Adjective
(mathematics) Being equivalent to the object whose constituents or parameters are reversed in order and mapped by some involution, particularly the inversion operator of a quasigroup.
Of a polynomial, being equivalent to the polynomial with reversed and additively inverted coefficients:
∑
i
=
0
n
a
i
x
i
{\displaystyle \sum _{i=0}^{n}a_{i}x^{i}}
is antipalindromic iff
∑
i
=
0
n
a
i
x
i
=
∑
i
=
0
n
−
a
n
−
i
x
i
⟺
a
i
=
−
a
n
−
i
{\displaystyle \sum _{i=0}^{n}a_{i}x^{i}=\sum _{i=0}^{n}-a_{n-i}x^{i}\Longleftrightarrow a_{i}=-a_{n-i}}
.
x
2
−
1
{\displaystyle x^{2}-1}
is an antipalindromic polynomial.
Of a natural number, with respect to base
b
{\displaystyle b}
, being equivalent to the natural number whose digits are reversed and subtracted from
b
−
1
{\displaystyle b-1}
:
∑
i
=
0
n
a
i
b
i
{\displaystyle \sum _{i=0}^{n}a_{i}b^{i}}
is antipalindromic iff
∑
i
=
0
n
a
i
b
i
=
∑
i
=
0
n
(
b
−
1
−
a
n
−
i
)
b
i
⟺
a
i
=
b
−
1
−
a
n
−
i
{\displaystyle \sum _{i=0}^{n}a_{i}b^{i}=\sum _{i=0}^{n}\left(b-1-a_{n-i}\right)b^{i}\Longleftrightarrow a_{i}=b-1-a_{n-i}}
.
1728
10
{\displaystyle 1728_{10}}
is an antipalindromic number with respect to base 10.