A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

0, 2, 7, 12, 26, 34, 45, 56, 100, 114, 131, 148, 174, 194, 217, 240, 392, 418, 447, 476, 514, 546, 581, 616, 684, 722, 763, 804, 854, 898, 945, 992, 1552, 1602, 1655, 1708, 1770, 1826, 1885, 1944, 2036, 2098, 2163, 2228, 2302, 2370, 2441, 2512, 2712, 2786, 2863

Offset: 0

Author

Kevin Ryde, Dec 14 2021

Keywords

• base
• easy
• nonn

Comments

The effect of n OR k is to force a 1-bit at all bit positions where n has a 1-bit, which means n*(n+1) in the sum. Bits of k where n has a 0-bit are NOT(n) AND k = n CNIMPL k so that a(n) = A350094(n) + n*(n+1).

Links

• Winston de Greef, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003986 (bitwise OR), A001196 (bit doubling).

Row sums of A080098.

Other sums: A222423 (AND), A224915 (XOR), A265736 (IMPL), A350094 (CNIMPL).

Programs

PARI

a(n) = (3*(n^2 + fromdigits(binary(n),4)) + 2*n) >> 2;

Formula

a(n) = ((3*n+2)*n + A001196(n)) / 4.

a(2*n) = 4*a(n) - n.

a(2*n+1) = 4*a(n) + 2*n + 2.

a(n) = A222423(n) + A224915(n), being OR = AND + XOR.