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A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

%I A164894 #32 Jul 11 2025 14:06:03
%S A164894 1,6,52,840,26896,1721376,220336192,56406065280,28879905423616,
%T A164894 29573023153783296,60565551418948191232,248076498612011791288320,
%U A164894 2032242676629600594233921536,33296264013899376135928570454016,1091051979207454757222107396637212672
%N A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.
%C A164894 These numbers are half the sum of powers of 2 indexed by differences of a triangular number and each smaller triangular number (e.g., 21 - 15 = 6, 21 - 10 = 11, ..., 21 - 0 = 21).
%C A164894 This suggests another way to think about these numbers: consider the number triangle formed by the characteristic function of the triangular numbers (A010054), join together the first n rows (the very first row is row 0) as a single binary string and that gives the (n + 1)th term of this sequence. - _Alonso del Arte_, Nov 15 2013
%C A164894 Numbers k such that the k-th composition in standard order (row k of A066099) is an initial interval. - _Gus Wiseman_, Apr 02 2020
%H A164894 Harvey P. Dale, <a href="/A164894/b164894.txt">Table of n, a(n) for n = 1..81</a>
%F A164894 a(n) = Sum_{k=0..n-1} 2^((n^2 + n)/2 - (k^2 + k)/2 - 1). - _Alonso del Arte_, Nov 15 2013
%F A164894 Intersection of A333255 and A333217. - _Gus Wiseman_, Apr 02 2020
%F A164894 a(n) = Sum_{k=0..n-1} 2^(k*(2*n-k-1)/2+n-1). - _Chai Wah Wu_, Jul 11 2025
%e A164894 a(1) = 1, also 1 in binary.
%e A164894 a(2) = 6, or 110 in binary.
%e A164894 a(3) = 52, or 110100 in binary.
%e A164894 a(4) = 840, or 1101001000 in binary.
%t A164894 Table[Sum[2^((n^2 + n)/2 - (k^2 + k)/2 - 1), {k, 0, n - 1}], {n, 25}] (* _Alonso del Arte_, Nov 14 2013 *)
%t A164894 Module[{nn=15,t},t=Table[10^n,{n,0,nn}];Table[FromDigits[Flatten[IntegerDigits/@Take[t,k]],2],{k,nn}]] (* _Harvey P. Dale_, Jan 16 2024 *)
%o A164894 (Python)
%o A164894 def a(n): return int("".join("1"+"0"*i for i in range(n)), 2)
%o A164894 print([a(n) for n in range(1, 16)]) # _Michael S. Branicky_, Jul 05 2021
%o A164894 (Python)
%o A164894 def A164894(n): return sum(1<<(k*((n<<1)-k-1)>>1)+n-1 for k in range(n)) # _Chai Wah Wu_, Jul 11 2025
%Y A164894 The version for prime (rather than binary) indices is A002110.
%Y A164894 The non-strict generalization is A225620.
%Y A164894 The reversed version is A246534.
%Y A164894 Standard composition numbers of permutations are A333218.
%Y A164894 Standard composition numbers of strict increasing compositions are A333255.
%Y A164894 Cf. A000120, A029931, A048793, A066099, A070939, A124768, A233564, A272919, A333217, A333220, A333379.
%K A164894 base,easy,nonn
%O A164894 1,2
%A A164894 _Gil Broussard_, Aug 29 2009