A131179 a(n) = if n mod 2 == 0 then n*(n+1)/2, otherwise (n-1)*n/2 + 1.

0, 1, 3, 4, 10, 11, 21, 22, 36, 37, 55, 56, 78, 79, 105, 106, 136, 137, 171, 172, 210, 211, 253, 254, 300, 301, 351, 352, 406, 407, 465, 466, 528, 529, 595, 596, 666, 667, 741, 742, 820, 821, 903, 904, 990, 991, 1081, 1082, 1176, 1177, 1275, 1276, 1378, 1379, 1485

Offset: 0

Author

Philippe LALLOUET, Sep 16 2007

Keywords

• nonn
• easy

Comments

From Wesley Ivan Hurt, Jun 24 2024: (Start)

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. For n > 0, a(n) is row 1 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=1, n>0. (End)

Examples

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...
- _Wesley Ivan Hurt_, Jun 24 2024
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A128918.

For rows k = 1..10: this sequence (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

Programs

Haskell

a131179 n = (n + 1 - m) * n' + m  where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 12 2013

Magma

[(n^2+1+(n-1)*(-1)^n )/2: n in [0..60]]; // Vincenzo Librandi, Feb 12 2016

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 3, 4, 10}, 60] (* Jean-François Alcover, Feb 12 2016 *)
Table[If[EvenQ[n],(n(n+1))/2,(n(n-1))/2+1],{n,0,60}] (* Harvey P. Dale, Jul 25 2024 *)

Python

def A131179(n): return n*(n+1)//2 + (1-n)*(n % 2) # Chai Wah Wu, May 24 2022

Formula

G.f.: -x*(1+2*x-x^2+2*x^3)/((1+x)^2*(x-1)^3). - R. J. Mathar, Sep 05 2012

a(n) = ( n^2+1+(n-1)*(-1)^n )/2. - Luce ETIENNE, Aug 19 2014