A137362 - OEIS

%I #12 Jun 05 2017 19:01:38

%S 4,7,8,11,14,17,18,21,24,25,28,31,34,35,38,41,42,45,48,49,52,55,58,59,

%T 62,65,66,69,72,75,76,79,82,83,86,89,92,93,96,99,100,103,106,107,110,

%U 113,116,117,120,123,124,127,130,133,134,137,140,141,144,147,148,151,154,157,158,161,164,165,168

%N Positions at which the truncated square root of triangular numbers is unique.

%C For any term p of the sequence B(p)=1+B(p-1)=-1+B(p+1), where B(p) = floor(p*(p-1)/2).

%C For any of others p, one of these equalities is wrong.

%C The difference between two successive isolated terms of the sequence is always 3 or 7 (a(4)-a(1)=11-4=7, a(5)-a(4)=14-11=3)

%C The difference between the first or second terms of two successive pairs of the sequence is always 7 or 10 (a(6)-a(3)=17-7=7=a(7)-a(3)=18-8=10, a(9-a(6)=24-17=a(10)-a(7)=25-18=7)

%C For any n, a(n+13)-a(n) is always equal to 31 or 33. a(14)-a(1)=35-4=31, a(16)-a(3)=41-8=33.

%C Consider the slowly rising step function A061288 of truncated square roots. It attains unique (non-repeated) values A061288(j)=2,4,5,7,9,11,12,... once, whereas all others (1,3,6,8,10,..) occur at least twice. The values j+1 of the associated indices j=3,6,7,10,13,16 are listed here. - _R. J. Mathar_, May 05 2008

%C n such that A161680(n-1) < (A061288(n)-1)^2. - _Robert Israel_, Jun 05 2017

%H Robert Israel, <a href="/A137362/b137362.txt">Table of n, a(n) for n = 1..10000</a>

%p R:= map(n -> floor(sqrt(n*(n-1)/2)), [$1..100]):

%p select((t -> R[t] > R[t-1] and R[t] < R[t+1], [$2..99]); # _Robert Israel_, Jun 05 2017

%Y Cf. A061288, A161680

%K easy,nonn

%O 1,1

%A Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 26 2008, Jun 06 2008

%E Edited by _R. J. Mathar_, May 05 2008

%E Edited by, and more terms from, _Robert Israel_, Jun 05 2017