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%I #66 Jun 23 2022 22:24:04
%S 0,0,1,2,3,5,8,11,14,18,23,28,33,39,46,53,60,68,77,86,95,105,116,127,
%T 138,150,163,176,189,203,218,233,248,264,281,298,315,333,352,371,390,
%U 410,431,452,473,495,518,541,564,588,613,638,663,689,716,743,770,798
%N a(n) = ceiling(n*(n-1)/4).
%C Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
%C Form the clockwise spiral starting 0,1,2,....; then A054925(n+1) interleaves 2 horizontal (A033951, A033991) and 2 vertical (A007742, A054552) branches. A bisection is A014848. - _Paul Barry_, Oct 08 2007
%C 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
%C _Jon Perry_'s JavaScript code is explained by A238604. - _Michael Somos_, Mar 01 2014
%C Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - _Wesley Ivan Hurt_, Jun 09 2014
%C Ceiling of one-half of each triangular number. - _Harvey P. Dale_, Oct 03 2016
%C For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - _Eric W. Weisstein_, Jul 14 2017
%H Ivan Panchenko, <a href="/A054925/b054925.txt">Table of n, a(n) for n = 0..10000</a>
%H Craig Knecht, <a href="/A054925/a054925.jpg">Binary fill patterns in a triangle refilled with natural numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCoverNumber.html">Edge Cover Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - _Michael Somos_, Sep 02 2006
%F 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
%F a(n + 4) = a(n) + 2*n + 3. - _Michael Somos_, Mar 01 2014
%F a(n+1) = floor( sqrt( A238604(n))). - _Michael Somos_, Mar 01 2014
%F a(n) = A011848(n) + A133872(n+2). - _Wesley Ivan Hurt_, Jun 09 2014
%e a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
%e 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 + ...
%p seq(ceil(binomial(n,2)/2), n=0..57); # _Zerinvary Lajos_, Jan 12 2009
%t Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* _Wesley Ivan Hurt_, Nov 01 2013 *)
%t LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* _Vincenzo Librandi_, Jul 14 2015 *)
%t Join[{0},Ceiling[#/2]&/@Accumulate[Range[0,60]]] (* _Harvey P. Dale_, Oct 03 2016 *)
%o (PARI) {a(n) = ceil( n * (n-1)/4)}; /* _Michael Somos_, Feb 11 2004 */
%o (Sage) [ceil(binomial(n,2)/2) for n in range(0,58)] # _Zerinvary Lajos_, Dec 01 2009
%o (JavaScript)
%o p=new Array(0,0,0,0);
%o for (a=0;a<100;a++) {
%o p[a%4]+=a;
%o document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
%o } /* _Jon Perry_, Apr 16 2013 */
%o (Magma) [ Ceiling(n*(n-1)/4) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o (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
%Y Cf. A054924, A054925 + A011848 = C(n, 2).
%Y Cf. A213172, A238604.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, May 24 2000
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