cp's OEIS Frontend

A337672 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.

%I A337672 #7 Sep 20 2020 01:30:36
%S A337672 0,9,150,153,165,195,2268,2282,2289,2364,2394,2406,2409,2454,2457,
%T A337672 2469,2499,2618,2646,2649,2661,2702,2709,2723,2829,2835,3126,3129,
%U A337672 3150,3157,3171,3213,3219,3339,3591,34680,34740,34764,34770,34785,35576,35700,35756
%N A337672 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 A337672 Leading 0's in binary expansions are ignored.
%C A337672 Positive terms are digitally balanced (A031443).
%C A337672 If m belongs to the sequence, then A056539(m) also belongs to the sequence.
%C A337672 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 A337672 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A337672 The first 16 integers, alongside their binary representations and associate polynomials, are:
%e A337672   k   bin(k)  P(k)
%e A337672   --  ------  --------------
%e A337672    0       0               0
%e A337672    1       1            -X^2
%e A337672    2      10           2*X+1
%e A337672    3      11    -2*X^2-2*X-1
%e A337672    4     100       X^2+6*X+5
%e A337672    5     101      -X^2-2*X-3
%e A337672    6     110      -X^2+2*X+3
%e A337672    7     111    -3*X^2-6*X-5
%e A337672    8    1000   2*X^2+12*X+14
%e A337672    9    1001              -4
%e A337672   10    1010           4*X+6
%e A337672   11    1011   -2*X^2-8*X-12
%e A337672   12    1100          8*X+12
%e A337672   13    1101    -2*X^2-4*X-6
%e A337672   14    1110        -2*X^2+4
%e A337672   15    1111  -4*X^2-12*X-14
%e A337672 We have constant polynomials for k = 0 and k = 9, so a(1) = 0 and a(2) = 9.
%o A337672 (PARI) is(n) = { my (b=Vecrev(binary(n))); poldegree(p=sum(k=1, #b, ('X+k-1)^2 * (-1)^b[k]))<=0 }
%Y A337672 Cf. A031443, A056539, A133468.
%K A337672 nonn,base
%O A337672 1,2
%A A337672 _Rémy Sigrist_, Sep 15 2020