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

1,1

Comments

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

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A062219 A091103 A363409 * A045802 A006133 A317860

Adjacent sequences: A292307 A292308 A292309 * A292311 A292312 A292313

Keyword

nonn

Author

Antonio Roldán, Sep 14 2017

Extensions

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

Status

approved