A087060 -id:A087060 - cp's OEIS frontend

A087056 Difference between 2 * n^2 and the next smaller square number.

1, 4, 2, 7, 1, 8, 17, 7, 18, 4, 17, 32, 14, 31, 9, 28, 2, 23, 46, 16, 41, 7, 34, 63, 25, 56, 14, 47, 1, 36, 73, 23, 62, 8, 49, 92, 34, 79, 17, 64, 113, 47, 98, 28, 81, 7, 62, 119, 41, 100, 18, 79, 142, 56, 121, 31, 98, 4, 73, 144, 46, 119, 17, 92, 169, 63, 142, 32, 113, 196, 82

Offset: 1

Jens Voß, Aug 07 2003

The difference x - y between the legs of primitive Pythagorean triangles x^2 + y^2 = z^2 with even y is D(n, m) = n^2 - m^2 - 2*n*m (see A249866 for the restrictions on n and m to have primitive triangles, which are not used here except for 1 < = m <= n-1). Here a(n) is for positive D values the smallest number in row n, namely D(n, floor(n/(1 + sqrt(2)))), for n >= 3. For the smallest value |D| for negative D in row n >= 2 see A087059. - Wolfdieter Lang, Jun 11 2015

			a(10) = 4 because the difference between 2*10^2 = 200 and the next smaller square number (196) is 4.

Cf. A001951, A087055, A087057, A087058, A087059, A087060.

a(n) = 2*n^2 - sqrtint(2*n^2)^2; \\ Michel Marcus, Jul 08 2020

a(n) = 2*n^2 - A087055(n) = 2*n^2 - A001951(n)^2 = 2*n^2 - (floor[n*sqrt(2)])^2

a(n) = (n - f(n))^2 - 2*f(n)^2 with f(n) = floor(n/(1 + sqrt(2))), for n >= 1 (the values for n = 1, 2 have here been included). See comment above. - Wolfdieter Lang, Jun 11 2015

A087057 Smallest number whose square is larger than 2*n^2.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 94, 95, 97, 98, 99, 101

Offset: 1

Jens Voß, Aug 07 2003

Integer solutions to the equation x=ceiling(r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 14 2004

			a(10) = 15 because the 15^2 = 225 is the smallest square number greater than 2*10^2 = 200.
Can be built by recursive removals:
start with 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
get a(1) := 2 and remove the 2nd term (= 4):
[2] _ 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
get a(2) := 3 and remove the 3rd term (= 7):
[2, 3] _ 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...
get a(3) := 5 and remove the 5th term (= 11):
[2, 3, 5] _ 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ...
get a(4) := 6 and remove the 6th term (= 14):
[2, 3, 5, 6] _ 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, ...
get a(5) := 8 and remove the 8th term (= 18):
[2, 3, 5, 6, 8] _ 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, ...
get a(6) = 9 and remove the 9th term (= 21), etc.
- _Reinhard Zumkeller_, Feb 04 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001951, A087055, A087056, A087058, A087059, A087060.

a087057 n = a087057_list !! (n-1)
a087057_list = f [2..] where
   f (x:xs) = x : f (us ++ vs) where (us, _ : vs) = splitAt (x - 1) xs
-- Reinhard Zumkeller, Feb 04 2014

Ceiling[Range[110]Sqrt[2]] (* Harvey P. Dale, Oct 30 2013 *)

a(n)=ceil(n*sqrt(2)) \\ Charles R Greathouse IV, Oct 24 2011

a(n)=sqrtint(2*n^2+sqrtint(8*n^2)+1) \\ Charles R Greathouse IV, Oct 24 2011

a(n) = 1 + A001951(n) = 1 + floor(n*sqrt(2)) = sqrt(A087058(n)).

a(n) = ceiling(n*sqrt(2)). - Vincenzo Librandi, Oct 22 2011

A087059 Difference between 2*n^2 and the next greater square number.

Original entry on oeis.org

2, 1, 7, 4, 14, 9, 2, 16, 7, 25, 14, 1, 23, 8, 34, 17, 47, 28, 7, 41, 18, 56, 31, 4, 46, 17, 63, 32, 82, 49, 14, 68, 31, 89, 50, 9, 71, 28, 94, 49, 2, 72, 23, 97, 46, 124, 71, 16, 98, 41, 127, 68, 7, 97, 34, 128, 63, 161, 94, 25, 127, 56, 162, 89, 14, 124, 47, 161, 82, 1, 119

Offset: 1

Jens Voß, Aug 07 2003

For n >= 2, a(n) is also the smallest absolute value of all negative values in row n of the triangle D(n, m) = n^2 - m^2 - 2*n*m, for 2 <= m + 1 <= n. The negative values in row n start with m = floor(n/(1 + sqrt(2))) + 1 = ceiling(n/(1 + sqrt(2))). See also a comment on A087056 for the smallest positive numbers in row n >= 3. - Wolfdieter Lang, Jun 11 2015

			a(10) = 25 because the difference between 2*10^2 = 200 and the next greater square number (225) is 25.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001951, A087055, A087056, A087057, A087058, A087060.

