A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

2, 9, 22, 41, 68, 106, 154, 212, 285, 368, 477, 598, 741, 898, 1076, 1286, 1524, 1785, 2068, 2379, 2741, 3131, 3554, 4002, 4497, 5044, 5644, 6274, 6939, 7653, 8445, 9295, 10186, 11117, 12113, 13192, 14355, 15556, 16807, 18147, 19570, 21089, 22673, 24300, 26029, 27865, 29821, 31822, 33894, 36088

Offset: 0

Author

Marius A. Burtea, Feb 12 2019

Keywords

• nonn

Examples

			For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2.
For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.
		

References

• Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 14, p. 85, pr. 32. (in Romanian).

Links

• Robert Israel, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000041, A001156, A016754, A025425, A025434, A033461, A035294, A078406, A090677, A167661, A167700.

Programs

Magma

[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];

Maple

S:= proc(n, k, m)
   option remember;
   local p,j;
   if k = 0 then if n = 0 then return 1 else return 0 fi
   elif m < 1 then return 0
   elif n < k then return 0
   elif n > k*m^2 then return 0
   fi;
   if m^2 > n then
     p:= floor(sqrt(n));
     if p::even then p:= p-1 fi;
     return procname(n, k, p)
   fi;
   add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019

Mathematica

a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)