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A047461
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Numbers that are congruent to {1, 4} mod 8.
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21
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1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 44, 49, 52, 57, 60, 65, 68, 73, 76, 81, 84, 89, 92, 97, 100, 105, 108, 113, 116, 121, 124, 129, 132, 137, 140, 145, 148, 153, 156, 161, 164, 169, 172, 177, 180, 185, 188, 193, 196, 201, 204, 209, 212, 217, 220, 225, 228, 233
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OFFSET
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1,2
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COMMENTS
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Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008
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LINKS
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FORMULA
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G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(1)=1, a(2)=4, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jun 18 2013
E.g.f.: (8 - exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/4 + sqrt(2)*arccoth(sqrt(2))/8. - Amiram Eldar, Dec 11 2021
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MAPLE
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seq(coeff(series(factorial(n)*((8-exp(-x)+(8*x-7)*exp(x))/2), x, n+1), x, n), n=1..60); # Muniru A Asiru, Jul 23 2018
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MATHEMATICA
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Flatten[(#+{1, 4})&/@(8Range[0, 30])] (* or *) LinearRecurrence[ {1, 1, -1}, {1, 4, 9}, 60] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[ Series[(4x^2 + 3x + 1)/((x + 1) (x - 1)^2), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 24 2018 *)
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PROG
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(GAP) Filtered([1..250], n->n mod 8=1 or n mod 8 =4); # Muniru A Asiru, Jul 23 2018
(Magma) [4*n-3 - ((n+1) mod 2): n in [1..70]]; // G. C. Greubel, Mar 15 2024
(SageMath) [4*n-3 - ((n+1)%2) for n in range(1, 71)] # G. C. Greubel, Mar 15 2024
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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