A277640 a(n) is the integer r with |r| < prime(n)/2 such that (T(prime(n)^2)-T(prime(n)))/prime(n)^4 == r (mod prime(n)), where T(k) denotes the central trinomial coefficient A002426(k).
-2, 1, -3, -1, 7, -1, 6, 4, -15, -15, -13, 1, -23, 1, 8, -15, -22, 13, -33, 27, 25, 11, -17, 24, -32, -53, 31, 42, -19, 18, -35, 55, -5, 38, -29, 76, 34, 44, -71, -21, -13, 16, 46, 70, 92, 70, -39, 88, -84, -118, -120, 64, 107, 111, -56, 124, -13, -23
Offset: 3
Keywords
Examples
a(3) = -2 since (T(prime(3)^2)-T(prime(3)))/prime(3)^4 = (T(25)-T(5))/5^4 = (82176836301-51)/5^4 = 131482938 is congruent to -2 modulo prime(3) = 5 with |-2| < 5/2.
Links
- Hao Pan and Zhi-Wei Sun, Supercongruences for central trinomial coefficients, arXiv:2012.05121 [math.NT], 2020.
- Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
- Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.
Programs
-
Mathematica
T[n_]:=T[n]=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,n/2}] rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2] Do[Print[n," ",rMod[(T[Prime[n]^2]-T[Prime[n]])/Prime[n]^4,Prime[n]]],{n,3,60}]
Comments