A137362 Positions at which the truncated square root of triangular numbers is unique.

4, 7, 8, 11, 14, 17, 18, 21, 24, 25, 28, 31, 34, 35, 38, 41, 42, 45, 48, 49, 52, 55, 58, 59, 62, 65, 66, 69, 72, 75, 76, 79, 82, 83, 86, 89, 92, 93, 96, 99, 100, 103, 106, 107, 110, 113, 116, 117, 120, 123, 124, 127, 130, 133, 134, 137, 140, 141, 144, 147, 148, 151, 154, 157, 158, 161, 164, 165, 168

Offset

1,1

Comments

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).

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

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)

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)

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.

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

n such that A161680(n-1) < (A061288(n)-1)^2. - Robert Israel, Jun 05 2017

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

R:= map(n -> floor(sqrt(n*(n-1)/2)), [$1..100]):
select((t -> R[t] > R[t-1] and R[t] < R[t+1], [$2..99]); # Robert Israel, Jun 05 2017

Crossrefs

Cf. A061288, A161680

Sequence in context: A047347 A188376 A356088 * A190765 A024621 A375572

Adjacent sequences: A137359 A137360 A137361 * A137363 A137364 A137365

Keyword

easy,nonn

Author

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

Extensions

Edited by R. J. Mathar, May 05 2008

Edited by, and more terms from, Robert Israel, Jun 05 2017

Status

approved