A051533 Numbers that are the sum of two positive triangular numbers.

2, 4, 6, 7, 9, 11, 12, 13, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Keywords

• easy
• nonn
• nice

Comments

Numbers n such that 8n+2 is in A085989. - Robert Israel, Mar 06 2017

Examples

			666 is in the sequence because we can write 666 = 435 + 231 = binomial(22,2) + binomial(30,2).
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem

Crossrefs

Cf. A000217, A020756 (sums of two triangular numbers), A001481 (sums of two squares), A007294, A051611 (complement).

Cf. A061336: minimal number of triangular numbers that sum up to n.

Cf. A085989.

Programs

Haskell

a051533 n = a051533_list !! (n-1)
a051533_list = filter ((> 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

Maple

isA051533 := proc(n)
    local a,ta;
    for a from 1 do
        ta := A000217(a) ;
        if 2*ta > n then
            return false;
        end if;
        if isA000217(n-ta) then
            return true;
        end if;
    end do:
end proc:
for n from 1 to 200 do
    if isA051533(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 16 2015

Mathematica

f[k_] := If[!
   Head[Reduce[m (m + 1) + n (n + 1) == 2 k && 0 < m && 0 < n, {m, n},
       Integers]] === Symbol, k, 0]; DeleteCases[Table[f[k], {k, 1, 108}], 0] (* Ant King, Nov 22 2010 *)
nn=50; tri=Table[n(n+1)/2, {n,nn}]; Select[Union[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i,nn}]]], #<=tri[[-1]] &]
With[{nn=70},Take[Union[Total/@Tuples[Accumulate[Range[nn]],2]],nn]] (* Harvey P. Dale, Jul 16 2015 *)

PARI

is(n)=for(k=ceil((sqrt(4*n+1)-1)/2),(sqrt(8*n-7)-1)\2, if(ispolygonal(n-k*(k+1)/2, 3), return(1))); 0 \\ Charles R Greathouse IV, Jun 09 2015

Formula

A053603(a(n)) > 0. - Reinhard Zumkeller, Jun 28 2013

A061336(a(n)) = 2. - M. F. Hasler, Mar 06 2017