cp's OEIS Frontend

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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A218126 Number of 9-ary sequences with primitive period n.

Original entry on oeis.org

1, 9, 72, 720, 6480, 59040, 530640, 4782960, 43040160, 387419760, 3486725280, 31381059600, 282428998560, 2541865828320, 22876787671920, 205891132034880, 1853020145805120, 16677181699666560, 150094634909047920, 1350851717672992080, 12157665455570137920
Offset: 0

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Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Column k=9 of A143324.

Programs

  • Maple
    with(numtheory):
a:= n-> `if`(n=0, 1, add(9^d*mobius(n/d), d=divisors(n))):
seq(a(n), n=0..30);
  • PARI
    a(n) = if (n==0, 1, sumdiv(n, d, 9^d*moebius(n/d))); \\ Michel Marcus, Apr 15 2021

Formula

a(n) = Sum_{d|n} 9^d * mu(n/d) for n>0, a(0) = 1.
G.f.: 1 + 9 * Sum_{k>=1} mu(k) * x^k / (1 - 9*x^k). - Ilya Gutkovskiy, Apr 15 2021

A218127 Number of 10-ary sequences with primitive period n.

Original entry on oeis.org

1, 10, 90, 990, 9900, 99990, 998910, 9999990, 99990000, 999999000, 9999899910, 99999999990, 999998990100, 9999999999990, 99999989999910, 999999999899010, 9999999900000000, 99999999999999990, 999999998999001000, 9999999999999999990, 99999999989999990100
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Column k=10 of A143324.

Programs

  • Maple
    with(numtheory):
a:= n-> `if`(n=0, 1, add(10^d*mobius(n/d), d=divisors(n))):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{d|n} 10^d * mu(n/d) for n>0, a(0) = 1.
G.f.: 1 + 10 * Sum_{k>=1} mu(k) * x^k / (1 - 10*x^k). - Ilya Gutkovskiy, Apr 15 2021

A218132 Number of length 9 primitive (=aperiodic or period 9) n-ary words.

Original entry on oeis.org

0, 0, 504, 19656, 262080, 1953000, 10077480, 40353264, 134217216, 387419760, 999999000, 2357946360, 5159778624, 10604497176, 20661044040, 38443356000, 68719472640, 118587871584, 198359284536, 322687690920, 511999992000, 794280037320, 1207269207144
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Row n=9 of A143324.

Programs

  • Maple
    a:= n-> (n^6-1)*n^3:
seq(a(n), n=0..30);
  • Mathematica
    Table[n^9-n^3,{n,0,40}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,0,504,19656,262080,1953000,10077480,40353264,134217216,387419760},40] (* Harvey P. Dale, Feb 11 2015 *)

Formula

G.f.: 504*x^2*(x^6+29*x^5+175*x^4+310*x^3+175*x^2+29*x+1)/(x-1)^10.
a(n) = n^9-n^3.
a(0)=0, a(1)=0, a(2)=504, a(3)=19656, a(4)=262080, a(5)=1953000, a(6)=10077480, a(7)=40353264, a(8)=134217216, a(9)=387419760, a(n)=10*a(n-1)- 45*a(n-2)+120*a(n-3)-210*a(n-4)+252*a(n-5)- 210*a(n-6)+ 120*a(n-7)-45*a(n-8)+10*a(n-9)-a(n-10). - Harvey P. Dale, Feb 11 2015

A218133 Number of length 10 primitive (=aperiodic or period 10) n-ary words.

Original entry on oeis.org

0, 0, 990, 58800, 1047540, 9762480, 60458370, 282458400, 1073709000, 3486725280, 9999899910, 25937263440, 61917115260, 137858120400, 289254116970, 576649631040, 1099510578960, 2015992480320, 3570465336750, 6131063781360, 10239996799620, 16679876893680
Offset: 0

Views

Author

Alois P. Heinz, Oct 21 2012

Keywords

Links

Crossrefs

Row n=10 of A143324.

Programs

  • Maple
    a:= n-> (((n^5-1)*n^3-1)*n+1)*n:
seq(a(n), n=0..30);

Formula

G.f.: -30 *(x+1) *x^2 *(35*x^6 +1556*x^5 +13619*x^4 +30064*x^3 +13609*x^2 +1564*x +33) / (x-1)^11.
a(n) = n^10-n^5-n^2+n.

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.

A370821 Number of minimal deterministic Mealy automata with n states outputting ternary strings.

Original entry on oeis.org

3, 12, 54, 210, 798, 2850, 10038, 34410, 116406, 388362, 1283430, 4203786, 13675038, 44211570, 142202574, 455299242, 1451997726, 4614253122, 14617620726, 46177325994, 145505603694, 457437342546, 1435074324006, 4493508791754, 14045385985902
Offset: 1

Views

Author

Lucas B. Vieira, Mar 02 2024

Keywords

Comments

a(n) counts the minimal number of ternary words w = uv, with |w| = n, such that u is an irreducible prefix and v a primitive word. This defines a minimal "pattern", written as "u(v)", describing the behavior of a minimal n-state deterministic Mealy automaton outputting a string from a ternary alphabet, where u is the transient output, and v the cyclic output, possibly truncated. Used in the definition of the Deterministic Complexity (DC) of strings (Vieira and Budroni, 2022).

Examples

			a(1) = 3 as there are only 3 deterministic Mealy automata with 1 state producing ternary words, corresponding to the 3 patterns (0), (1) and (2), generating the strings w=0^L, w=1^L, and w=2^L for L >= 1.
a(2) = 12, since there are 12 minimal ternary patterns: (01), 0(1), (02), 0(2), (10), 1(0), (12), 1(2), (20), 2(0), (21), 2(1).
E.g.: The ternary string w = 000120120 can be described by the pattern 00(012), where the parentheses indicate the repeating part, up to truncation. This pattern is minimal, with 5 symbols (ignoring the parentheses). It describes the behavior of a minimal deterministic Mealy automaton producing the string w, leading to its Deterministic Complexity (DC) to be DC(w) = 5.

References

  • M. Domaratzki, D. Kisman, and J. Shallit, On the number of distinct languages accepted by finite automata with n states, J. Autom. Lang. Combinat. 7 (2002) 4-18, Section 6, f_1(n).

Links

Crossrefs

Cf. A059412 for the case of binary strings.
Cf. A143324 for psi(k,n).

Programs

  • Mathematica
    NumPrimitiveWords[k_, n_] := Sum[MoebiusMu[d] k^(n/d), {d, Divisors[n]}];
a[n_] := NumPrimitiveWords[3, n] + Sum[(3 - 1) 3^(i - 1) NumPrimitiveWords[3, n - i], {i, 1, n - 1}]

Formula

a(n) = psi(3, n) + Sum_{i=1..n-1} (3-1)*3^(i-1)*psi(3, n-i), where psi(k,n) is the number of primitive words of length n on a k-letter alphabet (Cf. A143324).
Previous Showing 21-26 of 26 results.

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