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A292310 Triangular numbers that are equidistant from two other triangular numbers. 4
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 (list; graph; refs; listen; history; text; internal format)
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 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

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: A074217 A062219 A091103 * A045802 A006133 A039766

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

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Last modified January 21 13:38 EST 2018. Contains 297994 sequences.