A323891 - OEIS

%I #21 Sep 19 2022 06:18:42

%S 2,9,22,41,68,106,154,212,285,368,477,598,741,898,1076,1286,1524,1785,

%T 2068,2379,2741,3131,3554,4002,4497,5044,5644,6274,6939,7653,8445,

%U 9295,10186,11117,12113,13192,14355,15556,16807,18147,19570,21089,22673,24300,26029,27865,29821,31822,33894,36088

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

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

%H Robert Israel, <a href="/A323891/b323891.txt">Table of n, a(n) for n = 0..2000</a>

%e 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.

%e 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.

%p S:= proc(n, k, m)

%p    option remember;

%p    local p,j;

%p    if k = 0 then if n = 0 then return 1 else return 0 fi

%p    elif m < 1 then return 0

%p    elif n < k then return 0

%p    elif n > k*m^2 then return 0

%p    fi;

%p    if m^2 > n then

%p      p:= floor(sqrt(n));

%p      if p::even then p:= p-1 fi;

%p      return procname(n, k, p)

%p    fi;

%p    add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)

%p end proc:

%p seq(S(72*n+42, 10, 72*n+42), n=0..100); # _Robert Israel_, Feb 24 2019

%t a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Sep 19 2022 *)

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

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

%K nonn

%O 0,1

%A _Marius A. Burtea_, Feb 12 2019