Search: a087059 -id:a087059
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A087056
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Difference between 2 * n^2 and the next smaller square number.
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+10
11
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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a(10) = 4 because the difference between 2*10^2 = 200 and the next smaller square number (196) is 4.
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PROG
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(PARI) a(n) = 2*n^2 - sqrtint(2*n^2)^2; \\ Michel Marcus, Jul 08 2020
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A087057
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Smallest number whose square is larger than 2*n^2.
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+10
10
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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
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OFFSET
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1,1
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COMMENTS
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Integer solutions to the equation x=ceiling(r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 14 2004
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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PROG
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(Haskell)
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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A120861
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Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.
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+10
9
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1, 2, 7, 3, 12, 41, 4, 19, 70, 239, 5, 24, 111, 408, 1393, 6, 31, 140, 647, 2378, 8119, 8, 36, 181, 816, 3771, 13860, 47321, 9, 48, 210, 1055, 4756, 21979, 80782, 275807, 10, 53, 280, 1224, 6149, 27720, 128103, 470832, 1607521, 11, 60, 309, 1632, 7134
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OFFSET
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1,2
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COMMENTS
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For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2; in fact, j(n) = A087056(n) and k(n) = A087059(n).
Suppose g >= 1 and let k = k(g). The numbers in row g of array D are among those n for which (j + k + 1)^2 - 4*k = 8*n^2 for some j; that is, k stays fixed and j and n vary - hence the name "fixed-k dispersion". (The fixed-j dispersion for Q = 8 is A120860.)
Every positive integer occurs exactly once in array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.
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LINKS
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FORMULA
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Define f(n) = 3*n + 2*floor(n*sqrt(2)) + 2. Let D(g,h) be the term in row g and column h of the array to be defined:
D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 6*D(1,h-1) - D(1,h-2) for h >= 3.
For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1,2,...,g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 6*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected by Petros Hadjicostas, Jul 07 2020]
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EXAMPLE
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Northwest corner:
1, 7, 41, 239, 1393, 8119, 47321, ...
2, 12, 70, 408, 2378, 13860, 80782, ...
3, 19, 111, 647, 3771, 21979, 128103, ...
4, 24, 140, 816, 4756, 27720, 161564, ...
5, 31, 181, 1055, 6149, 35839, 208885, ...
6, 36, 210, 1224, 7134, 41580, 242346, ...
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PROG
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(PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ Michel Marcus, Jul 08 2020
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CROSSREFS
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Cf. A087056, A087059, A120858, A120859, A120860, A120862, A120863, A336109 (first column), A002315 (first row), A001542 (2nd row), A253811 (3rd row).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A087060
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Difference between 2n^2 and the nearest square number.
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+10
7
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1, 1, 2, 4, 1, 8, 2, 7, 7, 4, 14, 1, 14, 8, 9, 17, 2, 23, 7, 16, 18, 7, 31, 4, 25, 17, 14, 32, 1, 36, 14, 23, 31, 8, 49, 9, 34, 28, 17, 49, 2, 47, 23, 28, 46, 7, 62, 16, 41, 41, 18, 68, 7, 56, 34, 31, 63, 4, 73, 25, 46, 56, 17, 89, 14, 63, 47, 32, 82, 1, 82, 36, 49, 73, 14, 103, 23, 68
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OFFSET
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1,3
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COMMENTS
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max(a(n)/n) approaches sqrt(2), and the indices of the maxima are apparently in A227792. - Ralf Stephan, Sep 23 2013
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LINKS
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FORMULA
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a(n) = min [A087056(n), A087059(n)] = min [2*n^2 - (floor[n*sqrt(2)])^2, (1 + floor[n*sqrt(2)])^2 - 2*n^2]
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EXAMPLE
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a(10) = 4 because the difference between 2*10^2 = 200 and the nearest square number (196) is 4.
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MATHEMATICA
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dnsn[n_]:=Module[{c=2n^2, a, b}, a=Floor[Sqrt[c]]^2; b=Ceiling[Sqrt[c]]^2; Min[c-a, b-c]]; Array[dnsn, 80] (* Harvey P. Dale, Jul 01 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A120860
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Fixed-j dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.
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+10
7
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1, 2, 5, 3, 10, 29, 4, 17, 58, 169, 6, 22, 99, 338, 985, 7, 34, 128, 577, 1970, 5741, 8, 39, 198, 746, 3363, 11482, 33461, 9, 46, 227, 1154, 4348, 19601, 66922, 195025, 11, 51, 268, 1323, 6726, 25342, 114243, 390050, 1136689, 12, 63, 297, 1562, 7711
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OFFSET
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1,2
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COMMENTS
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For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2; in fact, j(n) = A087056(n) and k(n) = A087059(n). Suppose g >= 1 and let j = j(g).
The numbers in row g of array D are among those n for which (j + k + 1)^2 - 4*k = 8*n^2 for some k; that is, j stays fixed and k and n vary - hence the name "fixed-j dispersion". (The fixed-k dispersion for Q = 8 is A120861.)
Every positive integer occurs exactly once in D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row. Possibly, D is the dispersion of A098021.
