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A108211 a(n) = 16*n^2 + 1. 2
17, 65, 145, 257, 401, 577, 785, 1025, 1297, 1601, 1937, 2305, 2705, 3137, 3601, 4097, 4625, 5185, 5777, 6401, 7057, 7745, 8465, 9217, 10001, 10817, 11665, 12545, 13457, 14401, 15377, 16385, 17425, 18497, 19601, 20737, 21905, 23105, 24337, 25601 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Area of a Maltese cross conventionally inscribed in a 5n X 5n-grid.
Areas of some other crosses, each made from unit squares, as shown in Weisstein's illustrations: Greek Cross = x-pentomino = 5. Latin Cross = 6. Saint Andrew's cross = crux decussata = 9. Saint Anthony's Cross = tau cross = crux commissa = 10. Gaullist Cross = cross of Lorraine or patriarchal cross = 13. Papal Cross = 22. - Jonathan Vos Post, Jun 18 2005
The identity (16*n^2+1)^2-(64*n^2+8)*(2*n)^2 = 1 can be written as a(n)^2-A158488(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012
Sequence found by reading the line from 17, in the direction 17, 65,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
Conjecture: a(n) = floor(1/((4n) - log(2) + 1/(n+1) + 1/(n+2) + ... + 1/(2n)). - Clark Kimberling, Sep 09 2014
LINKS
Eric Weisstein's World of Mathematics, Maltese Cross.
Eric Weisstein's World of Mathematics, Gaullist Cross.
Eric Weisstein's World of Mathematics, Greek Cross.
Eric Weisstein's World of Mathematics, Latin Cross.
FORMULA
a(n) = A002522(4*n) = A016802(n) + 1.
G.f.: x*(17+14*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 08 2012
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*coth(Pi/4)/8 - 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - Pi*csch(Pi/4)/8. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/4)*sinh(Pi/sqrt(8)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/4)*csch(Pi/4). (End)
MAPLE
A108211:=n->16*n^2+1: seq(A108211(n), n=1..50); # Wesley Ivan Hurt, Sep 24 2014
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {17, 65, 145}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
PROG
(PARI) a(n)= 16*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011
(Magma) I:=[17, 65, 145]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012
CROSSREFS
Sequence in context: A054402 A086533 A226685 * A255885 A130885 A036545
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jun 15 2005
STATUS
approved

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Last modified March 29 08:49 EDT 2024. Contains 371268 sequences. (Running on oeis4.)