A298011 -id:A298011 - cp's OEIS frontend

A298043 If n = Sum_{i=1..h} 2^b_i with b_1 > ... > b_h >= 0, then a(n) = Sum_{i=1..h} i * 2^b_i.

0, 1, 2, 4, 4, 6, 8, 11, 8, 10, 12, 15, 16, 19, 22, 26, 16, 18, 20, 23, 24, 27, 30, 34, 32, 35, 38, 42, 44, 48, 52, 57, 32, 34, 36, 39, 40, 43, 46, 50, 48, 51, 54, 58, 60, 64, 68, 73, 64, 67, 70, 74, 76, 80, 84, 89, 88, 92, 96, 101, 104, 109, 114, 120, 64, 66

Offset: 0

Rémy Sigrist, Jan 11 2018

This sequence is similar to A298011.

			For n = 42:
  42 = 32 + 8 + 2,
  hence a(42) = 1*32 + 2*8 + 3*2 = 54.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A000120, A000295, A053645.

Cf. A083741, A298011.

a(n) = my (b=binary(n), z=0); for (i=1, #b, if (b[i], b[i] = z++)); return (fromdigits(b,2))

a(n) = Sum_{k = 0..A000120(n)-1} A053645^k(n) for any n > 0 (where A053645^k denotes the k-th iterate of A053645).
a(n) >= n with equality iff n = 0 or n = 2^k for some k >= 0.
a(2 * n) = 2 * a(n).
a(2^n - 1) = A000295(n + 1).
a(2 ^ i + n) = a(n) + 2 ^ i + n for 2 ^ i > n. - David A. Corneth, Jan 14 2018

A318303 a(0) = 0, a(n) = n + a(n-1) if n is odd, a(n) = -3*a(n/2) if n is even.

Original entry on oeis.org

0, 1, -3, 0, 9, 14, 0, 7, -27, -18, -42, -31, 0, 13, -21, -6, 81, 98, 54, 73, 126, 147, 93, 116, 0, 25, -39, -12, 63, 92, 18, 49, -243, -210, -294, -259, -162, -125, -219, -180, -378, -337, -441, -398, -279, -234, -348, -301, 0, 49, -75, -24, 117, 170, 36, 91, -189, -132, -276, -217, -54, 7, -147, -84, 729, 794, 630

Offset: 0

Altug Alkan, Aug 24 2018

Let g_k(0) = 0. g_k(n) = n + g_k(n-1) if n is odd, g_k(n) = k*a(n/2) if n is even. A228451(n) is g_1(n), A298011(n) is g_2(n). This sequence is a(n) = g_k(n) where k = -3.

Altug Alkan, Table of n, a(n) for n = 0..32767
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..1000000 (where the color is function of A262304(n))
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..1000000 (where the color is function of floor(n / 2^(A070939(n) - 6)))
Rémy Sigrist, A colored scatterplot of (A317825(n), abs(A318303(n))) for n = 1..2^20-1 (where the color is function of floor(n / 2^(A070939(n)-5)))
Altug Alkan, A scatterplot of (A317825(n), A318303(n)+A317825(n)) for n = 1..2^17-1

Cf. A070939, A228451, A262304, A298011, A305865, A317825, A318265.

Nest[Append[#1, If[OddQ@ #2, #2 + #1[[-1]], -3 #1[[#2/2 + 1]] ]] & @@ {#, Length@ #} &, {0}, 66] (* Michael De Vlieger, Aug 25 2018 *)

a(n)=if(n==0, 0, if(n%2, n+a(n-1), -3*a(n/2)));

A343934 Irregular triangle read by rows: row n gives the sequence of iterations of k - A006519(k), starting with k=n, until 0 is reached.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 4, 6, 4, 7, 6, 4, 8, 9, 8, 10, 8, 11, 10, 8, 12, 8, 13, 12, 8, 14, 12, 8, 15, 14, 12, 8, 16, 17, 16, 18, 16, 19, 18, 16, 20, 16, 21, 20, 16, 22, 20, 16, 23, 22, 20, 16, 24, 16, 25, 24, 16, 26, 24, 16, 27, 26, 24, 16, 28, 24, 16

Offset: 1

Christian Perfect, May 04 2021

Row n starts with n, then the highest power of 2 dividing n is subtracted to produce the next entry in the row.

n first appears at position A000788(n)+1.

			The triangle begins
1
2
3 2
4
5 4
6 4
7 6 4

Peter Kagey, Rows n = 1..1023, flattened

Cf. A000120 (row widths), A000788, A006519, A129760, A298011 (row sums).

Table[Most @ NestWhileList[# - 2^IntegerExponent[#, 2] &, n, # > 0 &], {n, 1, 30}] // Flatten (* Amiram Eldar, May 05 2021 *)

def gen_a():
    for n in range(1,100):
        k = n
        while k>0:
            yield k
            k = k & (k-1)
a = gen_a()

cp's OEIS Frontend

A298043 If n = Sum_{i=1..h} 2^b_i with b_1 > ... > b_h >= 0, then a(n) = Sum_{i=1..h} i * 2^b_i.

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A318303 a(0) = 0, a(n) = n + a(n-1) if n is odd, a(n) = -3*a(n/2) if n is even.

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A343934 Irregular triangle read by rows: row n gives the sequence of iterations of k - A006519(k), starting with k=n, until 0 is reached.

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Mathematica

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