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.

Previous Showing 31-34 of 34 results.

A181651 Eigentriangle of number triangle A070909.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 6, 2, 2, 1, 1, 9, 3, 3, 1, 1, 1, 18, 6, 6, 2, 2, 1, 1, 27, 9, 9, 3, 3, 1, 1, 1, 54, 18, 18, 6, 6, 2, 2, 1, 1, 81, 27, 27, 9, 9, 3, 3, 1, 1, 1, 162, 54, 54, 18, 18, 6, 6, 2, 2, 1, 1
Offset: 0

Views

Author

Paul Barry, Nov 03 2010

Keywords

Comments

First column is (essentially) A038754. Row sums are A068911. Inverse is A181652.
Generalized (conditional) Riordan array with k-th column generated by
x^k*(1+x-x^2)/(1-3x^2) if k is even,
(1+x-2x^2-x^3)/(1-3x^2) if k is odd.

Examples

			Triangle begins
    1;
    1,  1;
    2,  1,  1;
    3,  1,  1,  1;
    6,  2,  2,  1,  1;
    9,  3,  3,  1,  1,  1;
   18,  6,  6,  2,  2,  1,  1;
   27,  9,  9,  3,  3,  1,  1,  1;
   54, 18, 18,  6,  6,  2,  2,  1,  1;
   81, 27, 27,  9,  9,  3,  3,  1,  1,  1;
  162, 54, 54, 18, 18,  6,  6,  2,  2,  1,  1;

Links

Crossrefs

Cf. A038754, A068911, A070909, A181652.

A347825 Number of ways to cut a 2 X n rectangle into rectangles with integer sides up to symmetries of the rectangle.

Original entry on oeis.org

1, 2, 6, 17, 61, 220, 883, 3597, 15232, 65130, 282294, 1229729, 5384065, 23630332, 103922707, 457561989, 2016346540, 8890227762, 39212714934, 173001054449, 763388725141, 3368934926716, 14868728620387, 65626328874621, 289666423135048, 1278582804528090
Offset: 0

Views

Author

Thomas Garrison, Jan 26 2022

Keywords

Comments

If all rotations and reflections are considered, a(2)=4 instead of 6.

Examples

			The a(2) = 6 ways to partition are:
  +-------+    +---+---+    +-------+
  |       |    |   |   |    |       |
  |       |    |   |   |    +-------+
  |       |    |   |   |    |       |
  +-------+    +---+---+    +-------+
.
  +---+---+    +-------+    +---+---+
  |   |   |    |       |    |   |   |
  |   +---+    +---+---+    +---+---+
  |   |   |    |   |   |    |   |   |
  +---+---+    +---+---+    +---+---+

Links

Crossrefs

The 1 X n case is A005418.
Cf. A034999, A068911 (fully symmetric).

Programs

  • PARI
    \\ here c(n) is A034999(n)
c(n) = polcoef((1-x)*(1-3*x)/(1-6*x+7*x^2) + O(x*x^n), n)
a(n) = if(n==0, 1, (c(n) + 2*3^(n-1) + c((n+1)\2) + c((n+2)\2))/4) \\ Andrew Howroyd, Feb 08 2022
  • Python
    # By Soumil Mukherjee
# Algebraic solutions to the number of ways to tile a 2 X n grid
import sys
# Total number of tilings
# Counts different reflections and rotations as distinct
counts = [1,2,8]
def tilings(n):
    if (n < len(counts)): return counts[n]
    for i in range(len(counts), n+1):
        val = 6 * counts[i-1] - 7 * counts[i-2]
        counts.append(val)
    return counts[n]
def getCounts(n):
  return counts[n]
def horizontallySymmetric(i):
    if i == 0: return 1
    return 2 * (3 ** (i-1))
def verticallySymmetric(i):
    if i == 0: return 1
    k = i//2
    if (i % 2 == 0):
        return counts[k+1] - counts[k]
    else:
        return counts[k+1]
def rotationallySymmetric(i):
    if i == 0: return 1
    k = i//2
    if (i % 2 == 0):
        return 2 * counts[k]
    else:
        return counts[k+1]
def perfectlySymmetric(i):
    if i == 0: return 1
    k = i//2
    if (i % 2 == 0):
        return 4 * (3 ** (k-1))
    else:
        return 2 * (3 ** k)
def asymmetric(i):
    return (
        counts[i]
        - verticallySymmetric(i)
        - horizontallySymmetric(i)
        - rotationallySymmetric(i)
        + (2 * perfectlySymmetric(i))
    )
def equivalenceClasses(i):
    tilings(i)
    return (
        (
          counts[i]
          + verticallySymmetric(i)
          + horizontallySymmetric(i)
          + rotationallySymmetric(i)
        )//4
        )

