A054925 a(n) = ceiling(n*(n-1)/4).

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

Offset: 0

Author

N. J. A. Sloane, May 24 2000

Keywords

• nonn
• easy

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

Jon Perry's JavaScript code is explained by A238604. - Michael Somos, Mar 01 2014

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)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

Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 0} q^(3*n) * (Product_{k >= 2*n+1} 1 - q^k) = 1 - q - q^2 + q^3 + q^5 - q^8 - q^11 + + - - .... Cf. A039825. - Peter Bala, Apr 13 2025

Examples

			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 + ...
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..10000
• Craig Knecht, Binary fill patterns in a triangle refilled with natural numbers.
• 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).

Crossrefs

Cf. A007742, A033951, A033991, A039825, A054552, A054924, A054925 + A011848 = C(n, 2).

Cf. A213172, A238604.

Programs

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

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 *)

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

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)

a(n) = A011848(n) + A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014

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