A337672 - OEIS

%I #7 Sep 20 2020 01:30:36

%S 0,9,150,153,165,195,2268,2282,2289,2364,2394,2406,2409,2454,2457,

%T 2469,2499,2618,2646,2649,2661,2702,2709,2723,2829,2835,3126,3129,

%U 3150,3157,3171,3213,3219,3339,3591,34680,34740,34764,34770,34785,35576,35700,35756

%N 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.

%C Leading 0's in binary expansions are ignored.

%C Positive terms are digitally balanced (A031443).

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

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

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

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

%e   k   bin(k)  P(k)

%e   --  ------  --------------

%e    0       0               0

%e    1       1            -X^2

%e    2      10           2*X+1

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

%e    4     100       X^2+6*X+5

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

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

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

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

%e    9    1001              -4

%e   10    1010           4*X+6

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

%e   12    1100          8*X+12

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

%e   14    1110        -2*X^2+4

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

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

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

%Y Cf. A031443, A056539, A133468.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Sep 15 2020