A303539 - OEIS

A303539

Number of ordered pairs (k, m) with 0 <= k <= m such that n - binomial(2*k,k) - binomial(2*m,m) can be written as the sum of two squares.

0, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 6, 4, 2, 2, 4, 4, 2, 2, 5, 5, 5, 4, 4, 4, 4, 4, 5, 4, 5, 4, 4, 3, 4, 3, 4, 4, 5, 6, 5, 5, 5, 4, 7, 3, 3, 4, 6, 4, 2, 3, 5, 6, 3, 4, 5, 6, 5, 2, 5, 4, 5, 3, 2, 4, 5, 4, 3, 3, 3, 6, 7, 5, 5, 6, 10, 6, 3, 4, 8

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OFFSET

1,3

COMMENTS

Conjecture: a(n) > 0 for all n > 1.

a(n) > 0 for all n = 2..10^10.

See also A303540 and A303541 for related sequences.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

EXAMPLE

a(2) = 1 with 2 - binomial(2*0,0) - binomial(2*0,0) = 0^2 + 0^2.

a(3) = 2 with 3 - binomial(2*0,0) - binomial(2*0,0) = 0^2 + 1^2 and 3 - binomial(2*0,0) - binomial(2*1,1) = 0^2 + 0^2.

a(5) = 2 with 5 - binomial(2*0,0) - binomial(2*1,1) = 1^2 + 1^2 and 5 - binomial(2*1,1) - binomial(2*1,1) = 0^2 + 1^2.

MATHEMATICA

c[n_]:=c[n]=Binomial[2n, n];

f[n_]:=f[n]=FactorInteger[n];

g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;

QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);

tab={}; Do[r=0; k=0; Label[bb]; If[c[k]>n, Goto[aa]]; Do[If[QQ[n-c[k]-c[j]], r=r+1], {j, 0, k}]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

CROSSREFS

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303540, A303541, A303543.

Sequence in context: A118480 A104377 A109337 * A303540 A137266 A062948

Adjacent sequences: A303536 A303537 A303538 * A303540 A303541 A303542

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 25 2018

STATUS

approved