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.

Showing 1-3 of 3 results.

A200062 Meanders of length n and central angle < 360 degrees.

Original entry on oeis.org

0, 1, 1, 4, 1, 15, 1, 41, 23, 133, 1, 650, 1, 1725, 961, 6930, 1, 30323, 1, 99716, 40431, 352729, 1, 1709125, 35467, 5200315, 2008233, 20960538, 1, 93058849, 1, 312220259, 105533203, 1166803129, 20194059, 5478229800, 1, 17672631921, 5731781295, 71539226243, 1
Offset: 1

Views

Author

Peter Luschny, Nov 16 2011

Keywords

Comments

A meander is a closed curve drawn by arcs of equal length and central angles of equal magnitude, starting with a positively oriented arc.
a(n) = 1 if and only if n is prime.

Examples

			See the link for n = 6,8,9.

Links

Crossrefs

Cf. A198060, A199932.

Programs

  • Maple
    A200062 := proc(n) local i;
add(A198060(i-1,n/i-1),i=numtheory[divisors](n)) - 2^(n-1) end: seq(A200062(i),i=1..41);
  • Mathematica
    A198060[m_, n_] := Sum[ Sum[ Sum[ (-1)^(j+i)*Binomial[i, j]*Binomial[n, k]^(m+1) * (n+1)^j * (k+1)^(m-j) / (k+1)^m, {i, 0, m}], {j, 0, m}], {k, 0, n}]; a[n_] := Sum[ A198060[d-1, n/d-1], {d, Divisors[n]}] - 2^(n-1); Table[a[n], {n, 1, 41}] (* Jean-François Alcover, Jun 27 2013 *)
  • PARI
    A200062(n) = { D = divisors(n);
sum(m = 2, #D, d = D[m];
   sum(k=0,n/d-1,binomial(n/d-1,k)^d*
      sum(j=0,d-1,((n/d)/(k+1))^j*
          sum(i=0,d-1,(-1)^(j+i)*binomial(i,j)
))))}

Formula

a(n) = Sum_{d|n} A198060(d-1,n/d-1) - 2^(n-1).

A200583 Table read by rows, n >= 1, 1 <= k <= card(divisors(n)), T(n,k) meanders of length n and central angle of 360/d degrees, d the k-th divisor of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 8, 3, 1, 16, 1, 32, 10, 4, 1, 64, 1, 128, 35, 5, 1, 256, 22, 1, 512, 126, 6, 1, 1024, 1, 2048, 462, 134, 46, 7, 1, 4096, 1, 8192, 1716, 8, 1, 16384, 866, 94, 1, 32768, 6435, 485, 9, 1, 65536, 1, 131072, 24310, 5812, 190, 10, 1, 262144, 1, 524288
Offset: 1

Views

Author

Peter Luschny, Nov 20 2011

Keywords

Comments

A meander is a closed curve drawn by arcs of equal length and central angle of equal magnitude, starting with a positively oriented arc.

Examples

			[ 1]            1
[ 2]          2, 1
[ 3]          4, 1
[ 4]        8, 3, 1
[ 5]         16, 1
[ 6]      32, 10, 4, 1
[ 7]         64, 1
[ 8]     128, 35, 5, 1
[ 9]       256, 22, 1
[10]     512, 126, 6, 1
[11]        1024, 1
[12] 2048, 462, 134, 46, 7, 1

Links

Crossrefs

Cf. A198060, A199932, A200062.

Programs

  • Maple
    A200583_row := proc(n) local i;
seq(A198060(i-1,n/i-1),i=numtheory[divisors](n)) end:
seq(print(A200583_row(i)),i=1..12);
  • Mathematica
    A198060[m_, n_] := Sum[Sum[Sum[(-1)^(j+i)*Binomial[i, j]*Binomial[n, k]^(m+1)*(n+1)^j*(k+1)^(m-j)/(k+1)^m, {i, 0, m}], {j, 0, m}], {k, 0, n}]; row[n_] := Table[A198060[d-1, n/d-1], {d, Divisors[n]}]; Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

Formula

T(n,k) = A198060(d-1,n/d-1) where d is the k-th divisor of n (the divisors in natural order).

A361575 Number of Fibonacci meanders of length n.

Original entry on oeis.org

1, 3, 5, 11, 13, 30, 34, 71, 97, 177, 233, 481, 610, 1157, 1677, 3027, 4181, 8016, 10946, 20379, 29534, 52461, 75025, 140748, 196778, 355979, 526123, 933044, 1346269, 2469992, 3524578, 6342729, 9400985, 16487211
Offset: 1

Views

Author

Peter Luschny, Mar 16 2023

Keywords

Comments

For an overview of the terms and functions used, compare A361574. The corresponding sequence counting meanders without the requirement to be Fibonacci is A199932.

Examples

			Fibonacci meanders with length 6 can have the central angle 360/m, where m is in divisors(6) = {1, 2, 3, 6}. In total there are a(6) = 30 such meanders, the list shows their binary representation together with the multiplicity with which they appear.
100000 x 1, 100001 x 2, 100010 x 1, 100100 x 2, 100101 x 1, 101000 x 1,
101001 x 1, 101010 x 1, 110000 x 2, 110001 x 2, 110010 x 1, 110100 x 1,
110101 x 1, 111000 x 2, 111001 x 2, 111010 x 1, 111100 x 2, 111101 x 1,
111110 x 1, 111111 x 4.

Links

Crossrefs

Cf. A361574, A199932, A198060.

Programs

  • Maple
    # The list A was computed with the functions given in A361574. They correspond to the columns in the table shown in the reference.
A := [[1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577, 5702886, 9227464, 14930351], [1, 3, 6, 13, 30, 70, 167, 405, 992, 2450, 6090, 15214, 38165, 96069, 242530, 613811, 1556856], [1, 3, 8, 21, 68, 242, 861, 3151, 11874, 45192, 173496], [1, 3, 10, 35, 154, 858, 4723, 25625], [1, 3, 12, 61, 360, 3058], [1, 3, 14, 111, 878], [1, 3, 16, 209], [1, 3, 18, 403], [1, 3, 20], [1, 3, 22], [1, 3, 24], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1]];
with(LinearAlgebra):  # a(n) is the sum of row n of this table.
row := k -> [seq(`if`(irem(n, k) <> 0, 0, A[k][n/k]), n = 1..34)]:
M := Transpose(Matrix([seq(row(n), n = 1..34)])):
seq(add(m, m = Row(M, n)), n = 1..34);
Showing 1-3 of 3 results.

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