A374004 - OEIS

A374004

a(n) = (1 + (n+3)^2 - (n-4)*(-1)^n)/2.

7, 14, 18, 25, 33, 40, 52, 59, 75, 82, 102, 109, 133, 140, 168, 175, 207, 214, 250, 257, 297, 304, 348, 355, 403, 410, 462, 469, 525, 532, 592, 599, 663, 670, 738, 745, 817, 824, 900, 907, 987, 994, 1078, 1085, 1173, 1180, 1272, 1279, 1375, 1382, 1482, 1489, 1593

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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 4 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=4.

LINKS

Table of n, a(n) for n=1..53.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

FORMULA

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

G.f.: -x*(7*x^4-7*x^3-10*x^2+7x+7)/((x+1)^2*(x-1)^3).

a(n) = A373663(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[-(7*x^4 - 7*x^3 - 10*x^2 + 7 x + 7)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]

k := 4; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

PROG

(Magma) [(1 + (n+3)^2 - (n-4)*(-1)^n)/2: n in [1..80]];

CROSSREFS

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), this sequence (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

Row 4 of the table in A056011.

Column 4 of the rectangular array in A056023.

Sequence in context: A326767 A131439 A213604 * A102041 A232488 A351063

Adjacent sequences: A374001 A374002 A374003 * A374005 A374006 A374007

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Jun 24 2024

STATUS

approved