A087278 -id:A087278 - cp's OEIS frontend

A087279 Nonnegative numbers whose distance to the nearest positive square equals exactly 1.

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

Reinhard Zumkeller, Aug 28 2003

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A004526, A000035, A087278.

Cf. A005563, A002522.

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

&cat[[n^2-1,n^2+1]: n in [1..30]]; // Bruno Berselli, Apr 21 2011

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 *)

a(n)=if(n%2,(n+1)^2/4-1,n^2/4+1) \\ Charles R Greathouse IV, Apr 25 2012

def A087279(n): return ((n+(b:=n&1))**2>>2)+1-(b<<1) # Chai Wah Wu, Aug 03 2022

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)

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

A050271 Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 14, 15, 16, 23, 24, 25, 34, 35, 36, 47, 48, 49, 62, 63, 64, 79, 80, 81, 98, 99, 100, 119, 120, 121, 142, 143, 144, 167, 168, 169, 194, 195, 196, 223, 224, 225, 254, 255, 256, 287, 288, 289, 322, 323, 324, 359, 360, 361, 398, 399, 400

Offset: 1

Benoit Cloitre, May 10 2003

Is a(n) asymptotic to C*n^(3/2) where 1/2 < C < 1?

Consists exactly of numbers of the forms j^2 - 2, j^2 - 1, and j^2. As such, is asymptotic to (1/9)*n^2. - Ivan Neretin, Feb 08 2017

Ivan Neretin, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000290, A005563, A008865, A087278.

a:=n->floor((n+4)/3)^2+irem(n+1,3)-2:
seq(a(n),n=1..58); # Lorenzo Sauras Altuzarra, Jan 31 2023

Select[Range@400, Floor[(r = Sqrt@#)*Ceiling@r] == # &] (* Ivan Neretin, Feb 08 2017 *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,3,4,7,8,9},60] (* Harvey P. Dale, Aug 10 2025 *)

isok(n) = floor(sqrt(n)*ceil(sqrt(n))) == n; \\ Michel Marcus, Nov 22 2013

Vec(x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^100)) \\ Colin Barker, Feb 09 2017

def A050271(n):
    a, b = divmod(n+4,3)
    return a**2+b-2 # Chai Wah Wu, Aug 02 2022

a(n) = floor((n + 4)/3)^2 + ((n + 1) mod 3) - 2. - Ivan Neretin, Feb 08 2017
From Colin Barker, Feb 09 2017: (Start)
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.
G.f.: x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2).
(End)
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 2 + Pi^2/6 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 + cosec(sqrt(2)*Pi)*Pi/(2*sqrt(2)). (End)
a(n) = A087278(n+1) - 1 if n > 0. - Lorenzo Sauras Altuzarra, Jan 31 2023

Data corrected by Michel Marcus and Benoit Cloitre, Nov 22 2013

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A087279 Nonnegative numbers whose distance to the nearest positive square equals exactly 1.

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A050271 Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).

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