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%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
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