It appears that the first column of the dispersion array D is A187968. That is, the first column of D consists of those positive integers m such that A187967(m) = 1; i.e., those m for which floor(sqrt(2)*m + 2*sqrt(2)) - floor(sqrt(2)*m) - floor(2*sqrt(2)) = 1.
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LINKS
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FORMULA
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Define f(n) = 3*n + 2*floor(n*2^(1/2)). Let D(g,h) be the term in row g and column h of the array to be defined:
D(1,1) = 1; D(1,2) = f(1); D(1,h) = 6*D(1,h-1) - D(1,h-2) for h >= 3.
For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1, 2, ..., g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 6*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected by Petros Hadjicostas, Jul 07 2020]
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EXAMPLE
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Northwest corner:
1, 5, 29, 169, 985, 5741, 33461, 195025, ...
2, 10, 58, 338, 1970, 11482, 66922, 390050, ...
3, 17, 99, 577, 3363, 19601, 114243, 665857, ...
4, 22, 128, 746, 4348, 25342, 147704, 860882, ...
6, 34, 198, 1154, 6726, 39202, 228486, 1331714, ...
7, 39, 227, 1323, 7711, 44943, 261947, 1526739, ...
...
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PROG
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(PARI) f(n) = 3*n + 2*sqrtint(2*n^2); \\ A098021
unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ Michel Marcus, Jul 08 2020
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CROSSREFS
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Cf. A087056, A087059, A098021, A120858, A120859, A120861, A120862, A120863, A120871, A187967, A187968.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A087055
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Largest square number less than 2*n^2.
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+10
6
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = A001951(n)^2 = floor(n*sqrt(2))^2.
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EXAMPLE
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a(10) = 196 because 196 is the largest square less than 2*10^2 = 200.
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MATHEMATICA
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Table[Floor[Sqrt[2n^2-1]], {n, 100}]^2 (* Harvey P. Dale, Mar 26 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A087058
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Smallest square number greater than 2*n^2.
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+10
6
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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A087058(10) = 225 because 225 is the smallest square number greater than 2*10^2 = 200.
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MATHEMATICA
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Table[Ceiling[Sqrt[2n^2]]^2, {n, 50}] (* Harvey P. Dale, Jan 22 2015 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A121490
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Rectangular array T by antidiagonals: T(n,k) = [y+1]^2-y^2, where y=n*sqrt(k) and [ ] denotes the floor function.
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+10
1
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3, 2, 5, 1, 1, 7, 5, 4, 7, 9, 4, 9, 9, 4, 11, 3, 5, 13, 1, 14, 13, 2, 1, 4, 17, 6, 9, 15, 1, 8, 10, 1, 21, 13, 2, 17, 7, 4, 1, 4, 19, 25, 22, 16, 19, 6, 13, 9, 9, 19, 16, 29, 4, 7, 21, 5, 9, 19, 16, 21, 9, 11, 33, 13, 25, 23, 4, 5, 10, 25, 25, 4, 30, 4, 37, 24, 14, 25, 3, 1, 1, 9, 31, 1, 18, 16
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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T(n,k) = [y+1]^2-y^2
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EXAMPLE
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Northwest corner:
3 2 1 5 4
5 1 4 9 5
7 7 9 13 4
9 4 1 17 1
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KEYWORD
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AUTHOR
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STATUS
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approved
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A120872
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a(n) is the value of k for row n of the fixed-k dispersion for Q = 8.
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+10
0
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2, 1, 7, 4, 14, 9, 16, 7, 25, 14, 23, 8, 34, 17, 47, 28, 41, 18, 56, 31, 46, 17, 63, 32, 82, 49, 68, 31, 89, 50, 71, 28, 94, 49, 72, 23, 97, 46, 124, 71, 98, 41, 127, 68, 97, 34, 128, 63, 161, 94, 127, 56, 162, 89, 124, 47, 161, 82, 119, 36, 158, 73, 199, 112
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OFFSET
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1,1
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COMMENTS
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This sequence results from A087059 by deleting duplicates.
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LINKS
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EXAMPLE
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For each positive integer n, there is a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2. This representation is used to define the fixed-k dispersion for Q=8, given by A120861, having northwest corner:
1, 7, 41, 239, ...
2, 12, 70, 408, ...
3, 19, 111, 647, ...
4, 24, 140, 816, ...
...
The pair (j,k) for each n, shown in the position occupied by n in the above array, is shown here:
(1,2), (17,2), (43,2), (673,2), ...
(4,1), (32,1), (196,1), (1152,1), ...
(2,7), (46,7), (306,7), (1822,7), ...
(7,4), (63,4), (391,4), (2303,4), ...
...
The fixed-k for row 1 is a(1) = 2;
the fixed-k for row 2 is a(2) = 1; etc.
(For example, (46 + 7 + 1)^2 - 4*7 = 8*19^2.)
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PROG
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(PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ A120860
q(n) = (1 + sqrtint(2*n^2))^2 - 2*n^2; \\ A087059
lista(nn) = my(m=D(nn)); vector(nn, n, q(m[n, 1])); \\ Michel Marcus, Jul 10 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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