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

0,1

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A343850 A235707 A056105 * A212069 A106058 A086718

Adjacent sequences: A323888 A323889 A323890 * A323892 A323893 A323894

Keyword

nonn

Author

Marius A. Burtea, Feb 12 2019

Status

approved