A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A080838 Orchard crossing number of complete bipartite graph K_{1,n}.
0, 0, 0, 2, 5, 12, 21, 36, 54, 80, 110, 150, 195, 252, 315, 392, 476, 576, 684, 810, 945, 1100, 1265, 1452, 1650, 1872, 2106, 2366, 2639, 2940, 3255, 3600, 3960, 4352, 4760, 5202, 5661, 6156, 6669, 7220, 7790, 8400, 9030, 9702, 10395, 11132, 11891
Offset: 1
Comments
Also the minimum number of transitive triples in a tournament on n nodes, i.e., a(n) = C(n,3) - A006918(n-2). - Leen Droogendijk, Nov 10 2014
a(n) = the number of binary strings of length n+1 with exactly one pair of adjacent 0's and exactly two pairs of adjacent 1's. - Jeremy Dover, Jul 07 2016
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- D. Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003.
- M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
- Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).
Crossrefs
Third column of A274228. - Jeremy Dover, Jul 07 2016
Essentially partial sums of A211539.
Programs
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Magma
[n/16*(2*n^2 - 8*n + 7 + (-1)^n): n in [1..50]]; // Vincenzo Librandi, May 17 2013
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Mathematica
CoefficientList[Series[(x^4 + 2 x^3) / (1 - x)^4 / (1 + x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, May 17 2013 *) Table[n/16*(2 n^2 - 8 n + 7 + (-1)^n), {n, 47}] (* Michael De Vlieger, Aug 01 2016 *) -
PARI
for(n=1,100,print1(if(n%2,n*(n-1)*(n-3)/8,n*(n-2)^2/8)","))
Formula
a(n) = (n/16) * (2*n^2 - 8*n + 7 + (-1)^n).
G.f.: (x^5 + 2*x^4) / (1-x)^4 / (1+x)^2.
For n odd, a(n) = A060423(n). - Gerald McGarvey, Sep 14 2008
A294285 Sum of the larger parts of the partitions of n into two distinct parts with larger part squarefree.
0, 0, 2, 3, 3, 5, 11, 18, 18, 13, 23, 28, 28, 34, 48, 63, 63, 80, 80, 89, 89, 99, 121, 144, 144, 131, 157, 143, 143, 157, 187, 218, 218, 234, 268, 303, 303, 321, 359, 398, 398, 418, 460, 481, 481, 458, 504, 551, 551, 551, 551, 576, 576, 629, 629, 684, 684
Offset: 1
Comments
Sum of the lengths of the distinct rectangles with squarefree length and positive integer width such that L + W = n, W < L. For example, a(14) = 34; the rectangles are 1 X 13, 3 X 11, 4 X 10. The sum of the lengths is then 13 + 11 + 10 = 34. - Wesley Ivan Hurt, Nov 01 2017
Examples
10 can be partitioned into two distinct parts as follows: (1, 9), (2, 8), (3, 7), (4, 6). The squarefree larger parts are 6 and 7, which sum to a(10) = 13. - _David A. Corneth_, Oct 27 2017
Links
Programs
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Mathematica
Table[Sum[(n - i)*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}] Table[Total[Select[IntegerPartitions[n,{2}],DuplicateFreeQ[#]&&SquareFreeQ[#[[1]]]&][[;;,1]]],{n,60}] (* Harvey P. Dale, Aug 30 2025 *) -
PARI
first(n) = {my(res = vector(n, i, binomial(i, 2) - binomial(i\2+1, 2)), nsqrfr = List()); forprime(i=2, sqrtint(n), for(k = 1, n \ i^2, listput(nsqrfr, k*i^2))); listsort(nsqrfr, 1); for(i=1, #nsqrfr, for(m = nsqrfr[i]+1, min(2*nsqrfr[i]-1, n), res[m]-=nsqrfr[i])); res} \\ David A. Corneth, Oct 27 2017 -
PARI
a(n) = sum(i=1, (n-1)\2, (n-i)*moebius(n-i)^2); \\ Michel Marcus, Nov 08 2017
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} (n - i) * mu(n - i)^2, where mu is the Möbius function (A008683).
A338544 a(n) = (5*floor((n-1)/2)^2 + (4+(-1)^n)*floor((n-1)/2)) / 2.
0, 0, 0, 4, 5, 13, 15, 27, 30, 46, 50, 70, 75, 99, 105, 133, 140, 172, 180, 216, 225, 265, 275, 319, 330, 378, 390, 442, 455, 511, 525, 585, 600, 664, 680, 748, 765, 837, 855, 931, 950, 1030, 1050, 1134, 1155, 1243, 1265, 1357, 1380, 1476, 1500, 1600, 1625, 1729, 1755, 1863
Offset: 0
Comments
Sum of the largest side lengths of all integer-sided triangles with perimeter 3n whose side lengths are in arithmetic progression (for example, when n=5 there are two triangles with perimeter 3*5 = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(5) = 7+6 = 13).
Links
- Wikipedia, Integer triangle
- Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Programs
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Mathematica
Table[(5 Floor[(n - 1)/2]^2 + Floor[(n - 1)/2] (4 + (-1)^n))/2, {n, 0, 100}]
Formula
From Stefano Spezia, Nov 01 2020: (Start)
G.f.: x^3*(4 + x)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)
16*a(n) = -14*n-1+10*n^2+(-1)^n-6*(-1)^n*n . - R. J. Mathar, Aug 19 2022
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)