OFFSET
1,1
COMMENTS
Any square can trivially be written as sum of two consecutive triangular numbers T = A000217, since T(n-1) + T(n) = n(n-1)/2 + n(n+1)/2 = n*2n/2 = n^2. But it seems nontrivial to determine the squares that can be written as sum of more than 2 consecutive triangular numbers.
Some of these have two different decompositions of this form, e.g., 286^2 = T(13)+...+T(78) = T(75)+...+T(96), 826^2 = T(13)+...+T(159) = T(43)+...+T(160). What is the sequence of these numbers?
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000
EXAMPLE
8^2 = T(5)+T(6)+T(7), 10^2 = T(5)+T(6)+T(7)+T(8), 19^2 = T(14)+T(15)+T(16), 26^2 = T(3)+...+T(15), 44^2 = T(13)+...+T(23), ...
PROG
(PARI) {a=[]; (S(n)=binomial(n+2, 3)); for(n=1, 999, for(k=1, n-3, issquare(S(n)-S(k))&&a=concat(a, sqrtint(S(n)-S(k))))); Set(a)[1..50]}
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, May 02 2015
EXTENSIONS
a(14), a(43)-a(54) from Chai Wah Wu, Jan 20 2016
STATUS
approved