

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
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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 300153 = 1536 = 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*ntilow) 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, i1, 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 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



