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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120, 65280, 19656, 990, 0
Offset: 1

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Author

Alois P. Heinz, Aug 07 2008

Keywords

Comments

Column k is Dirichlet convolution of mu(n) with k^n.
The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.

Examples

			T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,  2,   3,    4,    5, ...
  0,  2,   6,   12,   20, ...
  0,  6,  24,   60,  120, ...
  0, 12,  72,  240,  600, ...
  0, 30, 240, 1020, 3120, ...

Links

Crossrefs

Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127.
Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133.
Main diagonal gives A252764.
Cf. A074650, A143325, A008683, A054525.

Programs

  • Maple
    with(numtheory): f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T:= (n,k)-> f0(n)(k); seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
  • Mathematica
    f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

Formula

T(n,k) = Sum_{d|n} k^d * mu(n/d).
T(n,k) = k^n - Sum_{d
T(n,k) = A143325(n,k) * k.
T(n,k) = A074650(n,k) * n.
So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021

A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 6, 6, 4, 1, 0, 0, 12, 24, 12, 5, 1, 0, 0, 30, 72, 60, 20, 6, 1, 0, 0, 54, 240, 240, 120, 30, 7, 1, 0, 0, 126, 696, 1020, 600, 210, 42, 8, 1, 0, 0, 240, 2184, 4020, 3120, 1260, 336, 56, 9, 1
Offset: 0

Views

Author

Peter Luschny, Jul 04 2023

Keywords

Comments

Row n gives the number of n-ary sequences with primitive period k.
See A074650 and A143324 for combinatorial interpretations.

Examples

			Array A(n, k) starts:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  0,   0,    0,     0,      0,       0,        0, ... A019590
[2] 1, 2,  2,   6,   12,    30,     54,     126,      240, ... A027375
[3] 1, 3,  6,  24,   72,   240,    696,    2184,     6480, ... A054718
[4] 1, 4, 12,  60,  240,  1020,   4020,   16380,    65280, ... A054719
[5] 1, 5, 20, 120,  600,  3120,  15480,   78120,   390000, ... A054720
[6] 1, 6, 30, 210, 1260,  7770,  46410,  279930,  1678320, ... A054721
[7] 1, 7, 42, 336, 2352, 16800, 117264,  823536,  5762400, ... A218124
[8] 1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, ... A218125
A000012|A002378| A047928   |   A218130     |      A218131
    A001477,A007531,    A061167,        A133499,   (diagonal A252764)
.
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 0,  2,   1;
[4] 0, 0,  2,   3,   1;
[5] 0, 0,  6,   6,   4,   1;
[6] 0, 0, 12,  24,  12,   5,  1;
[7] 0, 0, 30,  72,  60,  20,  6, 1;
[8] 0, 0, 54, 240, 240, 120, 30, 7, 1;

Links

Crossrefs

Variant: A143324.
Rows: A000007 (n=0), A019590 (n=1), A027375 (n=2), A054718 (n=3), A054719 (n=4), A054720, A054721, A218124, A218125.
Columns: A000012 (k=0), A001477 (k=1), A002378 (k=2), A007531(k=3), A047928, A061167, A218130, A133499, A218131.
Cf. A252764 (main diagonal), A074650, A363914.

Programs

  • Maple
    A363916 := (n, k) -> local d; add(A363914(k, d) * n^d, d = 0 ..k):
for n from 0 to 9 do seq(A363916(n, k), k = 0..8) od;
  • SageMath
    def A363916(n, k): return sum(A363914(k, d) * n^d for d in range(k + 1))
for n in range(9): print([A363916(n, k) for k in srange(9)])
def T(n, k): return A363916(k, n - k)

Formula

If k > 0 then k divides A(n, k), see the transposed array of A074650.
If k > 0 then n divides A(n, k), see the transposed array of A143325.

A378203 Number of palindromic n-ary words of length n that include the last letter of their respective alphabet.

Original entry on oeis.org

1, 1, 1, 5, 7, 61, 91, 1105, 1695, 26281, 40951, 771561, 1214423, 26916709, 42664987, 1087101569, 1732076671, 49868399761, 79771413871, 2560599031177, 4108933742199, 145477500542221, 234040800869107, 9059621800971105, 14605723004036255, 613627780919407801
Offset: 0

Views

Author

John Tyler Rascoe, Nov 19 2024

Keywords

Examples

			a(0) = 1: ().
a(1) = 1: (a).
a(2) = 1: (b,b).
a(3) = 5: (a,c,a), (b,c,b), (c,a,c), (c,b,c), (c,c,c).

Crossrefs

Cf. A045531, A047969, A248828, A252764, A292845.

Programs

  • Maple
    a:= n-> (h-> n^h-`if`(n=0, 0, (n-1)^h))(ceil(n/2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 21 2024
  • Mathematica
    h[n_] := Ceiling[n/2];a[n_] := n^h[n] - (n - 1)^h[n];Join[{1},Table[a[n],{n,25}]] (* James C. McMahon, Nov 21 2024 *)
  • PARI
    h(n) = {ceil(n/2)}
a(n) = {n^h(n)-(n-1)^h(n)}
  • Python
    def A378203(n): return n**(m:=n+1>>1)-(n-1)**m if n else 1 # Chai Wah Wu, Nov 21 2024

Formula

a(n) = n^h(n) - (n-1)^h(n) for n > 0, where h(n) = ceiling(n/2).
a(n) = A047969(n-1,h(n)-1) for n > 0.
Showing 1-3 of 3 results.

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