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A024206
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Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).
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46
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0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 63, 71, 80, 89, 99, 109, 120, 131, 143, 155, 168, 181, 195, 209, 224, 239, 255, 271, 288, 305, 323, 341, 360, 379, 399, 419, 440, 461, 483, 505, 528, 551, 575, 599, 624, 649, 675, 701, 728, 755, 783, 811, 840
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OFFSET
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1,3
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COMMENTS
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a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.
[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.
First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .
Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002
a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.
Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010
Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010
Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015
Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016
Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020
a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2] [0 2 2] [1 0 2] [0 0 2] [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021
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REFERENCES
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O. Giering, Vorlesungen über höhere Geometrie, Vieweg, Braunschweig, 1982. See p. 59.
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LINKS
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FORMULA
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G.f.: x^2*(1+x-x^2)/((1-x^2)*(1-x)^2) = x^2*(1+x-x^2) / ( (1+x)*(1-x)^3 ).
a(0)=0, a(n) = floor(a(n-1) + sqrt(a(n-1)) + 1) for n > 0. - Gerald McGarvey, Jul 30 2004
a(n) = a(n-1) + [side length of the least square > a(n-1) ], that is a(n) = a(n-1) + ceiling(sqrt(a(n-1) + 1)). - Ctibor O. Zizka, Oct 06 2009
For a(1)=0, a(2)=1, a(n) = 2*a(n-1) - a(n-2) + 1 if n is odd; a(n) = 2*a(n-1) - a(n-2) if n is even. - Vincenzo Librandi, Dec 23 2010
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4); a(1)=0, a(2)=1, a(3)=3, a(4)=5. - Harvey P. Dale, Jun 14 2013
a(n) = a(-n-2) = n-1 + floor( (n-1)^2/4 ). - Bruno Berselli, Feb 03 2015
a(n) = (1/4)*(n+3)^2 - (1/8)*(1 + (-1)^n) - 1. - Serkan Sonel, Aug 26 2020
Sum_{n>=2} 1/a(n) = 7/4 + tan(sqrt(5)*Pi/2)*Pi/sqrt(5). - Amiram Eldar, Dec 10 2022
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EXAMPLE
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There are five 2 X 3 binary matrices with no zero rows or columns up to row and column permutation:
[1 0 0] [1 0 0] [1 1 0] [1 1 0] [1 1 1]
[0 1 1] [1 1 1] [0 1 1] [1 1 1] [1 1 1].
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MAPLE
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MATHEMATICA
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f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ Floor[ f[2, n^2 + 2 n - 2] /2], {n, 57}] (* Robert G. Wilson v, Aug 11 2010 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 1, 3, 5}, 60] (* Harvey P. Dale, Jun 14 2013 *)
Rest[CoefficientList[Series[x^2 (1 + x - x^2)/((1 - x^2) (1 - x)^2), {x, 0, 70}], x]] (* Vincenzo Librandi, Oct 02 2015 *)
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PROG
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(PARI) x='x+O('x^99); concat(0, Vec(x^2*(1+x-x^2)/ ((1-x^2)*(1-x)^2))) \\ Altug Alkan, Apr 05 2016
(Haskell)
a024206 n = (n - 1) * (n + 3) `div` 4
a024206_list = scanl (+) 0 $ tail a008619_list
(Magma) [(2*n^2+4*n-7-(-1)^n)/8 : n in [1..100]]; // Wesley Ivan Hurt, Jul 22 2014
(GAP) a:=[0, 1, 3, 5];; for n in [5..65] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; a; # Muniru A Asiru, Oct 23 2018
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CROSSREFS
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A row or column of the array A196416 (possibly with 1 subtracted from it).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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