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A373662
a(n) = (1 + (n+1)^2 - (n-2)*(-1)^n)/2.
10
2, 5, 9, 12, 20, 23, 35, 38, 54, 57, 77, 80, 104, 107, 135, 138, 170, 173, 209, 212, 252, 255, 299, 302, 350, 353, 405, 408, 464, 467, 527, 530, 594, 597, 665, 668, 740, 743, 819, 822, 902, 905, 989, 992, 1080, 1083, 1175, 1178, 1274, 1277, 1377, 1380, 1484, 1487, 1595
OFFSET
1,1
COMMENTS
Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 2 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=2.
FORMULA
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = A131179(n+1) + (-1)^n.
G.f.: -x*(2*x^4-3*x^3+3*x+2)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 12 2024
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 ...
...
MATHEMATICA
k := 2; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]
PROG
(Magma) [(1 + (n+1)^2 - (n-2)*(-1)^n)/2: n in [1..80]];
(Python)
def A373662(n): return ((n+1)*(n+2)-1 if n&1 else n*(n+1)+5)>>1 # Chai Wah Wu, Jun 23 2024
CROSSREFS
For rows k = 1..10: A131179 (k=1) n>0, this sequence (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).
Row n=2 of A056011.
Column k=2 of A056023.
Sequence in context: A070828 A112988 A218706 * A270950 A139405 A360899
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 12 2024
STATUS
approved