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A047461
Numbers that are congruent to {1, 4} mod 8.
28
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
OFFSET
1,2
COMMENTS
Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008
FORMULA
From R. J. Mathar, Oct 29 2008: (Start)
G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(n) + a(n+1) = A004770(n).
a(n+1) - a(n) = A010703(n). (End)
a(n) = 8*floor((n-1)/2) + 4 - 3*(n mod 2). - Reinhard Zumkeller, Dec 15 2008
a(n) = A153125(n,n). - Reinhard Zumkeller, Dec 20 2008
a(n) = 8*n - a(n-1) - 11 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (7 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
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
a(n) = 1 + A004526(n)*3 + A004526(n-1)*5. - Gregory R. Bryant, Apr 16 2014
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 1.
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(Maxima) makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */
(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
(Python)
def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024
KEYWORD
nonn,easy
STATUS
approved