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A054925
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a(n) = ceiling(n*(n-1)/4).
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11
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0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798
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OFFSET
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0,4
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COMMENTS
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Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
Consider the standard 4-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive t-axis, and then back round to the x-axis. This sequence gives the floor of the Euclidean distance to the origin after n steps. - Jon Perry, Apr 16 2013
Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Ceiling of one-half of each triangular number. - Harvey P. Dale, Oct 03 2016
For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
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LINKS
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FORMULA
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Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
G.f.: x^2 * (x^2 - x + 1) / ((1 - x)^3 * (1 + x^2)) = x^2 * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)). a(1-n) = a(n). A011848(n) = a(-n). - Michael Somos, Feb 11 2004
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EXAMPLE
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a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
G.f. = x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 18*x^9 + 23*x^10 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* Vincenzo Librandi, Jul 14 2015 *)
Join[{0}, Ceiling[#/2]&/@Accumulate[Range[0, 60]]] (* Harvey P. Dale, Oct 03 2016 *)
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PROG
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(PARI) {a(n) = ceil( n * (n-1)/4)}; /* Michael Somos, Feb 11 2004 */
(Sage) [ceil(binomial(n, 2)/2) for n in range(0, 58)] # Zerinvary Lajos, Dec 01 2009
(JavaScript)
p=new Array(0, 0, 0, 0);
for (a=0; a<100; a++) {
p[a%4]+=a;
document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
(Magma) I:=[0, 0, 1, 2, 3]; [n le 5 select I[n] else 3*Self(n-1)-4*Self(n-2)+4*Self(n-3)-3*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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