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A257274 Numbers whose square can be written as sum of at least 3 consecutive triangular numbers (A000217). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)