A170893
First differences of the toothpick sequence A170892.
Original entry on oeis.org
0, 1, 1, 2, 4, 4, 4, 8, 10, 10, 4, 8, 10, 12, 12, 22, 26, 18, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 34, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 36, 12, 22, 28, 32, 42, 66, 68, 48, 42, 68, 84, 102, 146, 194, 162, 66, 4, 8, 10, 12, 12, 22, 26, 20, 12, 22, 28, 32, 42, 66, 66, 36, 12, 22, 28, 32, 42, 66, 68, 48, 42, 68, 84
Offset: 0
From _Omar E. Pol_, Jan 30 2013 (Start):
If written as an irregular triangle in which rows 0..2 have length 1, it appears that row j has length 2^(j-3), if j >= 3.
0;
1;
1;
2;
4,4;
4,8,10,10;
4,8,10,12,12,22,26,18;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,34;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,36,12,22,28,32,42,66,68,48,42,68,84,102,146,194,162,66;
4,8,10,12,12,22,26,20,12,22,28,32,42,66,66,36,12,22,28,32,42,66,68,48,42,68,84,...
(End)
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A170893(n, print_all=0)={my( ee=[[2*I, I]], p=Set( concat( vector( 2*n-(n>0), k, k-n-abs(k-n)*I ), I ))); print_all & print1("1,1"); for(i=3, n, p=setunion(p, Set(Mat(ee~)[, 1])); my(c, d, ne=[]); for( k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); print_all & print1(","#ee)); (n>0)*#ee} \\ M. F. Hasler, Jan 30 2013
A170886
Similar to A160406, but always staying outside the wedge, starting at stage 1 with a toothpick whose midpoint touches the vertex of the wedge.
Original entry on oeis.org
0, 1, 3, 5, 7, 11, 17, 23, 29, 37, 49, 55, 63, 75, 93, 107, 121, 141, 161, 167, 175, 187, 205, 219, 235, 259, 285, 301, 325, 363, 409, 447, 489, 541, 577, 583, 591, 603, 621, 635, 651, 675, 701, 717, 741, 779, 825, 863, 907, 963, 1005, 1021, 1045, 1083, 1129
Offset: 0
A170888
Similar to A160406, but always staying outside the wedge, starting at stage 0 with a vertical half-toothpick which protrudes from the vertex of the wedge.
Original entry on oeis.org
0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 59, 69, 81, 101, 127, 143, 147, 153, 163, 173, 185, 205, 231, 249, 261, 281, 309, 339, 381, 445, 511, 543, 547, 553, 563, 573, 585, 605, 631, 649, 661, 681, 709, 739, 781, 845, 911, 945, 957, 977, 1005, 1035, 1077, 1141
Offset: 0
A170890
Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.
Original entry on oeis.org
0, 1, 2, 4, 7, 10, 14, 21, 29, 37, 43, 53, 61, 71, 83, 103, 123, 139, 151, 165, 173, 183, 195, 215, 235, 253, 271, 295, 317, 345, 385, 441, 493, 531, 559, 581, 589, 599, 611, 631, 651, 669, 687, 711, 733, 761, 801, 857, 909, 949, 983, 1015, 1037, 1065, 1105, 1161
Offset: 0
From _M. F. Hasler_, Jan 29 2013: (Start)
The first steps are illustrated as follows, where two vertical "|" or three horizontal "_" correspond to one single full toothpick:
: ___ ___ |___ ___|
: ___ |___| |___| | |___| |
: _ |_ |_ | |_| | |_| | | |_|
: /\ |/\ |/\ |/\ ¯¯¯|/\ |¯¯¯|/\
: / \ / \ / \ / \ / \ / \
:
: a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(5)=7, a(6)=10, ... (End)
See
A170891 for the first differences.
-
A170890(n, print_all=0)={ my( cnt=n>0, ee=[[1,1]], p=Set(vector(2*n-cnt,k,k-n-abs(k-n)*I)), c, d); for(i=2, n, print_all & print1(cnt","); p=setunion(p, Set(Mat(ee~)[, 1])); my(ne=[]); for(k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee); cnt} \\ - M. F. Hasler, Jan 29 2013
Showing 1-4 of 4 results.
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