A190820 In base-2 lunar arithmetic, number of distinct squares of length 2n+1.

2, 2, 4, 8, 15, 29, 55, 105, 197, 367, 678, 1261, 2326, 4293, 7902, 14431, 26363, 47899

Offset: 0

Author

N. J. A. Sloane, Jun 01 2011

Keywords

• nonn
• base
• more

Links

• D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
• D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A067398, A191342.

Programs

Maple

rebase := proc(L)
    add( op(i,L)*2^(i-1),i=1..nops(L)) ;
end proc:
dismadd := proc(a,b)
    local adgs,bdgs,alen,blen,L,i ;
    adgs := convert(a,base,2) ;
    bdgs := convert(b,base,2) ;
    alen := nops(adgs) ;
    blen := nops(bdgs) ;
    L := [] ;
    for i from 1 to max(alen,blen) do
        if i <= alen and i <= blen then
            L := [op(L),max(op(i,adgs),op(i,bdgs))] ;
        elif i <= alen then
            L := [op(L),op(i,adgs)] ;
        else
            L := [op(L),op(i,bdgs)] ;
        end if;
    end do:
    rebase(L) ;
end proc:
dismult := proc(a,b)
    local bdgs,blen,c,i ;
    if b = 0 then
        return 0;
    end if;
    bdgs := convert(b,base,2) ;
    blen := nops(bdgs) ;
    if blen = 1 then
        # cannot be 0 (already above), so b=1
        return a;
    else
        c := 0 ;
        for i from 1 to blen do
            if op(i,bdgs) <> 0 then
                c := dismadd(c, 2^(i-1)*a ) ;
            end if;
        end do:
    end if;
    c ;
end proc:
A190820 := proc(n)
    local c,sq;
    sq := {} ;
    # length of square is 2*n+1 = 2*len(i)-1, so len(i)=n+1.
    for c from 2^n to 2^(n+1)-1 do
        sq := sq union { dismult(c,c) } ;
    end do:
    nops(sq) ;
end proc:

Formula

a(n) = #{k in A067398: A070939(k)=2*n+1}. - R. J. Mathar, Nov 23 2015

Extensions

a(16) and a(17) from R. J. Mathar, Nov 23 2015