OFFSET
0,4
COMMENTS
Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
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
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
Conjecture: For n>11, there always exists a prime number p such that a(n)<p<=a(n+1). - Raul Prisacariu, Sep 01 2024
For n = 1 up to at least n = 13, also the lower matching number of the triangular honeycomb bishop graph. - Eric W. Weisstein, Dec 13 2024
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
FORMULA
Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
From Michael Somos, Feb 11 2004: (Start)
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). (End)
From Michael Somos, Mar 01 2014: (Start)
a(n + 4) = a(n) + 2*n + 3.
a(n+1) = floor( sqrt( A238604(n))). (End)
Sum_{n>=2} 1/a(n) = 4 - Pi + 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2)+cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 23 2024
EXAMPLE
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 + ...
MAPLE
seq(ceil(binomial(n, 2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009
MATHEMATICA
Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* Wesley Ivan Hurt, Nov 01 2013 *)
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 *)
Ceiling[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Dec 13 2024 *)
Table[Ceiling[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
Table[(1 + (n - 1) n - Cos[n Pi/2] - Sin[n Pi/2])/4, {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
CoefficientList[Series[x^2 (-1 + x - x^2)/((-1 + x)^3 (1 + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 13 2024 *)
PROG
(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]))+", ");
} /* Jon Perry, Apr 16 2013 */
(Magma) [ Ceiling(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 24 2000
STATUS
approved