A359495 - OEIS

A359495

Sum of positions of 1's in binary expansion minus sum of positions of 1's in reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

0, 0, -1, 0, -2, 0, -2, 0, -3, 0, -2, 1, -4, -1, -3, 0, -4, 0, -2, 2, -4, 0, -2, 2, -6, -2, -4, 0, -6, -2, -4, 0, -5, 0, -2, 3, -4, 1, -1, 4, -6, -1, -3, 2, -5, 0, -2, 3, -8, -3, -5, 0, -7, -2, -4, 1, -9, -4, -6, -1, -8, -3, -5, 0, -6, 0, -2, 4, -4, 2, 0, 6

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OFFSET

0,5

COMMENTS

Also the sum of partial sums of reversed binary expansion minus sum of partial sums of binary expansion.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..16383

FORMULA

a(n) = A029931(n) - A230877(n).

If n = Sum_{i=1..k} q_i * 2^(i-1), then a(n) = Sum_{i=1..k} q_i * (2i-k-1).

EXAMPLE

The binary expansion of 158 is (1,0,0,1,1,1,1,0), with positions of 1's {1,4,5,6,7} with sum 23, reversed {2,3,4,5,8} with sum 22, so a(158) = 1.

MAPLE

a:= n-> (l-> add(i*(l[-i]-l[i]), i=1..nops(l)))(Bits[Split](n)):

seq(a(n), n=0..127); # Alois P. Heinz, Jan 09 2023

MATHEMATICA

sap[q_]:=Sum[q[[i]]*(2i-Length[q]-1), {i, Length[q]}];

Table[sap[IntegerDigits[n, 2]], {n, 0, 100}]

PROG

(Python)

def A359495(n):

k = n.bit_length()-1

return sum((i<<1)-k for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

CROSSREFS

Indices of positive terms are A359401.

Indices of 0's are A359402.

A030190 gives binary expansion, reverse A030308.

A070939 counts binary digits.

A230877 adds up positions of 1's in binary expansion, reverse A029931.

Cf. A000120, A048793, A053632, A222955, A231204, A291166, A326669, A326672, A326673, A359042, A359043.

Sequence in context: A183063 A318979 A172441 * A053399 A322996 A242096

Adjacent sequences: A359492 A359493 A359494 * A359496 A359497 A359498

KEYWORD

sign,look,base

AUTHOR

Gus Wiseman, Jan 05 2023

STATUS

approved