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A077043
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"Three-quarter squares": a(n)=n^2-A002620(n).
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14
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0, 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, 108, 127, 147, 169, 192, 217, 243, 271, 300, 331, 363, 397, 432, 469, 507, 547, 588, 631, 675, 721, 768, 817, 867, 919, 972, 1027, 1083, 1141, 1200, 1261, 1323, 1387, 1452, 1519, 1587, 1657, 1728, 1801, 1875, 1951
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 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 (ghodges14(AT)comcast.net), 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 formulae 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
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to compositions
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
| a(n) = ceiling[n^2*3/4] = A077042(n, 3). a(-n) = a(n).
Also can be computed from 1 * (n choose 0) + 2 * (n choose 1) + 2 * (n choose 2) - sum[(-2)^(k-3) C(n, k)] - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Nov 10 2002
a(n)=A002620(n-1)+A002620(n)+A002620(n+1) - Jon Perry (perry(AT)globalnet.co.uk), May 29 2003
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, ... - Jon Perry (perry(AT)globalnet.co.uk), May 29 2003
a(n+1) - a(n) = A001651(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2007
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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 10 2008]
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<=k<=n} a(2k+1) = (n+1)^3 , (see example). - DELEHAM Philippe, Dec 17 2011
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EXAMPLE
| a(4)=12 since the compositions of [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.
x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...
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; ...- DELEHAM Philippe, Dec 17 2011
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PROG
| (PARI) {a(n) = n^2 - (n^2 \ 4)} /* Michael Somos Jun 29 2011 */
(MAGMA) [Ceiling(n^2*3/4): n in [0..60]]; // Vincenzo Librandi, Jun 29 2011
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CROSSREFS
| Column 3 of A195040. - Omar E. Pol, Sep 28 2011
Sequence in context: A194102 A006317 A194147 * A022330 A024219 A025713
Adjacent sequences: A077040 A077041 A077042 * A077044 A077045 A077046
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Oct 22 2002
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