

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
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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

HsienKuei 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/wpcontent/files/2016/12/aathhrr1.pdf. Also Exact and Asymptotic Solutions of a DivideandConquer 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