(Floor[Sqrt[#]]+1)^2-#&/@Table[2n^2,{n,80}] (* Harvey P. Dale, Jan 15 2023 *)

a(n) = (1 + sqrtint(2*n^2))^2 - 2*n^2 \\ Michel Marcus, Jun 25 2013

a(n) = A087058(n) - 2*n^2 = A087057(n)^2 - 2*n^2 = (1 + A001951(n))^2 - 2*n^2 = (1 + floor(n*sqrt(2)))^2 - 2*n^2.

a(n) = 2*c(n)^2 - (n - c(n))^2, with c(n) := ceiling(n/(1 + sqrt(2))), n >= 1. - Wolfdieter Lang, Jun 11 2015

A087055 Largest square number less than 2*n^2.

Original entry on oeis.org

1, 4, 16, 25, 49, 64, 81, 121, 144, 196, 225, 256, 324, 361, 441, 484, 576, 625, 676, 784, 841, 961, 1024, 1089, 1225, 1296, 1444, 1521, 1681, 1764, 1849, 2025, 2116, 2304, 2401, 2500, 2704, 2809, 3025, 3136, 3249, 3481, 3600, 3844, 3969, 4225, 4356, 4489

Offset: 1

Jens Voß, Aug 07 2003

			a(10) = 196 because 196 is the largest square less than 2*10^2 = 200.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000290, A001951, A087056, A087057, A087058, A087059, A087060.

Table[Floor[Sqrt[2n^2-1]],{n,100}]^2  (* Harvey P. Dale, Mar 26 2011 *)

a(n) = sqrtint(2*n^2)^2; \\ Michel Marcus, Jul 08 2020

a(n) = A001951(n)^2 = floor(n*sqrt(2))^2.

A087058 Smallest square number greater than 2*n^2.

Original entry on oeis.org

4, 9, 25, 36, 64, 81, 100, 144, 169, 225, 256, 289, 361, 400, 484, 529, 625, 676, 729, 841, 900, 1024, 1089, 1156, 1296, 1369, 1521, 1600, 1764, 1849, 1936, 2116, 2209, 2401, 2500, 2601, 2809, 2916, 3136, 3249, 3364, 3600, 3721, 3969, 4096, 4356, 4489

Offset: 1

Jens Voß, Aug 07 2003

			A087058(10) = 225 because 225 is the smallest square number greater than 2*10^2 = 200.

Cf. A001951, A087055, A087056, A087057, A087059, A087060.

Table[Ceiling[Sqrt[2n^2]]^2,{n,50}] (* Harvey P. Dale, Jan 22 2015 *)

A087058(n) = A087057(n)^2 = (1 + A001951(n))^2 = (1 + floor[n*sqrt(2)])^2

A227790 Difference between 3n^2 and the nearest square number.

Original entry on oeis.org

1, 3, 2, 1, 6, 8, 3, 4, 13, 11, 2, 9, 22, 12, 1, 16, 26, 11, 6, 25, 27, 8, 13, 36, 26, 3, 22, 48, 23, 4, 33, 47, 18, 13, 46, 44, 11, 24, 61, 39, 2, 37, 71, 32, 9, 52, 66, 23, 22, 69, 59, 12, 37, 88, 50, 1, 54, 92, 39, 16, 73, 83, 26, 33, 94, 72, 11, 52, 117, 59, 6, 73, 111, 44, 25, 96, 98, 27, 46, 121

Offset: 1

Ralf Stephan, Sep 23 2013

nonn

max(a(n)/n) approaches sqrt(3), and the indices of the maxima are apparently in A041017.

			a(9) = 13 because the difference between 3*9^2 = 243 and the nearest square number (256) is 13.

Cf. A001951, A087060.

a(n)=min(3*n^2-(floor(n*sqrt(3)))^2,(1+floor(n*sqrt(3)))^2-3*n^2)

a(n) = min (A033428(n)-A022838(n)^2, (1+A022838(n))^2-A033428(n)) = min [3*n^2 - (floor[n*sqrt(3)])^2, (1 + floor[n*sqrt(3)])^2 - 3*n^2].

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A087056 Difference between 2 * n^2 and the next smaller square number.

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A087057 Smallest number whose square is larger than 2*n^2.

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A087059 Difference between 2*n^2 and the next greater square number.

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A087055 Largest square number less than 2*n^2.

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A087058 Smallest square number greater than 2*n^2.

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A227790 Difference between 3n^2 and the nearest square number.

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