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A379270 Numbers with only digits "1" and three digits "0".

%I A379270 #10 Dec 20 2024 20:42:59
%S A379270 1000,10001,10010,10100,11000,100011,100101,100110,101001,101010,
%T A379270 101100,110001,110010,110100,111000,1000111,1001011,1001101,1001110,
%U A379270 1010011,1010101,1010110,1011001,1011010,1011100,1100011,1100101,1100110,1101001,1101010,1101100
%N A379270 Numbers with only digits "1" and three digits "0".
%C A379270 Binary representation of A379269.
%C A379270 Numbers in A007088 with three 0 digits.
%F A379270 a(n) = A007088(A379269(n)).
%t A379270 Select[Range[10^7],Count[IntegerDigits[#],0]==3&&Max[IntegerDigits[#]]==1&] (* _James C. McMahon_, Dec 20 2024 *)
%o A379270 (Python)
%o A379270 from math import isqrt, comb
%o A379270 from sympy import integer_nthroot
%o A379270 def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(n<comb(m+2, 3))+2, 3)<<1))-((r<<2)<=(k<<2)*(k+1)+1)
%o A379270 def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
%o A379270 def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
%o A379270 def A379270(n):
%o A379270     a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+2
%o A379270     j = comb(a,4)-n
%o A379270     b, c, d = A360010(j+1)+1, A056557(j)+1, A333516(j+1)-1
%o A379270     return (10**a-1)//9-10**b-10**c-10**d
%Y A379270 Cf. A007088, A379269, A023416, A056557, A333516, A360010, A360573.
%K A379270 nonn,base
%O A379270 1,1
%A A379270 _Chai Wah Wu_, Dec 19 2024