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 A173318 Partial sums of A005811. 10
 0, 1, 3, 4, 6, 9, 11, 12, 14, 17, 21, 24, 26, 29, 31, 32, 34, 37, 41, 44, 48, 53, 57, 60, 62, 65, 69, 72, 74, 77, 79, 80, 82, 85, 89, 92, 96, 101, 105, 108, 112, 117, 123, 128, 132, 137, 141, 144, 146, 149, 153, 156, 160, 165, 169, 172, 174, 177, 181, 184, 186, 189 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partial sums of number of runs in binary expansion of n (n>0). Partial sums of number of 1's in Gray code for n. The subsequence of squares in this partial sum begins 0, 1, 4, 9, 144, 169, 256, 289, 324 (since we also have 32 and 128 I wonder about why so many powers). The subsequence of primes in this partial sum begins: 3, 11, 17, 29, 31, 37, 41, 53, 79, 89, 101, 137, 149, 181, 191, 197, 229, 271. Note: A227744 now gives the squares, which occur at positions given by A227743. - Antti Karttunen, Jul 27 2013 REFERENCES Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585 LINKS Antti Karttunen, Table of n, a(n) for n = 0..8191 FORMULA a(n) = sum(i=0..n) A005811(i) = sum(i=0..n) (A037834(i)+1) = sum(i=0..n) (A069010(i) + A033264(i)). a(A000225(n)) = A001787(n) = A000788(A000225(n)). - Antti Karttunen, Jul 27 2013 & Aug 09 2013 a(2n) = 2*a(n) + n - 2*(ceiling(A005811(n)/2) - (n mod 2)), a(2n+1) = 2*a(n) + n + 1. - Ralf Stephan, Aug 11 2013 EXAMPLE 1 has 1 run in its binary representation "1". 2 has 2 runs in its binary representation "10". 3 has 1 run in its binary representation "11". 4 has 2 runs in its binary representation "100". 5 has 3 runs in its binary representation "101". Thus a(1) = 1, a(2) = 1+2 = 3, a(3) = 1+2+1 = 4, a(4) = 1+2+1+2 = 6, a(5) = 1+2+1+2+3 = 9. MATHEMATICA Accumulate[Join[{0}, Table[Length[Split[IntegerDigits[n, 2]]], {n, 110}]]] (* Harvey P. Dale, Jul 29 2013 *) CROSSREFS Cf. A005811, A000788, A056539, A014707, A014577, A082410, A000975, A034947. Cf. also A227737, A227741, A227742. Sequence in context: A047415 A087805 A213040 * A153380 A268273 A034027 Adjacent sequences:  A173315 A173316 A173317 * A173319 A173320 A173321 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Feb 16 2010 STATUS approved

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Last modified April 11 16:42 EDT 2021. Contains 342888 sequences. (Running on oeis4.)