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A368842 a(n) gives the number of triples of equally spaced equal digits in the binary expansion of n (without leading zeros).

%I A368842 #17 Jan 10 2024 12:15:20
%S A368842 0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,2,2,1,0,0,0,1,0,2,1,0,0,0,1,2,2,4,4,2,
%T A368842 2,1,0,0,0,1,2,0,2,1,0,0,2,3,2,1,0,0,0,1,0,2,2,1,2,2,2,3,4,6,6,4,3,2,
%U A368842 2,2,1,2,2,1,1,1,0,1,1,3,3,2,0,0,2,3,1
%N A368842 a(n) gives the number of triples of equally spaced equal digits in the binary expansion of n (without leading zeros).
%C A368842 This sequence diverges to infinity by Van der Waerden's theorem.
%C A368842 A000225 \ {1, 3} corresponds to indices of records.
%H A368842 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A368842 a(2^k) = A002620(k - 1) for any k > 0.
%F A368842 a(2^k - 1) = A002620(k - 1) for any k > 0.
%F A368842 a(n) = A368843(n) + A368844(n).
%F A368842 a(floor(n/2)) <= a(n).
%e A368842 For n = 277:
%e A368842 - the binary expansion of 277 is "100010101",
%e A368842 - we have the following triples:  1   1   1
%e A368842                                    000
%e A368842                                    0 0 0
%e A368842                                      0 0 0
%e A368842                                       1 1 1
%e A368842 - so a(277) = 5.
%o A368842 (PARI) a(n, base=2) = { my (d = digits(n, base), v = 0); for (i = 1, #d-2, forstep (j = i+2, #d, 2, if (d[i]==d[j] && d[i]==d[(i+j)/2], v++;););); return (v); }
%o A368842 (Python)
%o A368842 def A368842(n):
%o A368842     l = len(s:=bin(n)[2:])
%o A368842     return sum(1 for i in range(l-2) for j in range(1,l-i+1>>1) if s[i:i+(j<<1)+1:j] in {'000','111'}) # _Chai Wah Wu_, Jan 10 2024
%Y A368842 Cf. A000225, A002620, A368841, A368843, A368844.
%K A368842 nonn,base,easy
%O A368842 0,16
%A A368842 _Rémy Sigrist_, Jan 07 2024