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A214697 Least k > 1 such that tri(n)+ ... + tri(n+k-1) is a triangular number.

%I A214697 #33 Mar 21 2025 18:33:59
%S A214697 2,3,5,17,7,2,89,125,3,215,269,13,10,8,11,27,719,815,21,57,316,11,26,
%T A214697 1517,17,1799,30,26,7,5,2609,11,2975,10,2,76,3779,1251,208,4445,115,
%U A214697 4919,1045,5417,11,17,1205,6485,38,2860,7349,18,25,8267,8585,8909
%N A214697 Least k > 1 such that tri(n)+ ... + tri(n+k-1) is a triangular number.
%C A214697 tri(n) = n*(n+1)/2 is the n-th triangular number, A000217(n).
%C A214697 a(n) is how many consecutive triangular numbers starting from tri(n) are needed to sum up to tri(x) for some x. The requirement a(n) > 1 is needed, because otherwise all a(n) = 1.
%C A214697 Because an oblong number (A002378) is twice a triangular number, this sequence is also the least k > 1 such that oblong(n) + ... + oblong(n+k-1) is an oblong number.
%C A214697 a(n) is least k > 1 such that 12*k^3 + 36*n*k^2 + 36*k*n^2 - 12*k + 9 is a perfect square. - _Chai Wah Wu_, Mar 01 2016
%C A214697 a(n) <= 3*n^2 - 3*n - 1 for n > 1, since 12*k^3 + 36*n*k^2 + 36*k*n^2 - 12*k + 9 is a square when k = 3*n^2 - 3*n - 1. - _Robert Israel_, Mar 03 2016
%H A214697 Chai Wah Wu, <a href="/A214697/b214697.txt">Table of n, a(n) for n = 0..5000</a>
%e A214697 0+1 = 1 is a triangular number, two summands, so a(0)=2.
%e A214697 1+3+6 = 10 is a triangular number, three summands, so a(1)=3.
%e A214697 3+6+10+15+21 = 55 is a triangular number, five summands, so a(2)=5.
%e A214697 Starting from Triangular(5)=15:  15+21=36 is a triangular number, two summands, so a(5)=2.
%p A214697 f:= proc(n) local k;
%p A214697     for k from 2 do if issqr(12*k^3+36*k^2*n+36*k*n^2-12*k+9) then return k fi od
%p A214697 end proc:
%p A214697 map(f, [$0..100]); # _Robert Israel_, Mar 03 2016
%t A214697 triQ[n_] := IntegerQ[Sqrt[1+8*n]]; Table[k = n+1; s = k^2; While[! triQ[s], k++; s = s + k*(k+1)/2]; k - n + 1, {n, 0, 55}] (* _T. D. Noe_, Jul 26 2012 *)
%o A214697 (Python)
%o A214697 for n in range(77):
%o A214697     i = ti = n
%o A214697     sum = 0
%o A214697     tn_gte_sum = 0  # least oblong  number >= sum
%o A214697     while i-n<=1 or tn_gte_sum!=sum:
%o A214697         sum += i*(i+1)
%o A214697         i+=1
%o A214697         while tn_gte_sum<sum:
%o A214697             tn_gte_sum = ti*(ti+1)
%o A214697             ti+=1
%o A214697     print(i-n, end=', ')
%o A214697 (Python)
%o A214697 from math import sqrt
%o A214697 def A214697(n):
%o A214697     k, a1, a2, m = 2, 36*n, 36*n**2 - 12, n*(72*n + 144) + 81
%o A214697     while int(round(sqrt(m)))**2 != m:
%o A214697         k += 1
%o A214697         m = k*(k*(12*k + a1) + a2) + 9
%o A214697     return k # _Chai Wah Wu_, Mar 01 2016
%Y A214697 Cf. A000217, A000292, A214648, A214696.
%K A214697 nonn,look
%O A214697 0,1
%A A214697 _Alex Ratushnyak_, Jul 26 2012