A334896 - OEIS

%I #7 May 27 2020 21:47:26

%S 648,648,686,12096,12096,686,192000,139968,192000,139968,1866240,

%T 179712,179712,1866240,814968,2101248,102036672,331776000,102036672,

%U 331776000,2101248,814968,179712000,4423680000,1866240000,131010048,179712000,4423680000,1866240000,131010048

%N Read terms e = T(n,k) in A333624 as Product(prime(k)^e) for n in A334769.

%C Row a(n) of A067255 = row A334769(n) of A333624.

%C An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.

%C Let T(n,k) address the terms in the k-th position of row n in A333624.

%C This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) => A333624(A334769(n)), we use A067255(a(n)).

%H Michael De Vlieger, <a href="/A334896/b334896.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>

%e a(1) = 648, since b(A334769(1)) = b(151) = 10010111, which generates T(151) as shown below, replacing 1 with "@" and 0 with ".":

%e   @ . . @ . @ @ @

%e    @ . @ @ @ . .

%e     @ @ . . @ .

%e      . @ . @ @

%e       @ @ @ .

%e        . . @

%e         . @

%e          @

%e In this figure we see 3 zero-triangles of side length k = 1, and 4 of side length k = 2, therefore, T(1,1) = 3 and T(1,2) = 4. This becomes 2^3 * 3^4 = 8 * 81 = 648.

%e Relationship of this sequence to A334556 and A333624:

%e         n  A334769(n)  a(n)  Row n of A333624

%e       --------------------------------------

%e        1    151       648    3, 4

%e        2    233       648    3, 4

%e        3    543       686    1, 0, 0, 3

%e        4    599     12096    6, 3, 0, 1

%e        5    937     12096    6, 3, 0, 1

%e        6    993       686    1, 0, 0, 3

%e        7   1379    192000    9, 1, 3

%e        8   1483    139968    6, 7

%e        9   1589    192000    9, 1, 3

%e       10   1693    139968    6, 7

%e       11   2359   1866240    9, 6, 1

%e       12   2391    179712    9, 3, 0, 0, 0, 1

%e       13   3753    179712    9, 3, 0, 0, 0, 1

%e       14   3785   1866240    9, 6, 1

%e       15   8607    814968    3, 3, 0, 3, 1

%e       16   9559   2101248   12, 3, 0, 0, 0, 0, 0, 1

%e       ...

%t With[{s = Rest[Import["https://oeis.org/A334769/b334769.txt", "Data"][[All, -1]]]}, Map[With[{w = NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &]}, If[Length@ # == 0, 1, Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ ReplacePart[ConstantArray[0, Max@ #[[All, 1]]], Map[#1 -> #2 & @@ # &, #]]] &@ Tally@ Flatten@ Array[If[# == 1, Map[If[First@ # == 1, Nothing, Length@ #] &, Split@ w[[#]]], Map[If[First@ # == -1, Length@ #, Nothing] &, Split[w[[#]] - Most@ w[[# - 1]]]]] &, Length@ w]] /. -Infinity -> 0 &, s[[1 ;; 29]]]]

%Y Cf. A038554, A067255, A070939, A333624, A334591, A334556, A334769.

%K nonn

%O 1,1

%A _Michael De Vlieger_, May 23 2020