Robin Houston - cp's OEIS frontend

A335656 Number of distinct board states reachable in n jumps, in English Peg Solitaire.

1, 4, 12, 60, 296, 1338, 5648, 21842, 77559, 249690, 717788, 1834379, 4138302, 8171208, 14020166, 20773236, 26482824, 28994876, 27286330, 22106348, 15425572, 9274496, 4792664, 2120101, 800152, 255544, 68236, 14727, 2529, 334, 32, 5

Offset: 0

Robin Houston, Jun 16 2020

			Example: for n=1 the four states are:
      ***        ***        ***        ***
      *.*        ***        ***        ***
    ***.***    *******    *******    *******
    *******    ****..*    *******    *..****
    *******    *******    ***.***    *******
      ***        ***        *.*        ***
      ***        ***        ***        ***

Robin Houston, Code used to generate this sequence

Identifying positions that are related by a symmetry of the board gives A112737.

A308711 Left-truncatable primes in base-10 bijective numeration.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 503, 523, 547, 607, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 907, 937, 947, 953, 967, 983, 997

Offset: 1

Robin Houston, Jun 19 2019

Not identical to A033664; in fact a strict subsequence of A033664. For example, 2003 belongs to A033664 but not to this sequence, since in bijective numerals 2003 is 19X3, whose suffix 9X3 = 1003 = 17 * 59.

Robin Houston, Table of n, a(n) for n = 1..8391
Wikipedia, Bijective numeration

Cf. A024785, A033664.

DIGITS = "123456789X"
DECODE = {d: i + 1 for i, d in enumerate(DIGITS)}
def decode(s):
    return reduce(lambda n, c: 10 * n + DECODE[c], s, 0)
def search(s):
    n = decode(s)
    if n > 0:
        if not is_prime(n): return
        yield n
    for digit in DIGITS: yield from search(digit + s)
full = sorted(search(""))
full[:10]

A249753 Number of triangles in the complex obtained by starting with an isosceles right triangle, and dividing each cell into two similar isosceles right triangles n times.

Original entry on oeis.org

1, 3, 7, 17, 40, 99, 246, 642, 1690, 4554, 12436, 34132, 95230, 263934, 744956, 2075132, 5892430, 16456014, 46871196, 131068572, 373897870, 1046231694, 2986898716, 8360588572, 23878057870, 66847653774, 190955945756, 534633021212, 1527373517710, 4276471354254

Offset: 0

Robin Houston, Nov 04 2014

nonn

Colin Barker, Table of n, a(n) for n = 0..1000
"Marsland", Reddit discussion

a(0) = 1;
a(1) = 3;
a(2) = 7;
a(2k+1) = (140 2^(3k) + 318 2^(2k) + 60 2^(k) - 48 + 16 (-1)^(k)) / 144, for k > 0;
a(2k) = (50 2^(3k) + 147 2^(2k) + 60 2^(k) - 48 + 16 (-1)^(k)) / 144, for k > 1.
Empirical g.f.: -(16*x^11 +8*x^10 +14*x^9 -7*x^8 -33*x^7 -8*x^6 -13*x^5 -13*x^4 +16*x^3 +9*x^2 -2*x -1) / ((x -1)*(2*x -1)*(2*x +1)*(x^2 +1)*(2*x^2 -1)*(8*x^2 -1)). - Colin Barker, Nov 14 2014

A239492 The fifth bicycle lock sequence: a(n) is the maximum value of min{xy, (5-x)(n-y)} over 0 <= x <= 5, 0 <= y <= n for integers x, y.

Original entry on oeis.org

0, 0, 2, 3, 4, 6, 6, 8, 9, 10, 12, 12, 14, 15, 16, 18, 18, 20, 21, 22, 24, 24, 26, 27, 28, 30, 30, 32, 33, 34, 36, 36, 38, 39, 40, 42, 42, 44, 45, 46, 48, 48, 50, 51, 52, 54, 54, 56, 57, 58, 60, 60, 62, 63, 64, 66, 66, 68, 69, 70, 72, 72, 74, 75, 76, 78, 78, 80, 81, 82, 84, 84, 86, 87, 88, 90, 90

Offset: 0

Robin Houston, Mar 23 2014

The minimum number of turns that always suffice to open from any starting position a bicycle lock that has n-1 dials with 5 numbers on each dial.
The minimum number of turns that always suffice to open from any starting position a bicycle lock that has 4 dials with n numbers on each dial.
(A "turn" consists of simultaneously rotating any number of adjacent dials by one place.)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robin Houston, Symmetry of bicycle lock numbers.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

The fifth row of A238158.

