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A118312 Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square. 1
1, 8, 33, 76, 129, 196, 277, 372, 481, 604, 741, 892, 1057, 1236, 1429, 1636, 1857, 2092, 2341, 2604, 2881, 3172, 3477, 3796, 4129, 4476, 4837, 5212, 5601, 6004, 6421, 6852, 7297, 7756, 8229, 8716, 9217, 9732, 10261, 10804, 11361, 11932, 12517, 13116, 13729, 14356, 14997, 15652 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Related to A018842: a(n) = A018842(n) + A018842(n-2) + A018842(n-4) + ... .

REFERENCES

M. Petkovic, Mathematics and Chess, Dover Publications (2003), Problem 3.11.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Mordechai Katzman, Knight's moves on an infinite board

Mordechai Katzman, Counting monomials, arXiv:math/0504113 [math.AC], 2005.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = -3 + 4*n + 7*n^2 + 4*Sign[(n - 2)(n - 1)].

G.f.: (1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)/(1 - x)^3

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 09 2012

For n>=3, a(n) = A005892(n).

EXAMPLE

a(2)=33 because knight in 2 moves from square (0,0) can reach one of the following squares: {{0,0}, {-4,-2}, {-4,0}, {-4,2}, {-3,-3}, {-3,-1}, {-3,1}, {-3,3}, {-2,-4}, {-2,0}, {-2,4}, {-1,-3}, {-1,-1}, {-1,1}, {-1,3}, {0,-4}, {0,-2}, {0,2}, {0,4}, {1,-3}, {1,-1}, {1,1}, {1,3}, {2,-4}, {2,0}, {2,4}, {3,-3}, {3,-1}, {3,1}, {3,3}, {4,-2}, {4,0}, {4,2}}.

MATHEMATICA

Table[ -3 + 4*n + 7*n^2 + 4*Sign[(n - 2)(n - 1)], {n, 0, 100}]

CoefficientList[Series[(1+5*x+12*x^2-8*x^4+4*x^5)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 09 2012 *)

Join[{1, 8, 33}, LinearRecurrence[{3, -3, 1}, {76, 129, 196}, 50]] (* Harvey P. Dale, Dec 05 2014 *)

PROG

(MAGMA) I:=[1, 8, 33, 76, 129, 196, 277]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]: // Vincenzo Librandi, Jul 09 2012

(PARI) a(n)=7*n^2 + 4*n - 3 + 4*sign((n-2)*(n-1)) \\ Charles R Greathouse IV, Feb 09 2017

CROSSREFS

Cf. A018842 (squares in EXACTLY n moves), A018836 (squares in <=n moves).

Sequence in context: A107291 A044466 A022274 * A212679 A204468 A140867

Adjacent sequences:  A118309 A118310 A118311 * A118313 A118314 A118315

KEYWORD

easy,nice,nonn

AUTHOR

Anton Chupin (chupin(AT)icmm.ru), May 14 2006

EXTENSIONS

Link updated by Tristan Miller, Jun 13 2013

STATUS

approved

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Last modified November 14 06:14 EST 2018. Contains 317162 sequences. (Running on oeis4.)