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%I #154 Jan 16 2023 08:18:42
%S 0,1,3,7,12,19,27,37,48,61,75,91,108,127,147,169,192,217,243,271,300,
%T 331,363,397,432,469,507,547,588,631,675,721,768,817,867,919,972,1027,
%U 1083,1141,1200,1261,1323,1387,1452,1519,1587,1657,1728,1801,1875,1951
%N "Three-quarter squares": a(n) = n^2 - A002620(n).
%C Triangular numbers plus quarter squares: (n+1)*(n+2)/2 + floor(n^2/4) (i.e., A000217(n+1) + A002620(n)).
%C 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.
%C 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
%C 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:
%C \.\.\.../././
%C .\.\.\./././.
%C ..\.\.x././..
%C ---+-*-*-+---
%C ----*-*-*----
%C ---+-*-*-+---
%C .././.x.\.\..
%C ./././.\.\.\.
%C /././...\.\.\
%C where '*' = triple intersection, '+' and 'x' = double intersection.
%C 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
%C 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
%C 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
%C Also concentric triangular numbers. A033428 and A003215 interleaved. - _Omar E. Pol_, Sep 28 2011
%C Number of (w,x,y) with all terms in {0,...,n} and w=x>range{w,x,y}. - _Clark Kimberling_, Jun 02 2012
%C Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x<=y. - _Clark Kimberling_, Jul 02 2012
%C From _Bob Selcoe_, Aug 05 2013: (Start)
%C 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).
%C 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)
%C Number of partitions of 3n into exactly 3 parts. - _Wesley Ivan Hurt_, Jan 21 2014
%C Number of partitions of 3(n-1) into at most 3 parts. - _Colin Barker_, Mar 31 2015
%C Number of possible positions after n-1 steps on the lines of a hexagonal grid. - _Reg Robson_, Mar 08 2014
%C 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
%C Square of largest Euclidean distance from start point reachable by an n-step walk on a honeycomb lattice. - _Hugo Pfoertner_, Jun 21 2018
%H Vincenzo Librandi, <a href="/A077043/b077043.txt">Table of n, a(n) for n = 0..10000</a>
%H Ralf Hinze, <a href="https://www.cs.ox.ac.uk/people/ralf.hinze/publications/CSC.pdf">Concrete stream calculus: An extended study</a>, J. Funct. Progr. 20 (5-6) (2010) 463-535, <a href="https://doi.org/10.1017/S0956796810000213">doi</a>, Section 4.4.4.
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7 (2004), Article 04.1.6.
%H Gabriel Nivasch and Eyal Lev, <a href="https://www.jstor.org/stable/30044202">Nonattacking Queens on a Triangle</a>, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See page 402.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>.
%F a(n) = ceiling(n^2*3/4) = A077042(n, 3); a(-n) = a(n).
%F 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
%F a(n) = A002620(n-1) + A002620(n) + A002620(n+1). - _Jon Perry_, May 29 2003
%F From _Jon Perry_, May 29 2003: (Start)
%F a(2k) = a(2k-2) + 6k - 3,
%F a(2k+1) = a(2k-1) + 6k,
%F a(4n) = 12n^2,
%F a(4n+1) = a(4n) + 6n + 1,
%F a(4n+2) = a(4n+1) + 6n + 2,
%F a(4n+3) = a(4n+2) + 6n + 4,
%F a(4n+4) = a(4n+3) + 6n + 5.
%F Differences between alternate terms give 3, 6, 9, 12, ... (End)
%F a(n+1) - a(n) = A001651(n), partial sums of A001651. - _Reinhard Zumkeller_, Dec 28 2007
%F From _R. J. Mathar_, Nov 10 2008: (Start)
%F G.f.: x*(1+x+x^2)/((1+x)*(1-x)^3).
%F a(n) + a(n+1) = A005448(n+1).
%F The inverse binomial transform yields 0 followed by A141531. (End)
%F Euler transform of length 3 sequence [3, 1, -1]. - _Michael Somos_, Jun 29 2011
%F a(n) = 3*n^2/4 - ((-1)^n-1)/8. - _Omar E. Pol_, Sep 28 2011
%F Sum_{k=0..n} a(2k+1) = partial sums of A003215 = (n+1)^3 (see example). - _Philippe Deléham_, Dec 17 2011
%F 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
%F a(0)=0, a(1)=1, a(n) = 3*(n-1) + a(n-2). - _Reg Robson_, Mar 08 2014
%F a(2k) = 3k^2 = A033428(k), a(2k+1) = 3k^2 + 3k + 1 = A003215(k). - _Jon Perry_, Oct 25 2014
%F a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/2). - _Wesley Ivan Hurt_, Mar 12 2015
%F 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
%F a(n) = Sum_{k=1..n} floor((n+k)/2). - _Wesley Ivan Hurt_, Mar 31 2017
%F 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
%F E.g.f.: (1/8)*exp(-x)*(-1 + exp(2*x)*(1 + 6*x + 6*x^2)). - _Stefano Spezia_, Nov 29 2019
%F Sum_{n>=1} 1/a(n) = Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - _Amiram Eldar_, Jan 16 2023
%e G.f. = x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...
%e 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.
%e From _Philippe Deléham_, Dec 17 2011: (Start)
%e a(1) = 1 = 1^3;
%e a(1) + a(3) = 1 + 7 = 2^3;
%e a(1) + a(3) + a(5) = 1 + 7 + 19 = 3^3;
%e a(1) + a(3) + a(5) + a(7) = 1 + 7 + 19 + 37 = 4^3;
%e a(1) + a(3) + a(5) + a(7) + a(9) = 1 + 7 + 19 + 37 + 61 = 5^3; ... (End)
%p A077043:=n->ceil(3*n^2/4); seq(A077043(n), n=0..60); # _Wesley Ivan Hurt_, Jan 21 2014
%t 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 *)
%o (PARI) {a(n) = n^2 - (n^2 \ 4)}; /* _Michael Somos_, Jun 29 2011 */
%o (Magma) [Ceiling(n^2*3/4): n in [0..60]]; // _Vincenzo Librandi_, Jun 29 2011
%o (Haskell)
%o a077043 n = a077043_list !! n
%o a077043_list = scanl (+) 0 a001651_list
%o -- _Reinhard Zumkeller_, Jan 06 2014
%Y Column 3 of A195040. - _Omar E. Pol_, Sep 28 2011
%Y Cf. A019298 (partial sums).
%Y Cf. A002620, A033428, A003215.
%Y Equals one more than A331952 and one less than A084684. - _Greg Dresden_, Feb 22 2020
%K nonn,nice,easy
%O 0,3
%A _Henry Bottomley_, Oct 22 2002
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