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.

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

0,3

Comments

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.)

Links

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).

Formula

a(n) = max( min{x*y, (5-x)*(n-y) ) | 0 <= x <= 5, 0 <= y <= n ).

a(n) = n + floor(n/5) - [n == 1 mod 5]. - Ralf Stephan, Mar 29 2014

a(n) = 6*floor(n/5) + [0,0,2,3,4][n%5]. - Ralf Stephan, Mar 29 2014

G.f.: (2*x^5 + x^4 + x^3 + 2*x^2)/((1-x)*(1-x^5)). - Ralf Stephan, Mar 29 2014

a(n) = n - 1 + floor(n/5) + ceiling((n-1)/5) - floor((n-1)/5). - Wesley Ivan Hurt, Mar 29 2014

Maple

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

Mathematica

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 *)

Crossrefs

The fifth row of A238158.

Sequence in context: A335876 A306974 A278400 * A034298 A308162 A092404

Adjacent sequences: A239489 A239490 A239491 * A239493 A239494 A239495

Keyword

nonn,easy

Author

Robin Houston, Mar 23 2014

Status

approved