A239904 a(n) = n - wt(n) + (number of times 11 appears in binary expansion of n).

0, 0, 1, 2, 3, 3, 5, 6, 7, 7, 8, 9, 11, 11, 13, 14, 15, 15, 16, 17, 18, 18, 20, 21, 23, 23, 24, 25, 27, 27, 29, 30, 31, 31, 32, 33, 34, 34, 36, 37, 38, 38, 39, 40, 42, 42, 44, 45, 47, 47, 48, 49, 50, 50, 52, 53, 55, 55, 56, 57, 59, 59, 61, 62, 63, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 74, 74, 76

Offset

0,4

Comments

This is G_{2, 1/4}(n) in Prodinger's notation.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..10000

Helmut Prodinger, Generalizing the sum of digits function, SIAM J. Algebraic Discrete Methods 3 (1982), no. 1, 35--42. MR0644955 (83f:10009).

Formula

a(n) = n - A000120(n) + A014081(n).

Maple

A000120 := proc(n) add(i, i=convert(n, base, 2)) end:
# A014081:
cn := proc(v, k) local n, s, nn, i, j, som, kk;
som := 0;
kk := convert(cat(seq(1, j = 1 .. k)), string);
n := convert(v, binary);
s := convert(n, string);
nn := length(s);
for i to nn - k + 1 do
if substring(s, i .. i + k - 1) = kk then som := som + 1 fi od;
som; end;
[seq(n-A000120(n)+cn(n, 2), n=0..100)];

Mathematica

cn[n_, k_] := Count[Partition[IntegerDigits[n, 2], k, 1], Table[1, {k}]]; Table[n - DigitCount[n, 2, 1] + cn[n, 2], {n, 0, 78}] (* Michael De Vlieger, Sep 18 2015 *)

Prog

(PARI)
a(n) = n - hammingweight(n) + hammingweight(bitand(n, n>>1));
vector(79, i, a(i-1))  \\ Gheorghe Coserea, Sep 24 2015
(Python)
def A239904(n): return n-n.bit_count()+(n&(n>>1)).bit_count() # Chai Wah Wu, Feb 12 2023

Crossrefs

Cf. A000120, A014081, A239906, A239907.

Sequence in context: A081210 A285719 A070321 * A334819 A338375 A220838

Adjacent sequences: A239901 A239902 A239903 * A239905 A239906 A239907

Keyword

nonn,base

Author

N. J. A. Sloane, Apr 06 2014

Status

approved