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101100, 11011000, 1111001000, 111100010100, 11111100000100, 1111001001110000, 111100010111000100, 11111111000010000100, 1111100010011011000100, 111101101001110000100100
Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem below the root and then reflecting the underlying binary tree.
+10
28
0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100
COMMENTS
Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ...
Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e., A057164( A080068(n)) = A080068(n) (equivalently: A036044( A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.
EXAMPLE
This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
..............................................
.....\/................\/\/..........\/\/.....
......\/......\/\/......\/............\/......
.....\/........\/........\/..........\/.......
........................o.....................
........................|.....................
........o.....o.........o...o.........o.......
........|.....|..........\./..........|.......
....o...o.....o...o.......o.........o.o.o.....
.....\./.......\./........|..........\|/......
......*.........*.........*...........*.......
..[()(())]..[(())()]..[((())())]..[()(())()]..
...101100....110010....11100100....10110010...
42, 240, 916, 3748, 14960, 62104, 248176, 969304, 3876576, 15962544, 63772488, 248169896, 993554240, 4086635408, 16350541128, 63529835824, 254129143040, 1046249323840, 4184725760584, 16276030608712, 65054467548432, 267635134298624
COMMENTS
Question: to which Wolfram's class does this simple program belong, class 3 or class 4, or is such categorization at all applicable here?
PROG
(Python) # See the Links section
44, 232, 920, 3876, 14936, 60568, 248240, 996440, 3876264, 15524272, 63773584, 255477160, 993549616, 3970767760, 16350559552, 65386339632, 254129067336, 1016476056896, 4184726043136, 16740063237448, 65054466609736, 260416091191808
COMMENTS
Questions: to which Wolfram's class does this simple program belong, class 3 or class 4? (Is that classification applicable here? This is not 1D CA, although it may look like one).
Does the "central skyscraper" continue widening forever? (see the image for up to 16384th generation) At what specific points it widens? (A new sequence for that). How does that differ from A122242 and similar sister sequences, with different starting conditions?
0, 2, 10, 44, 178, 740, 2868, 11852, 47522, 190104, 735842, 3090116, 11777124, 48557252, 194656036, 778669672, 3117617996, 12677727330, 49850271300, 192901051976, 795560529352, 3243898094388, 12977884832332, 51055591319170
EXTENSIONS
Python program and Wolfram-like plot added by Antti Karttunen, Sep 14 2006.
0, 2, 12, 56, 228, 920, 3684, 14744, 58980, 235928, 943716, 3774872, 15099492, 60397976, 241591908, 966367640, 3865470564, 15461882264, 61847529060, 247390116248, 989560464996, 3958241859992, 15832967439972
COMMENTS
A simple formula exists, cf. A080675.
42, 212, 992, 3876, 15448, 64644, 252056, 989988, 4108676, 16147220, 63393540, 266083460, 1047285272, 4245874244, 16903342544, 67034166420, 274274527940, 1068738181764, 4246566244100, 17369295361736, 67322784388376, 269731897678032
52, 240, 964, 3972, 15556, 64532, 248288, 988964, 4164356, 15899248, 64719124, 257019652, 1070118936, 4197239188, 16299415152, 65592597568, 259741591312, 1093901323332, 4233842104068, 16616683414632, 70137217092164
COMMENTS
A122240 shows the same sequence in binary.
a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.
+10
3
1, 5, 24, 104, 279, 2001, 4131, 17453, 88826, 362532, 1655660, 6120642, 25376649, 128526482, 301370205, 1756488602, 8046359747, 30854867177, 73845140753, 488906501177, 2106640948770, 6573967883049, 29711211505300
COMMENTS
The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.
EXAMPLE
In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)= A165471(6)= A165471(7)=-1 and A165471(8)= A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.
PROG
(MIT/GNU Scheme)
(define ( A179417 n) (let ((ul ( A005408 n))) (let loop ((i ( A000290 n)) (j 0) (s 0)) (cond ((= j ul) s) ((= 0 (1+halved ( A165471 (1+ i)))) (loop (1+ i) (1+ j) s)) (else (loop (1+ i) (1+ j) (+ s (expt 2 j)))))))) (define (1+halved n) (floor->exact (/ (1+ n) 2)))
5, 18, 62, 180, 620, 1836, 5997, 23675, 76849, 263613, 923897, 3090855
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