A011848 a(n) = floor(binomial(n, 2)/2).

0, 0, 0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 663, 689, 715, 742, 770, 798

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn
• easy

Comments

Column sums of an array of the odd numbers repeatedly shifted 4 places to the right:

1 3 5 7 9 11 13 15 17...

1 3 5 7 9...

1...

.........................

-------------------------

1 3 5 7 10 14 18 22 27...

Floor of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014

Beginning with a(4)=3, the sequence might be called the "off-axis" Ulam-Spiral numbers because they are the numbers in ascending order on the horizontal and vertical spokes (heading outward) starting with the first turning points on the spiral (i.e., 3, 5, 7 and 10). That is, starting with: 3 (upward); 5 (leftward); 7 (downward) and 10 (rightward). These are A033991 (starting at a(1)), A007742 (starting at a(1)), A033954 (starting at a(1)) and A001107 (starting at a(2)), respectively. These quadri-sections are summarized in the formulas of Sep 26 2015. - Bob Selcoe, Oct 05 2015

Conjecture: For n = 2, a(n) is the greatest k such that A123663(k) < A000217(n - 2). - Peter Kagey, Nov 18 2016

a(n) is also the matching number of the n-triangular graph, (n-1)-triangular honeycomb queen graph, (n-1)-triangular honeycomb bishop graphs, and (for n > 7) (n-1)-triangular honeycomb obtuse knight graphs. - Eric W. Weisstein, Jun 02 2017 and Apr 03 2018

After 0, 0, 0, add 1, then add 2 three times, then add 3, then add 4 three times, then add 5, etc.; i.e., first differences are A004524 = (0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, ...). - M. F. Hasler, May 09 2018

Let s(0) = s(1) = 1, s(-1) = s(2) = x, and s(n+2)*s(n-2) = s(n+1)*s(n-1) + s(n)^2 for all n in Z. Then s(n) = p(n) / x^e(n) is a Laurent polynomial in x with p(n) a polynomial with nonnegative integer coefficients of degree a(n) for all n in Z. If x = 1, then s(n) = p(n) = A006720(n+1). - Michael Somos, Mar 22 2023

Examples

			G.f. = x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 14*x^8 + 18*x^9 + 22*x^10 + ...
p(0) = p(1) = 1, p(2) = 1 + x, p(3) = 1 + x + x^3, p(4) = 1 + 2*x + 2*x^2 + x^3 + x^5. - _Michael Somos_, Mar 22 2023
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
• Eric Weisstein's World of Mathematics, Matching Number.
• Eric Weisstein's World of Mathematics, Triangular Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Crossrefs

A column of triangle A011857.

First differences are in A004524.

Cf. A007318, A033991, A007742, A033954, A001107, A006720, A035608 (bisection), A156859 (bisection).

Programs

GAP

List([0..60],n->Int(Binomial(n,2)/2)); # Muniru A Asiru, Apr 05 2018

Haskell

a011848 n = if n < 2 then 0 else flip div 2 $ a007318 n 2
-- Reinhard Zumkeller, Mar 04 2015

Magma

[ Floor(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

seq(floor(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Mathematica

Table[Floor[n (n - 1)/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
CoefficientList[Series[x^3/((1 + x^2) (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 21 2013 *)
LinearRecurrence[{3, -4, 4, -4, 1}, {0, 0, 1, 3, 5}, {0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[Floor[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[1/4 (-1 + (-1 + n) n + Cos[n Pi/2] + Sin[n Pi/2]), {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Floor[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Apr 03 2018 *)

PARI

a(n) = binomial(n, 2)\2;

PARI

vector(100, n, n--; floor(n*(n-1)/4)) \\ Altug Alkan, Sep 30 2015

Python

def a(n): return n*(n-1)//4 # Christoph B. Kassir, Oct 07 2022

Sage

[floor(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

Formula

G.f.: x^3*(1-x^2)/((1-x)^3*(1-x^4)).

G.f.: x^3/((1+x^2)*(1-x)^3). - Jon Perry, Mar 31 2004

a(n) = +3*a(n-1) -4*a(n-2) +4*a(n-3) -3*a(n-4) +a(n-5). - R. J. Mathar, Apr 15 2010

a(n) = floor((n/(1+e^(1/n)))^2). - Richard R. Forberg, Jun 19 2013

a(n) = floor(n*(n-1)/4). - T. D. Noe, Jun 20 2013

a(n) = (1/4) * ( n^2 - n - 1 + (-1)^floor(n/2) ). - Ralf Stephan, Aug 11 2013

a(n) = A054925(n) - A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014

a(4*n) = A033991(n). a(4*n+1) = A007742(n). a(4*n+2) = A033954(n). a(4*n+3) = A001107(n+1). - Bob Selcoe, Sep 26 2015

E.g.f.: (sin(x) + cos(x) + (x^2 - 1)*exp(x))/4. - Ilya Gutkovskiy, Nov 18 2016

A054925(n) = a(-n). A035608(n) = a(2*n+1). Wesley Ivan Hurt, Jun 09 2014

A156859(n) = a(2*n+2). - Michael Somos, Nov 18 2016

Euler transform of length 4 sequence [ 3, -1, 0, 1]. - Michael Somos, Nov 18 2016

From Amiram Eldar, Mar 18 2022: (Start)

Sum_{n>=3} 1/a(n) = 40/9 - 2*Pi/3.

Sum_{n>=3} (-1)^(n+1)/a(n) = 32/9 - 4*log(2). (End)

0 = a(n+2)*(a(n)*(a(n) -6*a(n+1) +4*a(n+2)) +a(n+1)*(8*a(n+1) -10*a(n+2)) + 3*a(n+2)^2) +a(n+3)*(a(n)*(+a(n) -2*a(n+1)) +a(n+2)*(2*a(n+1) -a(n+2))) for all n in Z. - Michael Somos, Mar 22 2023

2*a(n) + 2*a(n-2) = (n-1)*(n-2). - R. J. Mathar, Feb 12 2024