

A014132


Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + nk)/2, n, k > 0, read by antidiagonals.


39



2, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79
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OFFSET

1,1


COMMENTS

Numbers that are not triangular (nontriangular numbers).
Also definable as follows: a(1)=2; for n>1, a(n) is smallest integer greater than a(n1) such that the condition "n and a(a(n)) have opposite parities" can always be satisfied.  Benoit Cloitre and Matthew Vandermast, Mar 10 2003
A248952(a(n)) < 0.  Reinhard Zumkeller, Oct 20 2014
Record values in A256188 that are greater than 1: a(n) = A256188(A004202(n)).  Reinhard Zumkeller, Mar 26 2015
Daniel Forgues, Apr 10 2015: (Start)
With n >= 1, k >= 1:
t(n+k)  k, 1 <= k <= n+k1, n >= 1;
t(n+k1) + n, 1 <= n <= n+k1, k >= 1;
where t(n+k) = t(n+k1) + (n+k) is (n+k)th triangular number, while the number of compositions of n+k into 2 parts is C(n+k1, 21) = n+k1, the number of nontriangular numbers between t(n+k1) and t(n+k), just right!
Related to Hilbert's Infinite Hotel:
0) All rooms, numbered through the positive integers, are full;
1) An infinite number of trains, each containing an infinite number of passengers, arrives: i.e., a 2D lattice of pairs of positive integers;
2) Move occupant of room m, m >= 1, to room t(m) = m*(m+1)/2, where t(m) is the mth triangular number;
3) Assign nth passenger from kth train to room t(n+k1) + n, 1 <= n <= n+k1, k >= 1;
4) Everybody has his or her own room, no room is empty, for m >= 1.
If situation 1 happens again, repeat steps 2 and 3, you're back to 4.
(End)
1711 + 2*a(n)*(58 + a(n)) is prime for n<=21. The terms that do not have this property start 29,32,34,43,47,58,59,60,62,63,65,68,70,73,...  Benedict W. J. Irwin, Nov 22 2016
Also numbers k with the property that in the symmetric representation of sigma(k) both Dyck paths of have a central peak or both Dych paths have a central valley. (Cf. A237593)  Omar E. Pol, Aug 28 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence arXiv:math/0305308 [math.NT], 2003.
Bakir Farhi, An explicit formula generating the nonFibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011. See Example 5 p. 456.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454458.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343347.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.


FORMULA

a(n) = n + round(sqrt(2*n)).
a(a(n)) = n + 2*floor(1/2 + sqrt(2n)) + 1.
a(n) = a(n1) + A035214(n), a(1)=2.
a(n) = A080036(n)  1.
a(n) = n + A002024(n).  Vincenzo Librandi, Jul 08 2010
A010054(a(n)) = 0.  Reinhard Zumkeller, Dec 10 2012
From Boris Putievskiy, Jan 14 2013: (Start)
a(n) = A007401(n)+1.
a(n) = A003057(n)^2  A114327(n).
a(n) = ((t+2)^2 + i  j)/2, where
i = nt*(t+1)/2,
j = (t*t+3*t+4)/2n,
t = floor((1+sqrt(8*n7))/2). (End)
From Robert Israel, Apr 20 2015 (Start):
a(n) = A118011(n)  n.
G.f.: x/(1x)^2 + x/(1x) * Sum(j>=0, x^(j*(j+1)/2)) = x/(1x)^2 + x^(7/8)/(22*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)


EXAMPLE

From Boris Putievskiy, Jan 14 2013: (Start)
Start of the sequence as a table (read by antidiagonals, right to left), where the kth row corresponds to the kth column of the rectangular triangle (shown thereafter):
2, 4, 7, 11, 16, 22, 29, ...
5, 8, 12, 17, 23, 30, 38, ...
9, 13, 18, 24, 31, 39, 48, ...
14, 19, 25, 32, 40, 49, 59, ...
20, 26, 33, 41, 50, 60, 71, ...
27, 34, 42, 51, 61, 72, 84, ...
35, 43, 52, 62, 73, 85, 98, ...
(...)
Start of the sequence as a rectangular triangle (read by rows), where the i elements of the ith row are t(i) + 1 up to t(i+1)  1, i >= 1:
2;
4, 5;
7, 8, 9;
11, 12, 13, 14;
16, 17, 18, 19, 20;
22, 23, 24, 25, 26, 27;
29, 30, 31, 32, 33, 34, 35;
(...)
Row number i contains i numbers, where t(i) = i*(i+1)/2:
t(i) + 1, t(i) + 2, ..., t(i) + i = t(i+1)  1
(End) [Edited by Daniel Forgues, Apr 11 2015]


MATHEMATICA

f[n_] := n + Round[Sqrt[2n]]; Array[f, 71] (* or *)
Complement[ Range[83], Array[ #(# + 1)/2 &, 13]] (* Robert G. Wilson v, Oct 21 2005 *)


PROG

(PARI) a(n)=if(n<1, 0, n+(sqrtint(8*n7)+1)\2)
(PARI) isok(n) = !ispolygonal(n, 3); \\ Michel Marcus, Mar 01 2016
(MAGMA) IsTriangular:=func< n  exists{ k: k in [1..Isqrt(2*n)]  n eq (k*(k+1) div 2)} >; [ n: n in [1..90]  not IsTriangular(n) ]; // Klaus Brockhaus, Jan 04 2011
(Haskell)
a014132 n = n + round (sqrt $ 2 * fromInteger n)
a014132_list = filter ((== 0) . a010054) [0..]
 Reinhard Zumkeller, Dec 12 2012


CROSSREFS

Cf. A000217, A006002, A035214, A080036, A002024, A007401, A003057, A114327, A002260, A004736, A118011.
Cf. A000124 (left edge: quasitriangular numbers), A000096 (right edge: almosttriangular numbers), A006002 (row sums), A001105 (central terms).
Cf. A242401 (subsequence).
Cf. A004202, A256188.
Cf. A145397 (the nontetrahedral numbers).
Sequence in context: A010423 A035235 A253723 * A184008 A183862 A254058
Adjacent sequences: A014129 A014130 A014131 * A014133 A014134 A014135


KEYWORD

nonn,easy,nice,tabl


AUTHOR

N. J. A. Sloane


EXTENSIONS

Following Alford Arnold's comment: keyword tabl and correspondent crossrefs added by Reinhard Zumkeller, Dec 12 2012
I restored the original definition.  N. J. A. Sloane, Jan 27 2019


STATUS

approved



