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 A006127 a(n) = 2^n + n. (Formerly M2547) 44
 1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, 16398, 32783, 65552, 131089, 262162, 524307, 1048596, 2097173, 4194326, 8388631, 16777240, 33554457, 67108890, 134217755, 268435484, 536870941, 1073741854, 2147483679, 4294967328, 8589934625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For numbers m=n+2^n such that equation x=2^(m-x) has solution x=2^n, see A103354. - Zak Seidov, Mar 23 2005 Primes of the form x^x+1 must be of the form 2^2^(a(n))+1, that is, Fermat number F_(a(n)) (Sierpiński 1958). - David W. Wilson, May 08 2005 a(n) = n-th Mersenne number + n + 1 = A000225(n) + n + 1. Partial sums of a(n) are A132925(n+1). - Jaroslav Krizek, Oct 16 2009 Intersection of A188916 and A188917: A188915(a(n)) = (2^n)^2 = 2^(2*n) = A000302(n). - Reinhard Zumkeller, Apr 14 2011 a(n) is also the number of all connected subtrees of a star tree, having n leaves. The star tree is a tree, where all n leaves are connected to one parent P. - Viktar Karatchenia, Feb 29 2016 REFERENCES John H. Conway, R. K. Guy, The Book of Numbers, Copernicus Press, p. 84. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..100 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 435 C. L. Mallows & N. J. A. Sloane, Emails, May 1991 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind Eric Weisstein's World of Mathematics, Star Graph Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph FORMULA Row sums of triangle A135227. - Gary W. Adamson, Nov 23 2007 Partial sums of A094373. G.f.: (1-x-x^2)/((1-x)^2(1-2x)). - Paul Barry, Aug 05 2004 Binomial transform of [1,2,1,1,1,1,1,...]. - Franklin T. Adams-Watters, Nov 29 2006 a(n) = 2*a(n-1) - n + 2 (with a(0)=1). - Vincenzo Librandi, Dec 30 2010 EXAMPLE From Viktar Karatchenia, Feb 29 2016: (Start) a(0) = 1. There are n=0 leaves, it is a trivial tree consisting of a single parent node P. a(1) = 3. There is n=1 leaf, the tree is P-A, the subtrees are: 2 singles: P, A; 1 pair: P-A; 2+1 = 3 subtrees in total. a(2) = 6. When n=2, the tree is P-A P-B, the subtrees are: 3 singles: P, A, B; 2 pairs: P-A, P-B; 1 triple: A-P-B (the whole tree); 3+2+1 = 6. a(3) = 11. For n=3 leaf nodes, the tree is P-A P-B P-C, the subtrees are: 4 singles: P, A, B, C; 3 pairs: P-A, P-B, P-C; 3 triples: A-P-B, A-P-C, B-P-C; 1 quad: P-A P-B P-C (the whole tree); 4+3+3+1 = 11 in total. a(4) = 20. For n=4 leaves, the tree is P-A P-B P-C P-D, the subtrees are: 5 singles: P, A, B, C, D; 4 pairs: P-A, P-B, P-C, P-D; 6 triples: A-P-B, A-P-C, B-P-C, A-P-D, B-P-D, C-P-D; 4 quads: P-A P-B P-C, P-A P-B P-D, P-A P-C P-D, P-B P-C P-D; the whole tree counts as 1; 5+4+6+4+1 = 20. In general, for n leaves, connected to the parent node P, there will be: (n+1) singles; (n, 1) pairs; (n, 2) triples; (n, 3) quads; ... ; (n, n-1) subtrees having (n-1) edges; 1 whole tree, having all n edges. Thus, the total number of all distinct subtrees is: (n+1) + (n, 1) + (n, 2) + (n, 3) + ... + (n, n-1) + 1 = (n + (n, 0)) + (n, 1) + (n, 2) + (n, 3) + ... + (n, n-1) + (n, n) = n + (sum of all binomial coefficients of size n) = n + (2^n). (End) MAPLE A006127:=(-1+z+z**2)/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA Table[2^n + n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *) Table[BitXOr(i, 2^i), {i, 1, 100}] (* Peter Luschny, Jun 01 2011 *) PROG (Haskell) a006127 n = a000079 n + n a006127_list = s  where    s xs = last xs : (s \$ zipWith (+) [1..] (xs ++ reverse xs)) Reinhard Zumkeller, May 19 2015, Feb 05 2011 (PARI) a(n)=1<

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Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)