A054208 Consider all integer triples (i,j,k), j >= k > 0, with i^3 = binomial(j+2,3) + binomial(k+2,3), ordered by increasing i; sequence gives i values.

2, 11, 45, 65, 109, 280, 644, 1079, 1309, 3180, 3355, 6864, 8284, 9700, 10681, 10856, 16775, 17094, 33506, 35650, 50435, 63196, 84330, 105731, 125220, 145200, 167986, 220545, 503690, 678730, 1046629, 1065426

Offset

0,1

Comments

a(n)^3 is the sum of two positive tetrahedrals A000292(j) + A000292(k). j values are A054209 and k values are A054210.

Another term (not necessarily the next one) is 2630045. - R. J. Mathar, Apr 07 2023

Links

Table of n, a(n) for n=0..31.

R. J. Mathar, C++ program for A054208

Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008; Theorem 5.4.

Example

2^3 = 8 = binomial(2+2,3) + binomial(2+2,3).
11^3 = 1331 = binomial(19+2,3) + binomial(3,3).

Maple

# is x(x+1)(x+2)/6= A000292(x)=n solvable?
# return true if yes.
isA000292 := proc(n)
    local x;
    if n = 0 then
        return true ;
    end if;
    x := iroot(6*n, 3) ;
    # newton algorithm
    while true do
        x := x-round((x*(x+1)*(x+2)-6*n)/(3*x^2+6*x+2)) ;
        if A000292(x) < n and A000292(x+1) >n then
            return false ;
        elif A000292(x) > n and A000292(x-1) <n then
            return false ;
        elif A000292(x) = n then
            return true;
        end if;
    end do:
end proc:
isA054208 := proc(n)
    local c, i, ti, tj;
    c := n^3 ;
    for i from 1 do
        ti := A000292(i) ;
        if ti > c/2 then
            return false ;
        end if ;
        tj := c-ti ;
        if isA000292(tj) then
            return true ;
        end if;
    end do:
end proc:
for n from 1 do
    if isA054208(n) then
        print(n)
    end if;
end do: # R. J. Mathar, Mar 17 2023

Mathematica

(* Range of j values is merely empirical *) jmin[k_] := Floor[Max[k, 1.86*k - 20000]]; jmax[k_] := Ceiling[1.86*k + 16000]; jmax[3005] = 10^5; ii = Reap[ Do[ Do[i = (Binomial[j+2, 3] + Binomial[k+2, 3])^(1/3); If[IntegerQ[i], Print[{i, j, k}]; Sow[i]; Break[]], {j, jmin[k], jmax[k]}], {k, 1, 40000}] ][[2, 1]]; A054208 = Union[ii] (* Jean-François Alcover, Dec 12 2012 *)

Crossrefs

Cf. A054209, A054210.

Sequence in context: A181270 A110679 A127109 * A257066 A209604 A120279

Adjacent sequences: A054205 A054206 A054207 * A054209 A054210 A054211

Keyword

nice,nonn,more

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 31 2000

Extensions

More terms from Sascha Kurz, Mar 22 2002

Corrected by T. D. Noe, Oct 25 2006

a(21)-a(26) from Sean A. Irvine, Jan 25 2022

Terms complete up to 1065426 from R. J. Mathar, Apr 07 2023

Status

approved