cp's OEIS Frontend

A278161 Run length transform of A008619 (floor(n/2)+1).

%I A278161 #21 Sep 28 2021 01:21:48
%S A278161 1,1,1,2,1,1,2,2,1,1,1,2,2,2,2,3,1,1,1,2,1,1,2,2,2,2,2,4,2,2,3,3,1,1,
%T A278161 1,2,1,1,2,2,1,1,1,2,2,2,2,3,2,2,2,4,2,2,4,4,2,2,2,4,3,3,3,4,1,1,1,2,
%U A278161 1,1,2,2,1,1,1,2,2,2,2,3,1,1,1,2,1,1,2,2,2,2,2,4,2,2,3,3,2,2,2,4,2,2,4,4,2,2,2,4,4,4,4,6,2,2,2,4,2,2,4,4,3
%N A278161 Run length transform of A008619 (floor(n/2)+1).
%H A278161 Antti Karttunen, <a href="/A278161/b278161.txt">Table of n, a(n) for n = 0..16383</a>
%H A278161 Chai Wah Wu, <a href="https://arxiv.org/abs/1610.06166">Sums of products of binomial coefficients mod 2 and run length transforms of sequences</a>, arXiv:1610.06166 [math.CO], 2016.
%H A278161 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A278161 a(n) = A046951(A005940(1+n)), a(A156552(n)) = A046951(n).
%F A278161 a(n) = Sum_{k=0..n} ((binomial(n+3k,6k)*binomial(n,k)) mod 2).  - _Chai Wah Wu_, Nov 19 2019
%e A278161 n=111 is "1101111" in binary, which has two runs of 1-bits: the other has length 2, and the other has length 4, thus we take the product A008619(2)*A008619(4) = (floor(2/2)+1) * (floor(4/2)+1) = 2*3, which is the result, so a(111) = 6.
%t A278161 f[n_] := Floor[n/2] + 1; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 120}] (* _Jean-François Alcover_, Jul 11 2017 *)
%o A278161 (Scheme)
%o A278161 (define (A278161 n) (fold-left (lambda (a r) (* a (A008619 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
%o A278161 (define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
%o A278161 ;; See A227349 for the required other functions.
%o A278161 (Python)
%o A278161 def A278161(n): return sum(int(not (~(n+3*k) & 6*k) | (~n & k)) for k in range(n+1)) # _Chai Wah Wu_, Sep 28 2021
%Y A278161 Cf. A005940, A008619, A046951, A156552.
%Y A278161 Cf. A106737, A227349 for other run length transforms, and also A278222.
%K A278161 nonn,base
%O A278161 0,4
%A A278161 _Antti Karttunen_, Nov 14 2016