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.

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

Offset: 1

Author

Rémy Sigrist, Sep 15 2020

Keywords

• nonn
• base

Comments

Leading 0's 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).

Examples

			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.
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A031443, A056539, A133468.

Programs

PARI

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