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 A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1. 16
 1, 6, 52, 840, 26896, 1721376, 220336192, 56406065280, 28879905423616, 29573023153783296, 60565551418948191232, 248076498612011791288320, 2032242676629600594233921536, 33296264013899376135928570454016, 1091051979207454757222107396637212672 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). 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 Numbers k such that the k-th composition in standard order (row k of A066099) is an initial interval. - Gus Wiseman, Apr 02 2020 LINKS FORMULA a(n) = Sum_{k=0..n-1} 2^((n^2 + n)/2 - (k^2 + k)/2 - 1). - Alonso del Arte, Nov 15 2013 Intersection of A333255 and A333217. - Gus Wiseman, Apr 02 2020 EXAMPLE a(1) = 1, also 1 in binary. a(2) = 6, or 110 in binary. a(3) = 52, or 110100 in binary. a(4) = 840, or 1101001000 in binary. MATHEMATICA Table[Sum[2^((n^2 + n)/2 - (k^2 + k)/2 - 1), {k, 0, n - 1}], {n, 25}] (* Alonso del Arte, Nov 14 2013 *) PROG (Python) def a(n): return int("".join("1"+"0"*i for i in range(n)), 2) print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jul 05 2021 CROSSREFS The version for prime (rather than binary) indices is A002110. The non-strict generalization is A225620. The reversed version is A246534. Standard composition numbers of permutations are A333218. Standard composition numbers of strict increasing compositions are A333255. Cf. A000120, A029931, A048793, A066099, A070939, A124768, A233564, A272919, A333217, A333220, A333379. Sequence in context: A271802 A097820 A166889 * A027835 A055973 A223345 Adjacent sequences:  A164891 A164892 A164893 * A164895 A164896 A164897 KEYWORD base,easy,nonn AUTHOR Gil Broussard, Aug 29 2009 STATUS approved

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)