A261120 The number of distinct triple points in the set of function values FLSN(m/6/7^n), m in 0, 1, 2... 6*7^n, where FLSN:[0,1] is the "flowsnake" plane filling curve.

2, 17, 134, 989, 7082, 50057, 351854, 2467349, 17284562, 121031297, 847337174, 5931714509, 41523064442, 290664639737, 2034662044094, 14242663006469, 99698727138722, 697891348251377, 4885240212600614, 34196683812727229, 239376793662659402, 1675637576559322217

Offset: 1

Author

Bradley Klee, Aug 08 2015

Keywords

• nonn
• easy

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

Examples

			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}.
		

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).

Crossrefs

Cf. A260482, A260747, A260748, A260749, A260750, A261180, A261185.

Programs

Magma

[1/14*(7-7*3^n+6*7^n): n in [1..25]]; // Vincenzo Librandi, Aug 10 2015

Maple

A261120:=n->(7-7*3^n+6*7^n)/14: seq(A261120(n), n=1..30); # Wesley Ivan Hurt, Aug 27 2015

Mathematica

1/14 (7 - 7*3^# + 6*7^#) & /@ Range[1, 20]
LinearRecurrence[{11, -31, 21}, {2, 17, 134}, 20]

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

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)