A247282 - OEIS

%I #25 Feb 04 2022 11:19:05

%S 1,1,1,3,1,1,3,5,1,1,1,3,3,3,5,15,1,1,1,3,1,1,3,5,3,3,3,9,5,5,15,17,1,

%T 1,1,3,1,1,3,5,1,1,1,3,3,3,5,15,3,3,3,9,3,3,9,15,5,5,5,15,15,15,17,51,

%U 1,1,1,3,1,1,3,5,1,1,1,3,3,3,5,15,1,1,1,3,1,1,3,5,3,3,3,9,5,5,15,17,3,3,3,9,3,3,9,15,3,3,3,9,9,9,15,45

%N Run Length Transform of A001317.

%C The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

%C This sequence is obtained by applying Run Length Transform to the right-shifted version of the sequence A001317: 1, 3, 5, 15, 17, 51, 85, 255, 257, ...

%H Antti Karttunen, <a href="/A247282/b247282.txt">Table of n, a(n) for n = 0..8192</a>

%F For all n >= 0, a(A051179(n)) = A246674(A051179(n)) = A051179(n).

%e 115 is '1110011' in binary. The run lengths of 1-runs are 2 and 3, thus a(115) = A001317(2-1) * A001317(3-1) = 3*5 = 15.

%e From _Omar E. Pol_, Feb 15 2015: (Start)

%e Written as an irregular triangle in which row lengths are the terms of A011782:

%e 1;

%e 1;

%e 1,3;

%e 1,1,3,5;

%e 1,1,1,3,3,3,5,15;

%e 1,1,1,3,1,1,3,5,3,3,3,9,5,5,15,17;

%e 1,1,1,3,1,1,3,5,1,1,1,3,3,3,5,15,3,3,3,9,3,3,9,15,5,5,5,15,15,15,17,51;

%e ...

%e Right border gives 1 together with A001317.

%e (End)

%t a1317[n_] := FromDigits[ Table[ Mod[Binomial[n-1, k], 2], {k, 0, n-1}], 2];

%t Table[ Times @@ (a1317[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* _Jean-François Alcover_, Jul 11 2017 *)

%o (MIT/GNU Scheme)

%o (define (A247282 n) (fold-left (lambda (a r) (* a (A001317 (- r 1)))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))

%o ;; Other functions as in A227349.

%o (Python)

%o # uses RLT function from A278159

%o def A247282(n): return RLT(n,lambda m: int(''.join(str(int(not(~(m-1)&k))) for k in range(m)),2)) # _Chai Wah Wu_, Feb 04 2022

%Y Cf. A003714 (gives the positions of ones).

%Y Cf. A001317, A051179.

%Y A001316 is obtained when the same transformation is applied to A000079, the powers of two.

%Y Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246674, A246685.

%K nonn

%O 0,4

%A _Antti Karttunen_, Sep 22 2014