A158464 Number of distinct squares in row n of Pascal's triangle.

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 0

Author

Reinhard Zumkeller, Mar 19 2009

Keywords

• nonn

Comments

It seems that some subset of terms in A055997 (A115599) gives the positions of 3's. E.g., we have a(9) = a(50) = a(289) = a(9801) = 3, but on the other hand, a(1682) = a(57122) = 2. - Antti Karttunen, Nov 03 2017

Examples

			a(8) = #{1} = 1;
a(9) = #{1,9,36} = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..11111
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000290, A055997 (A115599).

Programs

Maple

A158464 := proc(n)
    local sqset,k ;
    sqset := {} ;
    for k from 0 to n do
        P := binomial(n,k) ;
        if issqr(P) then
            sqset := sqset union {P} ;
        end if;
    end do:
    nops(sqset) ;
end proc:
seq(A158464(n),n=0..120) ; # R. J. Mathar, Jul 09 2016

Mathematica

CountDistinct /@ Table[Sqrt@ Binomial[n, k] /. k_ /; ! IntegerQ@ k -> Nothing, {n, 0, 104}, {k, 0, n}] (* Michael De Vlieger, Nov 03 2017 *)

PARI

A158464(n) = sum(k=0,n\2,issquare(binomial(n,k))); \\ Antti Karttunen, Nov 03 2017

Formula

a(n) = Sum_{k=0..floor(n/2)} A010052(A007318(n,k));

a(A000290(n)) > 1 for n > 1.

Extensions

More terms from Antti Karttunen, Nov 03 2017