%I #6 Jun 24 2024 19:23:35

%S 29,44,48,63,71,86,98,113,129,144,164,179,203,218,246,261,293,308,344,

%T 359,399,414,458,473,521,536,588,603,659,674,734,749,813,828,896,911,

%U 983,998,1074,1089,1169,1184,1268,1283,1371,1386,1478,1493,1589,1604,1704,1719

%N a(n) = (1 + (n+7)^2 - (n-8)*(-1)^n)/2.

%C Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 8 of the boustrophedon-style array (see example).

%C In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=8.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

%F G.f.: -x*(29*x^4-15*x^3-54*x^2+15*x+29)/((x+1)^2*(x-1)^3).

%F a(n) = A374008(n+1) + (-1)^n.

%e [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]

%e [ 1] 1 3 4 10 11 21 22 36 37 55 56 78 ...

%e [ 2] 2 5 9 12 20 23 35 38 54 57 77 ...

%e [ 3] 6 8 13 19 24 34 39 53 58 76 ...

%e [ 4] 7 14 18 25 33 40 52 59 75 ...

%e [ 5] 15 17 26 32 41 51 60 74 ...

%e [ 6] 16 27 31 42 50 61 73 ...

%e [ 7] 28 30 43 49 62 72 ...

%e [ 8] 29 44 48 63 71 ...

%e [ 9] 45 47 64 70 ...

%e [10] 46 65 69 ...

%e [11] 66 68 ...

%e [12] 67 ...

%e ...

%t CoefficientList[Series[-(29*x^4 - 15*x^3 - 54*x^2 + 15*x + 29)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]

%t k := 8; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

%o (Magma) [(1 + (n+7)^2 - (n-8)*(-1)^n)/2: n in [1..80]];

%Y For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), this sequence (k=8), A374010 (k=9), A374011 (k=10).

%Y Row 8 of the table in A056011.

%Y Column 8 of the rectangular array in A056023.

%K nonn,easy

%O 1,1

%A _Wesley Ivan Hurt_, Jun 24 2024