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A228289 Determinant of the p_n X p_n matrix with (i,j)-entry equal to D(i+j) for all i,j = 0,...,p_n-1, where D(k) = A002895(k) is the k-th Domb number and p_n is the n-th prime. 2
12, 2448, 428587718400, 4994319435309277891448832, 191901511752240055024005979549622856313555581586068578283027431424, 637213222716753775758429677219909335140503764595701930312765250413280716374852064945052319744 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: If p_n == 1 (mod 3) and p_n = x^2 + 3*y^2 with x and y integers, then we have a(n) == (-1)^{(p_n-1)/2}*(4*x^2-2*p_n) (mod p_n^2). In the case p_n == 2 (mod 3), we have a(n) == 0 (mod p_n^2).

Zhi-Wei Sun also made the following similar conjecture:

If p is an odd prime and b(p) is the p X p determinant with (i,j)-entry equal to A053175(i+j) for all i,j = 0,...,p-1, then we have the supercongruence b(p) == (-1)^{(p-1)/2} (mod p^2).

REFERENCES

Zhi-Wei Sun, Conjectures and results on x^2 mod p^2 with 4*p = x^2 + d*y^2, in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Higher Education Press & International Press, Beijing and Boston, 2013, pp. 147-195.

LINKS

Table of n, a(n) for n=1..6.

MATHEMATICA

d[n_]:=Sum[Binomial[n, k]^2*Binomial[2k, k]Binomial[2(n-k), n-k], {k, 0, n}]

a[n_]:=Det[Table[d[i+j], {i, 0, Prime[n]-1}, {j, 0, Prime[n]-1}]]

Table[a[n], {n, 1, 8}]

CROSSREFS

Cf. A002895, A053175, A225776, A228143.

Sequence in context: A135398 A203426 A268589 * A306391 A195536 A010053

Adjacent sequences:  A228286 A228287 A228288 * A228290 A228291 A228292

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 19 2013

STATUS

approved

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Last modified January 22 16:37 EST 2020. Contains 331152 sequences. (Running on oeis4.)