Formula

Define V(n) to be the set of tilings that are vertically symmetric.
Define H(n) to be the set of tilings that are horizontally symmetric.
Define R(n) to be the set of tilings that are rotationally symmetric.
a(n) = (c(n) + |V(n)| + |H(n)| + |R(n)|)/4 for n > 0, where:
c(n) = A034999(n),
|H(n)| = 2 * (3^n-1),
|V(n)| = c(n/2+1) - c(n/2) for even n; otherwise c(floor(n/2)+1),
|R(n)| = 2*c(n/2) for even n; otherwise c(floor(n/2)+1).
From Andrew Howroyd, Feb 08 2022: (Start)
a(n) = (c(n) + 2*3^(n-1) + c(floor((n+1)/2)) + c(floor((n+2)/2)))/4 for n > 0, where c(n) = A034999(n).
a(n) = 9*a(n-1) - 19*a(n-2) - 33*a(n-3) + 143*a(n-4) - 63*a(n-5) - 175*a(n-6) + 147*a(n-7) for n > 7.
G.f.: (1 - 7*x + 7*x^2 + 34*x^3 - 55*x^4 - 31*x^5 + 66*x^6 - 7*x^7)/((1 - 3*x)*(1 - 6*x + 7*x^2)*(1 - 6*x^2 + 7*x^4)).
(End)
a(n) ~ k*(3 + sqrt(2))^n, where k = (4 + sqrt(2))/56. - Stefano Spezia, Feb 17 2022

A235701 Number of n step walks (each step +/- 1 starting from 0) which are never more than 6 or less than -6.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 126, 252, 490, 980, 1890, 3780, 7252, 14504, 27734, 55468, 105840, 211680, 403368, 806736, 1535954, 3071908, 5845406, 11690812, 22238062, 44476124, 84582428, 169164856, 321661970, 643323940, 1223146232, 2446292464, 4650833040
Offset: 0

Views

Author

Geoffrey Critzer, Jan 14 2014

Keywords

Links

Crossrefs

Cf. A068911, A068912, A216212, A216241.
Row n=6 of A068913.

Programs

  • Mathematica
    nn=30;r=Solve[{s==1 + x a + x b, a==x s + x c, b==x s +x d, c==x a +x e, d==x b + x f, e==x c+x g, f==x d+x h,g==x e+x i, h==x f+x j,i==x g+x k,j==x h+x l,k==x i,l==x j, z==x k + x l }, {s,a,b,c,d,e,f,g,h,i,j,k,l,z}]; CoefficientList[Series[s+a+b+c+d+e+f+g+h+i+j+k+l/.r,{x,0,nn}],x]

Formula

G.f.: (1 + x - 2*x^2 - x^3)^2/(1 - 7*x^2 + 14*x^4 - 7*x^6).

A294429 Number of zero-energy ground states for one-dimensional periodic fermions with n sites.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 36, 54, 108, 172, 324, 530, 984, 1672, 3028, 5232, 9388
Offset: 0

Views

Author

N. J. A. Sloane, Nov 07 2017

Keywords

Links

Crossrefs

Cf. A007040, A068911.
Previous Showing 31-34 of 34 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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