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A014409 Number of inequivalent ways (mod D_4) a pair of checkers can be placed on an n X n board. 27
0, 2, 8, 21, 49, 93, 171, 278, 446, 660, 970, 1347, 1863, 2471, 3269, 4188, 5356, 6678, 8316, 10145, 12365, 14817, 17743, 20946, 24714, 28808, 33566, 38703, 44611, 50955, 58185, 65912, 74648, 83946, 94384, 105453, 117801, 130853, 145331, 160590, 177430, 195132 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

Computed by Fred Hallden.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

FORMULA

a(2*n) = n/2*(2*n^3 + 3*n - 1); a(2*n+1) = n/2*(2*n^3 + 4*n^2 + 7*n + 3).

a(0)=0, a(1)=2, a(2)=8, a(3)=21, a(4)=49, a(5)=93, a(6)=171, a(7)=278, a(n)=2*a(n-1)+2*a(n-2)-6*a(n-3)+0*a(n-4)+6*a(n-5)-2*a(n-6)- 2*a(n-7)+ a(n-8). - Harvey P. Dale, May 06 2012

G.f.: -x^2*(x^5+x^4+3*x^3+x^2+4*x+2) / ((x-1)^5*(x+1)^3). - Colin Barker, Jul 11 2013

From James Stein, May 22 2014: (Start)

For odd n: a(n) = (n^4 + 8*n^2 - 8*n - 1)/16;

for even n: a(n) = n*(n^3 + 6*n - 4)/16. (End)

a(n) = A054252(n, 2), n >= 0. - Wolfdieter Lang, Oct 03 2016

MATHEMATICA

LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 2, 8, 21, 49, 93, 171, 278}, 40]

CoefficientList[Series[- x (x^5 + x^4 + 3 x^3 + x^2 + 4 x + 2)/((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)

PROG

(PARI) a(n)=if(n%2, n^4 + 8*n^2 - 8*n - 1, n^4 + 6*n^2 - 4*n)/16  \\ Charles R Greathouse IV, Feb 09 2017

CROSSREFS

Cf. A054252, A019318, A082966, A242279, A242358, A054247.

Sequence in context: A182704 A000160 A034519 * A303721 A109782 A237268

Adjacent sequences:  A014406 A014407 A014408 * A014410 A014411 A014412

KEYWORD

nonn,nice,easy

AUTHOR

Borghard, William (bogey(AT)hostare.att.com)

EXTENSIONS

More terms and formula from Hugo van der Sanden

More terms from Colin Barker, Jul 11 2013

STATUS

approved

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Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)