This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A063657 #68 Oct 23 2024 01:01:54
%S A063657 3,7,8,13,14,15,21,22,23,24,31,32,33,34,35,43,44,45,46,47,48,57,58,59,
%T A063657 60,61,62,63,73,74,75,76,77,78,79,80,91,92,93,94,95,96,97,98,99,111,
%U A063657 112,113,114,115,116,117,118,119,120,133,134,135,136,137,138,139,140
%N A063657 Numbers with property that truncated square root is unequal to rounded square root.
%C A063657 Also: skip 1, take 0, skip 2, take 1, skip 3, take 2, ...
%C A063657 Integers for which the periodic part of the continued fraction for the square root of n begins with a 1. - _Robert G. Wilson v_, Nov 01 2001
%C A063657 a(n) belongs to the sequence if and only if a(n) > floor(sqrt(a(n))) * ceiling(sqrt(a(n))), i.e. a(n) in (k*(k+1),k^2), k >= 0. - _Daniel Forgues_, Apr 17 2011
%C A063657 Any integer between (k - 1/2)^2 and k^2 exclusive, for k > 1, is in this sequence. If we take this sequence and remove each term that is one more than the previous term, we obtain the central polygonal numbers (A002061). If instead we remove each term that is one less than the next term, we obtain numbers that are one less than squares (A005563). - _Alonso del Arte_, Dec 28 2013
%H A063657 Seiichi Manyama, <a href="/A063657/b063657.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%H A063657 Roger B. Nelsen, <a href="https://www.jstor.org/stable/2691505">Proof without Words: Consecutive sums of consecutive integers</a>, Mathematics Magazine, Vol. 63, No. 1 (1990), p. 25.
%F A063657 a(n) = A217575(n) + 1. - _Reinhard Zumkeller_, Jun 20 2015
%F A063657 From _Stefano Spezia_, Oct 20 2024: (Start)
%F A063657 As a triangle:
%F A063657 T(n,k) = n^2 + n + k with 1 <= k <= n.
%F A063657 G.f.: x*y*(3 + x^2*(1 - 4*y) - x*(2 + y) + x^3*y*(1 + 2*y))/((1 - x)^3*(1 - x*y)^3). (End)
%e A063657 7 is in the sequence because its square root is 2.64575..., which truncates to 2 but rounds to 3.
%e A063657 8 is in the sequence because its square root is 2.828427..., which also truncates to 2 but rounds to 3.
%e A063657 9 is not in the sequence because its square root is 3 exactly, which truncates and rounds the same.
%e A063657 Here is the example per Lamoen's skip n, take n - 1 process: starting at 0, we skip one integer (0) but take zero integers for our sequence. Then we skip two integers (1 and 2) and take one integer (3) for our sequence. Then we skip three integers (4, 5, 6) and take two integers for our sequence (7 and 8, so the sequence now stands as 3, 7, 8). Then we skip four integers (9, 10, 11, 12) and so on and so forth.
%e A063657 From _Seiichi Manyama_, Sep 19 2017: (Start)
%e A063657 See R. B. Nelsen's paper.
%e A063657 k| A063656(n) | a(n)
%e A063657 -------------------------------------------------------------------
%e A063657 0| 0
%e A063657 1| 1 + 2 = 3
%e A063657 2| 4 + 5 + 6 = 7 + 8
%e A063657 3| 9 + 10 + 11 + 12 = 13 + 14 + 15
%e A063657 4| 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24
%e A063657 | ...
%e A063657 (End)
%e A063657 The triangle begins as:
%e A063657 3;
%e A063657 7, 8;
%e A063657 13, 14, 15;
%e A063657 21, 22, 23, 24;
%e A063657 31, 32, 33, 34, 35;
%e A063657 43, 44, 45, 46, 47, 48;
%e A063657 57, 58, 59, 60, 61, 62, 63;
%e A063657 73, 74, 75, 76, 77, 78, 79, 80;
%e A063657 91, 92, 93, 94, 95, 96, 97, 98, 99;
%e A063657 ... - _Stefano Spezia_, Oct 20 2024
%p A063657 A063657:=n->`if`(floor(floor(sqrt(n+1)) * (1+floor(sqrt(n+1)))/(n+1))=1, NULL, n+1); seq(A063657(n), n=1..200); # _Wesley Ivan Hurt_, Dec 28 2013
%t A063657 Select[ Range[200], Floor[ Sqrt[ # ]] != Floor[ Sqrt[ # ] + 1/2] & ] (* or *) Select[ Range[200], First[ Last[ ContinuedFraction[ Sqrt[ # ]]]] == 1 & ]
%o A063657 (PARI) { n=0; for (m=0, 10^9, if (sqrt(m)%1 > .5, write("b063657.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 27 2009
%o A063657 (Haskell)
%o A063657 a063657 n = a063657_list !! n
%o A063657 a063657_list = f 0 [0..] where
%o A063657 f k (_:xs) = us ++ f (k + 1) (drop (k + 1) vs) where
%o A063657 (us, vs) = splitAt k xs
%o A063657 -- _Reinhard Zumkeller_, Jun 20 2015
%Y A063657 Cf. A002620, A004201-A004202, A063656, A189151.
%Y A063657 Cf. A005563 (main diagonal), A059270 (row sums), A217575.
%K A063657 nonn,tabl
%O A063657 1,1
%A A063657 _Floor van Lamoen_, Jul 24 2001