A047461 Numbers that are congruent to {1, 4} mod 8.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A004526, A004770, A010703, A017077, A017113, A047398, A047452.

Cf. A047470, A047524, A047535, A047615, A047617, A153125, A342819.

Programs

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

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 *)

Maxima

makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

Python

def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024

SageMath

[4*n-3 - ((n+1)%2) for n in range(1,71)] # G. C. Greubel, Mar 15 2024

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