A375338 Number of distinct, irreducible ways that a Pythagorean hyperrectangle of 2 or more dimensions can produce diagonal length n.

0, 1, 1, 1, 1, 1, 3, 5, 8, 5, 12, 16, 14, 23, 17, 52, 57, 58, 108, 104, 113, 174, 248, 359, 217, 265, 406, 597, 731, 579, 1137, 1525, 1137, 1400, 1898, 2659, 3136, 3073, 3203, 5021, 4660, 5022, 8410, 8850, 7748, 10066, 15455, 18718, 17354, 12118, 18483, 22683

Offset

1,7

Comments

Here "irreducible" means the base and diagonal lengths are coprime, and no subset of bases form smaller Pythagorean hyperrectangles.

Subset of cases counted in A037444, which includes nonreduced cases and 1 dimensional cases.

Links

Table of n, a(n) for n=1..52.

Example

a(2)=1: {1, 1, 1, 1}
a(3)=1: {1, 2, 2}
...
a(7)=3: {1, 4, 4, 4}, {2, 2, 4, 5}, {2, 3, 6}
a(8)=5: {1, 1, 1, 5, 6}, {1, 1, 2, 3, 7}, {1, 2, 3, 5, 5}, {1, 3, 3, 3, 6}, {2, 2, 2, 3, 3, 3, 5}
...
Not counted:
{0} -> 0 (0 length base, <2 dimensions)
{1} -> 1 (<2 dimensions)
{1, 1, 1, 1, 1, 1, 1, 1, 1} -> 3 (contains subset {1, 1, 1, 1} -> 2)
{4, 3} -> 5 (indistinct from {3, 4} -> 5)
{6, 8} -> 10 (reduces to {3, 4} -> 5)
{5, 6, 6, 8, 8} -> 15 (reduces to {5, 10, 10} -> 15, then to {1, 2, 2} -> 3)

Prog

(PARI) a(n)=if(n, /* globals */ v=List([n]); c=n^2; a_n=0); if(c==0, if(#v>=3 && gcd(concat(v))==1, a_n++); return); my(b=if(#v>=2, v[#v], 1)-1); while(b++ && b^2<=c, if(hassub(b^2)==true, next); listput(v, b); c-=b^2; a(); c+=b^2; listpop(v)); a_n
hassub(b2)=my(r=false); for(t=1, 2^(#v-1)-1-if(b2==c, 1, 0), my(tb=binary(t)); if(issquare(b2+sum(i=0, #tb-1, tb[#tb-i]*v[#v-i]^2)), r=true; break)); r
for(n=1, 52, print(n, " ", a(n)))

Crossrefs

Cf. A037444.

Sequence in context: A021902 A136188 A073334 * A368741 A021740 A172370

Adjacent sequences: A375335 A375336 A375337 * A375339 A375340 A375341

Keyword

nonn

Author

Charles L. Hohn, Aug 12 2024

Status

approved