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A087279
Nonnegative numbers whose distance to the nearest positive square equals exactly 1.
3
0, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, 120, 122, 143, 145, 168, 170, 195, 197, 224, 226, 255, 257, 288, 290, 323, 325, 360, 362, 399, 401, 440, 442, 483, 485, 528, 530, 575, 577, 624, 626, 675, 677, 728, 730, 783, 785, 840
OFFSET
1,2
COMMENTS
Union of A005563 and A002522\{1}: a(2*k+1) = (k+1)^2 - 1 = A005563(k); a(2*k) = k^2 + 1 = A002522(k); positive square + 1 or positive square - 1.
FORMULA
a(1) = 0; a(2*k+1) = a(2*k) + 2*k-1; a(2*k) = a(2*k-1) + 2.
a(n-1) = floor((n+1)/2)^2+(-1)^(n mod 2).
From Bruno Berselli, Apr 21 2011: (Start)
G.f.: x^2*(2+x-2*x^2+x^3)/((1+x)^2*(1-x)^3).
a(n) = (2*n*(n+1) - (2*n-7)*(-1)^n+1)/8. (End)
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=2} 1/a(n) = coth(Pi)*Pi/2 + 1/4.
Sum_{n>=2} (-1)^n/a(n) = coth(Pi)*Pi/2 - 5/4. (End)
MATHEMATICA
Union[(r = Range[30]^2) - 1, r + 1] (* Jean-François Alcover, Oct 25 2013 *)
Flatten[#+{1, -1}&/@(Range[30]^2)]//Union (* Harvey P. Dale, Oct 15 2016 *)
PROG
(Magma) &cat[[n^2-1, n^2+1]: n in [1..30]]; // Bruno Berselli, Apr 21 2011
(PARI) a(n)=if(n%2, (n+1)^2/4-1, n^2/4+1) \\ Charles R Greathouse IV, Apr 25 2012
(Haskell)
a087279 n = a087279_list !! (n-1)
a087279_list = 0 : 2 : f (drop 2 a000290_list)
where f (x:xs) = x-1 : x+1 : f xs
-- Reinhard Zumkeller, Nov 01 2013
(Python)
def A087279(n): return ((n+(b:=n&1))**2>>2)+1-(b<<1) # Chai Wah Wu, Aug 03 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Aug 28 2003
EXTENSIONS
Franklin T. Adams-Watters pointed out on Jun 26 2007 that there were problems with the first couple of terms. I have made some changes, so now the definition matches the sequence. But some of the comments may need further minor adjustments. - N. J. A. Sloane, Jun 01 2008
STATUS
approved