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A034720 Number of different words that can be formed from an n X n grid of letters, reading horizontally, vertically or diagonally. 3
1, 16, 65, 168, 345, 616, 1001, 1520, 2193, 3040, 4081, 5336, 6825, 8568, 10585, 12896, 15521, 18480, 21793, 25480, 29561, 34056, 38985, 44368, 50225, 56576, 63441, 70840, 78793, 87320, 96441, 106176, 116545, 127568, 139265, 151656 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also variance of (i.e., expectation of the square of) area under a random walk with 2n steps whose first return to the origin is at point 2n. - Henry Bottomley, Apr 11 2002

Also number of numbers to be checked for primality in the n X n generalization of the Gordon Lee puzzle (see A109943). - Hugo Pfoertner, Jul 09 2005

LINKS

Table of n, a(n) for n=1..36.

FORMULA

n^2 + 4*(n+1)*Sum {(n-j-1); j=0..n-1} + 8*Sum {(n-k-j-1); j=0..n-2, k=1..n-1-j}.

a(n) = n*(2n-1)*(5n-2)/3. - Henry Bottomley, Apr 11 2002

EXAMPLE

For n=2, from the array

  ab

  cd

we get 16 words:

  a,b,c,d,ab,ba,cd,dc,ac,ca,bd,db,ad,da,cb,bc.

For n=3, from abc/def/ghi we get 9 of length 1, 40 of length 2, 16 of length 3, a total of 65.

Gordon Lee puzzle: In a 3 X 3 matrix ((1 2 3)(4 5 6)(7 8 9)) the following numbers have to be checked: 9 single-digit numbers 1...9;

40 2-digit numbers: row-wise 12, 21, 23, 32, 45, 54, 56, 65, 78, 87, 89, 98; column-wise 14, 41, 47, 74, 25, 52, 58, 85, 36, 63, 69, 96; diagonals 15, 51, 59, 95, 48, 84, 26, 62; antidiagonals 35, 53, 57, 75, 24, 42, 68, 86;

16 3-digit numbers: 123, 321, 456, 654, 789, 987, 147, 741, 258, 852, 369, 963, 159, 951, 357, 753.

CROSSREFS

Cf. A109943 [Number of primes in the solution of the Gordon Lee puzzle].

Sequence in context: A043402 A044154 A044535 * A232050 A119285 A253673

Adjacent sequences:  A034717 A034718 A034719 * A034721 A034722 A034723

KEYWORD

nonn

AUTHOR

Felice Russo

EXTENSIONS

More terms from Hugo Pfoertner, Jul 09 2005

STATUS

approved

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Last modified April 21 13:48 EDT 2021. Contains 343154 sequences. (Running on oeis4.)