A257274 Numbers whose square can be written as sum of at least 3 consecutive triangular numbers (A000217).

8, 10, 19, 26, 44, 58, 79, 84, 91, 105, 111, 121, 135, 140, 154, 172, 184, 188, 195, 203, 208, 212, 217, 222, 230, 240, 246, 265, 276, 286, 292, 316, 322, 329, 338, 364, 390, 426, 429, 462, 490, 498, 506, 548, 550, 605, 704, 714, 715, 763, 770, 780, 782, 807

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?

The terms > 2 of A129445, numbers k > 0 such that k^2 is a centered triangular number, form a subsequence: they correspond to k^2 = T(n-2) + T(n-1) + T(n), with n in sequence A129444: numbers n such that centered triangular number A005448(n) = 3n(n-1)/2 + 1 is a perfect square.

Terms > 2 of sequence A075870 also form a subsequence, namely the numbers whose square is the sum of four triangular numbers T(n-3)+...+T(n), with n given by twice the terms > 1 of A046090 or A182435.

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

Sequence in context: A157911 A090097 A236653 * A022322 A302637 A230862

Adjacent sequences: A257271 A257272 A257273 * A257275 A257276 A257277

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