proposed

approved

editing

proposed

Leading zeros 0's in binary expansions are ignored.

proposed

editing

editing

proposed

Cf. A031443, A056539, A133468.

allocated for Rémy Sigrist

Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant.

0, 9, 150, 153, 165, 195, 2268, 2282, 2289, 2364, 2394, 2406, 2409, 2454, 2457, 2469, 2499, 2618, 2646, 2649, 2661, 2702, 2709, 2723, 2829, 2835, 3126, 3129, 3150, 3157, 3171, 3213, 3219, 3339, 3591, 34680, 34740, 34764, 34770, 34785, 35576, 35700, 35756

1,2

Leading zeros in binary expansions are ignored.

Positive terms are digitally balanced (A031443).

If m belongs to the sequence, then A056539(m) also belongs to the sequence.

If m and n belong to the sequence, then their binary concatenation also belongs to the sequence (assuming the concatenation with 0 is neutral).

<a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

The first 16 integers, alongside their binary representations and associate polynomials, are:

k bin(k) P(k)

-- ------ --------------

0 0 0

1 1 -X^2

2 10 2*X+1

3 11 -2*X^2-2*X-1

4 100 X^2+6*X+5

5 101 -X^2-2*X-3

6 110 -X^2+2*X+3

7 111 -3*X^2-6*X-5

8 1000 2*X^2+12*X+14

9 1001 -4

10 1010 4*X+6

11 1011 -2*X^2-8*X-12

12 1100 8*X+12

13 1101 -2*X^2-4*X-6

14 1110 -2*X^2+4

15 1111 -4*X^2-12*X-14

We have constant polynomials for k = 0 and k = 9, so a(1) = 0 and a(2) = 9.

(PARI) is(n) = { my (b=Vecrev(binary(n))); poldegree(p=sum(k=1, #b, ('X+k-1)^2 * (-1)^b[k]))<=0 }

Cf. A031443, A056539.

allocated

nonn,base

Rémy Sigrist, Sep 15 2020

approved

editing

allocated for Rémy Sigrist

allocated

approved