OFFSET
1,1
COMMENTS
One derives recurrence equations for the numbers of tiles, internal edges, internal vertices, and triple point vertices--respectively t(n), e(n), v(n), a(n)--by creating a set of planar substitution rules and proving that two triple points occur on the interior of every supertile, and that other triple points only occur at the intersection of three supertiles.
Restricting the domain to [0,1] introduces flowsnake deceptions along the boundary: the set of function values FLSN(m/6/7^n), m in 0, 1, 2... 6*7^n contains some points that would be exactly triple points if [0,1] were extended to [-infinity,infinity]. Extending the system of linear recurrence equations constrains the deception-free count to equal a(n) + 3^n . - Bradley Klee, Aug 30 2015
This sequence counts all triple points of the Q-function, up to the boundary deceptions ( cf. Klee, "A Pit of Flowsnakes" ). - Bradley Klee, Aug 30 2015
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM, (1972), Item 115.
B. Klee, A Pit of Flowsnakes, Complex Systems, 24, 4 (2015).
B. Klee, Flowsnake Q-Function, Wolfram Demonstrations(2015).
Index entries for linear recurrences with constant coefficients, signature (11,-31,21).
FORMULA
t(0)=1, e(n)=v(n)=a(n)=0,
t(n)= 7 t(n-1),
e(n)= 12 t(n-1)+ 3 e(n-1),
v(n)= 6 t(n-1) + 2 e(n-1) + v(n-1),
a(n)= 2 t(n-1) + 1/2 v(n-1).
G.f.: 1/14 (7/(1 - x) - 7/(1 - 3 x) + 6/(1 - 7 x)).
From Colin Barker, Aug 17 2015: (Start)
a(n) = (7-7*3^n+6*7^n)/14.
a(n) = 11*a(n-1)-31*a(n-2)+21*a(n-3) for n>3.
G.f.: -x*(9*x^2-5*x+2) / ((x-1)*(3*x-1)*(7*x-1)).
(End)
EXAMPLE
Define one particular snowflake, or slowfake, or flowsnake on [0,1] with values:
FLSN(m/6) = {{0, 0}, {1/2, -Sqrt[3]/6}, {4/7, 2 Sqrt[3]/7}, {1/6, Sqrt[3]/6}, {1/2, Sqrt[3]/2}, {1, Sqrt[3]/3}, {1, 0}}.
There exists a(1) = 2 triple points when the denominator is 42 = 6*7:
FLSN(5/42) = FLSN(11/42) = FLSN(17/42) = {3/7, Sqrt[3]/21},
FLSN(13/42) = FLSN(31/42) = FLSN(37/42) = {5/7, 4 Sqrt[3]/21}.
MAPLE
MATHEMATICA
1/14 (7 - 7*3^# + 6*7^#) & /@ Range[1, 20]
LinearRecurrence[{11, -31, 21}, {2, 17, 134}, 20]
PROG
(Magma) [1/14*(7-7*3^n+6*7^n): n in [1..25]]; // Vincenzo Librandi, Aug 10 2015
(PARI) Vec(-x*(9*x^2-5*x+2)/((x-1)*(3*x-1)*(7*x-1)) + O(x^30)) \\ Colin Barker, Aug 17 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bradley Klee, Aug 08 2015
STATUS
approved