login
Irregular table read by rows: the first entry of n-th row is length of run of rightmost identical bits (either 0 or 1, equal to n mod 2), followed by length of the next run of bits, etc., in the binary representation of n, when scanned from the least significant to the most significant end.
22

%I #58 Apr 21 2024 22:23:03

%S 1,1,1,2,2,1,1,1,1,1,2,3,3,1,1,2,1,1,1,1,1,2,1,1,2,2,1,1,2,1,3,4,4,1,

%T 1,3,1,1,1,2,1,2,2,1,2,1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,3,2,1,2,2,1,1,1,

%U 2,2,1,2,2,3,1,1,3,1,4,5,5,1,1,4,1,1,1,3,1

%N Irregular table read by rows: the first entry of n-th row is length of run of rightmost identical bits (either 0 or 1, equal to n mod 2), followed by length of the next run of bits, etc., in the binary representation of n, when scanned from the least significant to the most significant end.

%C Row n has A005811(n) terms. In rows 2^(k-1)..2^k-1 we have all the compositions (ordered partitions) of k. Other orderings of compositions: A101211, A066099, A108244 and A124734.

%C Each row n (n>=1) contains the initial A005811(n) nonzero terms from the beginning of row n of A227186. A070939(n) gives the sum of terms on row n, while A167489(n) gives the product of its terms. A136480 gives the first column. A101211 lists the terms of each row in reverse order.

%H Antti Karttunen, <a href="/A227736/b227736.txt">The rows 1..1023 of the table, flattened</a>

%H Mikhail Kurkov, <a href="/A227736/a227736_1.txt">Comments on A227736</a> [verification needed]

%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Sep 09 2018. See Procedure 1.

%F a(n) = A227186(A227737(n), A227740(n)).

%F a(n) = A101211(A227741(n)).

%e Table begins as:

%e Row n in Terms on

%e n binary that row

%e 1 1 1;

%e 2 10 1,1;

%e 3 11 2;

%e 4 100 2,1;

%e 5 101 1,1,1;

%e 6 110 1,2;

%e 7 111 3;

%e 8 1000 3,1;

%e 9 1001 1,2,1;

%e 10 1010 1,1,1,1;

%e 11 1011 2,1,1;

%e 12 1100 2,2;

%e 13 1101 1,1,2;

%e 14 1110 1,3;

%e 15 1111 4;

%e 16 10000 4,1;

%e etc. with the terms of row n appearing in reverse order compared how the runs of the same length appear in the binary expansion of n (Cf. A101211).

%e From _Omar E. Pol_, Sep 08 2013: (Start)

%e Illustration of initial terms:

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

%e k m Diagram Composition

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

%e . _

%e 1 1 |_|_ 1;

%e 2 1 |_| | 1, 1,

%e 2 2 |_ _|_ 2;

%e 3 1 |_ | | 2, 1,

%e 3 2 |_|_| | 1, 1, 1,

%e 3 3 |_| | 1, 2,

%e 3 4 |_ _ _|_ 3;

%e 4 1 |_ | | 3, 1,

%e 4 2 |_|_ | | 1, 2, 1,

%e 4 3 |_| | | | 1, 1, 1, 1,

%e 4 4 |_ _|_| | 2, 1, 1,

%e 4 5 |_ | | 2, 2,

%e 4 6 |_|_| | 1, 1, 2,

%e 4 7 |_| | 1, 3,

%e 4 8 |_ _ _ _|_ 4;

%e 5 1 |_ | | 4, 1,

%e 5 2 |_|_ | | 1, 3, 1,

%e 5 3 |_| | | | 1, 1, 2, 1,

%e 5 4 |_ _|_ | | 2, 2, 1,

%e 5 5 |_ | | | | 2, 1, 1, 1,

%e 5 6 |_|_| | | | 1, 1, 1, 1, 1,

%e 5 7 |_| | | | 1, 2, 1, 1,

%e 5 8 |_ _ _|_| | 3, 1, 1,

%e 5 9 |_ | | 3, 2,

%e 5 10 |_|_ | | 1, 2, 2,

%e 5 11 |_| | | | 1, 1, 1, 2,

%e 5 12 |_ _|_| | 2, 1, 2,

%e 5 13 |_ | | 2, 3,

%e 5 14 |_|_| | 1, 1, 3,

%e 5 15 |_| | 1, 4,

%e 5 16 |_ _ _ _ _| 5;

%e .

%e Also irregular triangle read by rows in which row k lists the compositions of k, k >= 1.

%e Triangle begins:

%e [1];

%e [1,1], [2];

%e [2,1], [1,1,1], [1,2],[3];

%e [3,1], [1,2,1], [1,1,1,1], [2,1,1], [2,2], [1,1,2], [1,3], [4];

%e [4,1], [1,3,1], [1,1,2,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], [1,2,1,1], [3,1,1], [3,2], [1,2,2], [1,1,1,2], [2,1,2], [2,3], [1,1,3], [1,4], [5];

%e Row k has length A001792(k-1).

%e Row sums give A001787(k), k >= 1.

%e (End)

%t Array[Length /@ Reverse@ Split@ IntegerDigits[#, 2] &, 34] // Flatten (* _Michael De Vlieger_, Dec 11 2020 *)

%o (Scheme) (define (A227736 n) (A227186bi (A227737 n) (A227740 n))) ;; The Scheme-function for A227186bi has been given in A227186.

%o (Haskell)

%o import Data.List (group)

%o a227736 n k = a227736_tabf !! (n-1) !! (k-1)

%o a227736_row n = a227736_tabf !! (n-1)

%o a227736_tabf = map (map length . group) $ tail a030308_tabf

%o -- _Reinhard Zumkeller_, Aug 11 2014

%Y Cf. A227738 and also A227739 for similar table for unordered partitions.

%Y Cf. A030308, A245562, A245563.

%K nonn,base,tabf

%O 1,4

%A _Antti Karttunen_, Jul 25 2013