A292310 Triangular numbers that are equidistant from two other triangular numbers.

3, 21, 28, 36, 78, 105, 153, 171, 190, 210, 253, 325, 351, 378, 465, 528, 666, 703, 903, 946, 990, 1035, 1128, 1176, 1275, 1378, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2278, 2346, 2556, 2628, 2775, 2926, 3003, 3081, 3160, 3403, 3570, 3741, 3828, 4095, 4186, 4278, 4371, 4656

Offset: 1

Author

Antonio Roldán, Sep 14 2017

Keywords

• nonn

Comments

Triangular numbers which are the arithmetic mean of two other triangular numbers. - R. J. Mathar, Oct 01 2017

Examples

			3 is in the sequence because 0 = A000217(0), 6 = A000217(3), and the distances from 3 to 0 and 3 to 6 are the same.
153 is in the sequence because 153 = A000217(17), 6 = A000217(2), 300 = A000217(24), and the two distances 300-153 = 153-6 = 147 are the same.
		

Crossrefs

Cf. A000217, A292309, A292313, A292314, A292316.

Programs

Maple

isA292310 := proc(n)
    local ilow ;
    if isA000217(n) then
        for ilow from 0 do
            tilow := A000217(ilow) ;
            if tilow >= n then
                return false ;
            elif isA000217(2*n-tilow) then
                return true ;
            end if;
        end do:
    else
        false;
    end if;
end proc:
for n from 1 to 5000 do
    if isA292310(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 01 2017

Mathematica

Module[{t = 3, k = 2, i, e, v}, Reap[While[t <= 6000, i = k; e = 0; v = t + i; While[i > 0 && e == 0, If[IntegerQ@Sqrt[8v + 1], e = 1; Sow[t]]; i--; v += i]; k++; t += k]][[2, 1]]] (* Jean-François Alcover, Jun 25 2023, after first PARI code *)

PARI

t=3; k=2; while(t<=6000, i=k; e=0; v=t+i; while(i>0&&e==0, if(issquare(8*v+1), e=1; print1(t,", ")); i--; v+=i); k++; t+=k)

PARI

upto(n) = {my(t = 0, i = 0, triangulars = List([0]), res = List); while(t <= n, i++; t+=i; listput(triangulars, t)); for(i=2,#triangulars, tr = triangulars[i]<<1; for(j = 1, i-1, if(issquare(8 * (tr - triangulars[j]) + 1), listput(res, triangulars[i]); next(2)))); res} \\ David A. Corneth, Oct 04 2017

Formula

a(n) = A292309(n)/3.

Extensions

Term 105 added by David A. Corneth, Oct 04 2017