A239492:=n->n-1+floor(n/5)+ceil((n-1)/5)-floor((n-1)/5); seq(A239492(n), n=0..50); # Wesley Ivan Hurt, Mar 29 2014

a[n_] :=  Max[Table[Min[x*y, (5-x)*(n-y)], {x, 0, 5}, {y, 0, n}]]
Table[n - 1 + Floor[n/5] + Ceiling[(n - 1)/5] - Floor[(n - 1)/5], {n, 0, 50}] (* Wesley Ivan Hurt, Mar 29 2014 *)
CoefficientList[Series[(2 x^5 + x^4 + x^3 + 2 x^2)/((1 - x) (1 - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 30 2014 *)

a(n) = max( min{x*y, (5-x)*(n-y)} | 0 <= x <= 5, 0 <= y <= n ).
From Ralf Stephan, Mar 29 2014: (Start)
a(n) = n + floor(n/5) - [n == 1 mod 5].
a(n) = 6*floor(n/5) + [0,0,2,3,4][n%5].
G.f.: (2*x^5 + x^4 + x^3 + 2*x^2)/((1-x)*(1-x^5)). (End)
a(n) = n - 1 + floor(n/5) + ceiling((n-1)/5) - floor((n-1)/5). - Wesley Ivan Hurt, Mar 29 2014

A238158 Bicycle lock numbers: triangle T(n,k) with 1<=k<=n, where T(n,k) is the maximum value of min{xy, (n-x)(k-y)} over 0 <= x <= n, 0 <= y <= k for integers x, y.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 2, 2, 4, 0, 2, 3, 4, 6, 0, 3, 4, 6, 6, 9, 0, 3, 4, 6, 8, 9, 12, 0, 4, 5, 8, 9, 12, 12, 16, 0, 4, 6, 8, 10, 12, 15, 16, 20, 0, 5, 6, 10, 12, 15, 16, 20, 20, 25, 0, 5, 7, 10, 12, 15, 18, 20, 24, 25, 30, 0, 6, 8, 12, 14, 18, 20, 24, 25, 30

Offset: 1

Robin Houston, Mar 23 2014

Really an infinite symmetric matrix: the definition is symmetric in n and k. As a symmetric matrix, the first few rows are: A000004, A004526, A004523, A052928, A239492.
T(n+1, k) is the minimum number of turns that always suffice to open from any starting position a bicycle lock that has n dials with k numbers on each dial, where a turn consists of simultaneously rotating any number of adjacent dials by one place.
T(n, k) <= nk/4, with equality when n and k are both even.

			For n=5, k=4, the maximum value is attained at x=2, y=2, so T(5, 4) = 2*2 = 4. The first few rows of the triangle are:
0
0 1
0 1 2
0 2 2 4
0 2 3 4 6
0 3 4 6 6 9
0 3 4 6 8 9 12
0 4 5 8 9 12 12 16
0 4 6 8 10 12 15 16 20
0 5 6 10 12 15 16 20 20 25
0 5 7 10 12 15 18 20 24 25 30
0 6 8 12 14 18 20 24 25 30 30 36
0 6 8 12 15 18 21 24 28 30 35 36 42

Robin Houston, Symmetry of bicycle lock numbers.

Cf. A000004, A004526, A004523, A052928, A239492.

t[a_, b_] := Max[Table[Min[x*y, (a - x)*(b - y)], {x, 0, a}, {y, 0, b}]]

T(n,k) = max { min{xy, (n-x)(k-y)} | 0<=x<=n, 0<=y<=k; x, y integers }.

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User: Robin Houston

A335656 Number of distinct board states reachable in n jumps, in English Peg Solitaire.

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A308711 Left-truncatable primes in base-10 bijective numeration.

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Sage

A249753 Number of triangles in the complex obtained by starting with an isosceles right triangle, and dividing each cell into two similar isosceles right triangles n times.

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A239492 The fifth bicycle lock sequence: a(n) is the maximum value of min{x*y, (5-x)*(n-y)} over 0 <= x <= 5, 0 <= y <= n for integers x, y.

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Maple

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A238158 Bicycle lock numbers: triangle T(n,k) with 1<=k<=n, where T(n,k) is the maximum value of min{xy, (n-x)(k-y)} over 0 <= x <= n, 0 <= y <= k for integers x, y.

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Mathematica

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A239492 The fifth bicycle lock sequence: a(n) is the maximum value of min{xy, (5-x)(n-y)} over 0 <= x <= 5, 0 <= y <= n for integers x, y.