|
|
A153003
|
|
Toothpick sequence in the first three quadrants.
|
|
10
|
|
|
0, 1, 4, 7, 10, 16, 25, 31, 34, 40, 49, 58, 70, 91, 115, 127, 130, 136, 145, 154, 166, 187, 211, 226, 238, 259, 286, 316, 361, 427, 487, 511, 514, 520, 529, 538, 550, 571, 595, 610, 622, 643, 670, 700, 745, 811, 871, 898, 910, 931
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
On the infinite square grid, consider only the first three quadrants and count only the toothpicks of length 2.
At stage 0, we start from a vertical half toothpick at [(0,0),(0,1)]. This half toothpick represents one of the two components of the first toothpick placed in the toothpick structure of A139250, so a(0) = 0.
At stage 1, we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1. Also we place half toothpick at [(0,-1),(1,-1)]. This last half toothpick represents one of the two components of the third toothpick placed in the toothpick structure of A139250.
At stage 2, we place three toothpicks, so a(2) = 1+3 = 4.
In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end.
The sequence gives the number of toothpicks after n stages. A153004 (the first differences) gives the number of toothpicks added to the structure at n-th stage.
Note that this sequence is different from the toothpick "corner" sequence A153006. For more information see A139250. (End)
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (A139250(n+1)-3)*3/4 + 1, if n >= 1.
(End)
|
|
MATHEMATICA
|
A139250[n_] := A139250[n] = Module[{m, k}, If[n == 0, Return[0]]; m = 2^(Length[IntegerDigits[n, 2]] - 1); k = (2 m^2 + 1)/3; If[n == m, k, k + 2 A139250[n - m] + A139250[n - m + 1] - 1]];
a[n_] := If[n == 0, 0, (3/4)(A139250[n + 1] - 3) + 1];
|
|
PROG
|
(Python)
def msb(n):
t=0
while n>>t>0: t+=1
return 2**(t - 1)
def a139250(n):
k=(2*msb(n)**2 + 1)/3
return 0 if n==0 else k if n==msb(n) else k + 2*a139250(n - msb(n)) + a139250(n - msb(n) + 1) - 1
def a(n): return 0 if n==0 else (a139250(n + 1) - 3)*3/4 + 1
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|