OFFSET
0,3
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)
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).
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
EXAMPLE
[ 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
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 *)
PROG
(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
(Python)
def A131179(n): return n*(n+1)//2 + (1-n)*(n % 2) # Chai Wah Wu, May 24 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe LALLOUET, Sep 16 2007
STATUS
approved