A077043 - OEIS

OFFSET

0,3

COMMENTS

Triangular numbers plus quarter squares: (n+1)*(n+2)/2 + floor(n^2/4) (i.e., A000217(n+1) + A002620(n)).

Largest coefficient in the expansion of (1+x+x^2+...+x^(n-1))^3=((1-x^n)/(1-x))^3, i.e., the coefficient of x^floor[3(n-1)/2] and of x^ceiling[3(n-1)/2]; also number of compositions of [3(n+1)/2] into exactly 3 positive integers each no more than n.

A set of n independent statements a,b,c,d..., produces n^2 conditional statements of the form "If a, then b" (including self-implications such as "If a, then a"). If such statements are taken as equivalent to "It is not the case that the first statement is true and the second is false" (material implication), A077043(n) is the minimum number of the conditional statements that can be true. (The maximum number of false conditional statements is A002620(n), the maximum product of two integers whose sum is n.) - Matthew Vandermast, Mar 04 2003

This is also the maximum number of triple intersections between three sets of n lines, where the lines in each set are parallel to each other. E.g., for n=3:

\.\.\.../././

.\.\.\./././.

..\.\.x././..

---+-*-*-+---

----*-*-*----

---+-*-*-+---

.././.x.\.\..

./././.\.\.\.

/././...\.\.\

where '*' = triple intersection, '+' and 'x' = double intersection.

I am pretty sure that the hexagonal configuration of intersections shown above is the optimum and I get the formulas a(n) = (3n^2)/4 for n even and (3n^2+1)/4 for n odd. - Gabriel Nivasch (gnivasch(AT)yahoo.com), Jan 13 2004

For n > 1 the sequence represents the maximum number of points that can be placed in a plane such that the largest distances between any two points does not exceed the shortest of the distances between any two points by more than a factor n-1. - Johannes Koelman (Joc_kay(AT)hotmail.com), Apr 27 2006

This is also the number of distinct noncongruent isosceles triangles with side length up to n. - Patrick Hurst (patrick(AT)imsa.edu), May 14 2008

Also concentric triangular numbers. A033428 and A003215 interleaved. - Omar E. Pol, Sep 28 2011

Number of (w,x,y) with all terms in {0,...,n} and w=x>range{w,x,y}. - Clark Kimberling, Jun 02 2012

Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x<=y. - Clark Kimberling, Jul 02 2012

From Bob Selcoe, Aug 05 2013: (Start)

a(n) is the number of 3-member sets with non-repeating positive integer values (x,y,z) whose sums equal 3(n+1). Example: a(4)=12; thus there are 12 sets where x+y+z = 15: (1,2,12), (1,3,11), (1,4,10), (1,5,9), (1,6,8), (2,3,10), (2,4,9), (2,5,8), (2,6,7), (3,4,8), (3,5,7) and (4,5,6).

From above, the number of sets sharing minimum values (minvals) equals a(1)-a(0), a(2)-a(1), a(3)-a(2),... a(n)-a(n-1) which are the numbers not divisible by 3, in sequence (A001651), range n to 1. So in the above example, there is one set with minval 4, two sets with minval 3, four sets with minval 2 and five sets with minval 1. (End)

Number of partitions of 3n into exactly 3 parts. - Wesley Ivan Hurt, Jan 21 2014

Number of partitions of 3(n-1) into at most 3 parts. - Colin Barker, Mar 31 2015

Number of possible positions after n-1 steps on the lines of a hexagonal grid. - Reg Robson, Mar 08 2014

12*a(n) is a perfect square when n is even and 12*a(n) - 3 is a perfect square when n is odd. - Miquel Cerda, Jun 30 2016

Square of largest Euclidean distance from start point reachable by an n-step walk on a honeycomb lattice. - Hugo Pfoertner, Jun 21 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 4.4.4.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7 (2004), Article 04.1.6.

Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See page 402.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Index entries for sequences related to compositions.

FORMULA

a(n) = ceiling(n^2*3/4) = A077042(n, 3); a(-n) = a(n).

Also can be computed from 1 * C(n,0) + 2 * C(n,1) + 2 * C(n,2) - Sum((-2)^(k-3) C(n, k)). - Joshua Zucker, Nov 10 2002

a(n) = A002620(n-1) + A002620(n) + A002620(n+1). - Jon Perry, May 29 2003

From Jon Perry, May 29 2003: (Start)

a(2k) = a(2k-2) + 6k - 3,

a(2k+1) = a(2k-1) + 6k,

a(4n) = 12n^2,

a(4n+1) = a(4n) + 6n + 1,

a(4n+2) = a(4n+1) + 6n + 2,

a(4n+3) = a(4n+2) + 6n + 4,

a(4n+4) = a(4n+3) + 6n + 5.

Differences between alternate terms give 3, 6, 9, 12, ... (End)

a(n+1) - a(n) = A001651(n), partial sums of A001651. - Reinhard Zumkeller, Dec 28 2007

From R. J. Mathar, Nov 10 2008: (Start)

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

a(n) + a(n+1) = A005448(n+1).

The inverse binomial transform yields 0 followed by A141531. (End)

Euler transform of length 3 sequence [3, 1, -1]. - Michael Somos, Jun 29 2011

a(n) = 3*n^2/4 - ((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011

Sum_{k=0..n} a(2k+1) = partial sums of A003215 = (n+1)^3 (see example). - Philippe Deléham, Dec 17 2011

a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Dec 16 2012

a(0)=0, a(1)=1, a(n) = 3*(n-1) + a(n-2). - Reg Robson, Mar 08 2014

a(2k) = 3k^2 = A033428(k), a(2k+1) = 3k^2 + 3k + 1 = A003215(k). - Jon Perry, Oct 25 2014

a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/2). - Wesley Ivan Hurt, Mar 12 2015

a(n) = (3*n)^2/12 for n even and a(n) = ((3*n)^2 + 3)/12 for n odd. - Miquel Cerda, Jun 30 2016

a(n) = Sum_{k=1..n} floor((n+k)/2). - Wesley Ivan Hurt, Mar 31 2017

0 = 1 +a(n)*(+a(n+1) -a(n+2)) +a(n+1)*(-3 -a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Apr 02 2017

E.g.f.: (1/8)*exp(-x)*(-1 + exp(2*x)*(1 + 6*x + 6*x^2)). - Stefano Spezia, Nov 29 2019

Sum_{n>=1} 1/a(n) = Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Jan 16 2023

EXAMPLE

G.f. = x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...

a(4)=12 since the compositions of floor(3*(4+1)/2) = 7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+3, 2+4+1, 3+1+3, 3+2+2, 3+3+1, 4+1+2, 4+2+1.

From Philippe Deléham, Dec 17 2011: (Start)

a(1) = 1 = 1^3;

a(1) + a(3) = 1 + 7 = 2^3;

a(1) + a(3) + a(5) = 1 + 7 + 19 = 3^3;

a(1) + a(3) + a(5) + a(7) = 1 + 7 + 19 + 37 = 4^3;

a(1) + a(3) + a(5) + a(7) + a(9) = 1 + 7 + 19 + 37 + 61 = 5^3; ... (End)

MAPLE

A077043:=n->ceil(3*n^2/4); seq(A077043(n), n=0..60); # Wesley Ivan Hurt, Jan 21 2014

MATHEMATICA

Table[Ceiling[(3n^2)/4], {n, 0, 60}] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 3, 7}, 60] (* Harvey P. Dale, Dec 16 2012 *)

PROG

(PARI) {a(n) = n^2 - (n^2 \ 4)}; /* Michael Somos, Jun 29 2011 */