[seqfan] Re: FYI - conjecture: n=x^2+T_y+F_m

[Richard Mathar]

ap> From seqfan-bounces at list.seqfan.eu Mon Dec 22 15:53:50 2008
ap> Date: Mon, 22 Dec 2008 09:51:03 -0500
ap> From: "Alexander Povolotsky" <apovolot at gmail.com>
ap> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
ap> Subject: [seqfan]  FYI - conjecture: n=x^2+T_y+F_m
ap> 
ap> FYI - Perhaps someone might be interested to generate the sequence,
ap> based on the number of ways to write n in the form x^2+T_y+F_m
ap> with x,y,m non-negative integers - per below.
ap> 
ap> ARP

One could start with the simpler sequence for F_m=0, that is
"Number of ways to write n as an ordered sum x^2+T_y of
a square and a triangular number, allowing x=0 or T_y=0", similar to A000925, starting
at n=0 as

1,2,1,1,2,1,1,2,0,1,3,1,1,0,1,2,2,1,0,3,0,1,2,0,1,2,2,0,2,1,1,2,1,0,0,
1,2,4,0,1,2,0,1,0,1,2,3,0,0,2,1,1,2,1,1,2,1,1,0,2,0,2,0,0,4,1,1,2,0,0,
4,1,1,0,1,1,0,1,1,2,1,2,3,0,1,2,0,2,0,0,0,4,2,0,2,1,1,0,0,0,3,

Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1.
a(26)=2 because 26=25+1=16+10.

in Maple:
A000217 := proc(n)
        n*(n+1)/2 ;
end:
A := proc(n)
        local a,y,t ;
        a := 0 ;
        for y from 0 do
                t := A000217(y) ;
                if n-t < 0 then
                        RETURN(a) ;
                else
                        if issqr(n-t) then
                                a := a+1 ;
                        fi;
                fi;
        od:
end:

for n from 0 to 100 do
        printf("%a,",A(n)) ;
od: