A173209 Partial sums of A000069.

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset

1,2

Comments

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

Formula

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

Example

a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Mathematica

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

Prog

(PARI) a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

Crossrefs

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Sequence in context: A310278 A154772 A156859 * A331240 A146931 A176675

Adjacent sequences: A173206 A173207 A173208 * A173210 A173211 A173212

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Feb 12 2010

Status

approved