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A374005
a(n) = (1 + (n+4)^2 + (n-5)*(-1)^n)/2.
9
15, 17, 26, 32, 41, 51, 60, 74, 83, 101, 110, 132, 141, 167, 176, 206, 215, 249, 258, 296, 305, 347, 356, 402, 411, 461, 470, 524, 533, 591, 600, 662, 671, 737, 746, 816, 825, 899, 908, 986, 995, 1077, 1086, 1172, 1181, 1271, 1280, 1374, 1383, 1481, 1490, 1592, 1601
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 5 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=5.
FORMULA
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(6*x^4+2*x^3-21*x^2+2*x+15)/((x+1)^2*(x-1)^3).
a(n) = A374004(n+1) - (-1)^n.
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
CoefficientList[Series[-(6*x^4 + 2*x^3 - 21*x^2 + 2*x + 15)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 5; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]
PROG
(Magma) [(1 + (n+4)^2 + (n-5)*(-1)^n)/2: n in [1..80]];
CROSSREFS
For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), this sequence (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Row 5 of the table in A056011.
Column 5 of the rectangular array in A056023.
Sequence in context: A124334 A002155 A290749 * A091017 A157716 A113968
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 24 2024
STATUS
approved