], {n, 35}] (* Michael De Vlieger, Mar 18 2019 *)
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a(n) = sumdiv(n, d, (2^d + 2*(-1)^d)*eulerphi(n/d)); \\ Michel Marcus, Mar 16 2019
A306898
a(n) = Sum_{d|n} 2^d*phi(2*n/d).
Original entry on oeis.org
2, 8, 12, 32, 40, 96, 140, 320, 540, 1120, 2068, 4320, 8216, 16688, 32880, 66176, 131104, 263376, 524324, 1050880, 2097480, 4198480, 8388652, 16786368, 33554600, 67125344, 134218836, 268468928, 536870968, 1073809920, 2147483708, 4295099648, 8589938808, 17180131456, 34359739120, 68720011776
Offset: 1
A309673
Number of n-bead necklace structures using a maximum of four different colored beads and no adjacent beads having the same color.
Original entry on oeis.org
0, 1, 1, 3, 2, 9, 13, 41, 94, 257, 671, 1881, 5110, 14301, 39871, 112281, 316520, 897297, 2548819, 7265383, 20754748, 59437181, 170549237, 490338539, 1412147684, 4073528481, 11767897903, 34042917197, 98606864030, 285960106473, 830206177801, 2412787265021
Offset: 1
A327397
Number of n-bead necklace structures with beads of exactly three colors and no adjacent beads having the same color.
Original entry on oeis.org
0, 0, 1, 1, 1, 3, 3, 7, 11, 19, 31, 63, 105, 201, 367, 695, 1285, 2451, 4599, 8775, 16651, 31837, 60787, 116639, 223697, 430395, 828525, 1598227, 3085465, 5965999, 11545611, 22370999, 43383571, 84217615, 163617805, 318150719, 619094385, 1205614053, 2349384031
Offset: 1
Necklace structures for n=3..8 are:
a(3) = 1: ABC;
a(4) = 1: ABAC;
a(5) = 1: ABABC;
a(6) = 3: ABABAC, ABACBC, ABCABC;
a(7) = 3: ABABABC, ABABCAC, ABACABC;
a(8) = 7: ABABABAC, ABABACAC, ABABACBC, ABABCABC, ABABCBAC, ABACABAC, ABACBABC.
A306899
a(n) = Sum_{d|n} (2^d - (-1)^d)*phi(3*n/d).
Original entry on oeis.org
6, 12, 36, 48, 90, 180, 294, 576, 1134, 2160, 4158, 8496, 16458, 33096, 65880, 131712, 262242, 525852, 1048686, 2099520, 4195296, 8392824, 16777350, 33564672, 67109250, 134234256, 268438860, 536904480, 1073741994, 2147556240, 4294967478, 8590066944
Offset: 1
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See A306888.
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Table[DivisorSum[n, (2^# - (-1)^#) EulerPhi[3 n/#] &], {n, 10^4}] (* Michael De Vlieger, Mar 18 2019 *)
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a(n) = sumdiv(n, d, (2^d - (-1)^d)*eulerphi(3*n/d)); \\ Michel Marcus, Mar 16 2019
Showing 1-6 of 6 results.
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