This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211540 #78 Sep 08 2022 08:46:02
%S A211540 0,0,0,0,0,1,1,2,3,4,5,7,8,10,12,14,16,19,21,24,27,30,33,37,40,44,48,
%T A211540 52,56,61,65,70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,
%U A211540 161,169,176,184,192,200,208,217,225,234,243,252,261,271,280,290
%N A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.
%C A211540 For a guide to related sequences, see A211422.
%C A211540 Also the number of partitions of n+1 into three parts, where each part > 1. - _Peter Woodward_, May 25 2015
%C A211540 a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - _Giovanni Resta_, May 26 2015
%C A211540 Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - _Ece Uslu_ and Esin Becenen, Dec 31 2015
%C A211540 After the first three terms, partial sums of A008615. - _Robert Israel_, Dec 31 2015
%C A211540 For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - _Gus Wiseman_, Oct 11 2020
%H A211540 Robert Israel, <a href="/A211540/b211540.txt">Table of n, a(n) for n = 0..10000</a>
%H A211540 Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
%H A211540 Ece Uslu and Esin Becenen, <a href="https://web.archive.org/web/20160412082710/http://matematikprojesi.com/dosyalar/4360aEce%20Uslu%20Esin%20Becenen%20Tubitak%20Project.pdf">Identical Object Distributions</a>.
%H A211540 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A211540 a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
%F A211540 a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - _Alois P. Heinz_, Nov 03 2012
%F A211540 a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - _Ece Uslu_, Esin Becenen, Dec 31 2015
%F A211540 a(n) = A004526(n-2) + a(n-2) for n > 2. - _Ece Uslu_, Esin Becenen, Dec 31 2015
%F A211540 G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - _Robert Israel_, Dec 31 2015
%F A211540 a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - _Wesley Ivan Hurt_, Apr 27 2019
%F A211540 From _Gus Wiseman_, Oct 11 2020: (Start)
%F A211540 a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
%F A211540 a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
%F A211540 a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
%F A211540 a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
%F A211540 A000217(n-2) counts 3-part compositions.
%F A211540 a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
%F A211540 A000217(n-5) counts 3-part compositions with no 1's.
%F A211540 a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
%F A211540 (End)
%e A211540 a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
%e A211540 a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
%e A211540 a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
%e A211540 G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
%e A211540 From _Gus Wiseman_, Oct 11 2020: (Start)
%e A211540 The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
%e A211540 222 322 332 333 433 443 444 544 554 555 655
%e A211540 422 432 442 533 543 553 644 654 664
%e A211540 522 532 542 552 643 653 663 754
%e A211540 622 632 633 652 662 744 763
%e A211540 722 642 733 743 753 772
%e A211540 732 742 752 762 844
%e A211540 822 832 833 843 853
%e A211540 922 842 852 862
%e A211540 932 933 943
%e A211540 A22 942 952
%e A211540 A32 A33
%e A211540 B22 A42
%e A211540 B32
%e A211540 C22
%e A211540 The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
%e A211540 432 532 542 543 643 653 654 754 764 765 865
%e A211540 632 642 652 743 753 763 854 864 874
%e A211540 732 742 752 762 853 863 873 964
%e A211540 832 842 843 862 872 954 973
%e A211540 932 852 943 953 963 982
%e A211540 942 952 962 972 A54
%e A211540 A32 A42 A43 A53 A63
%e A211540 B32 A52 A62 A72
%e A211540 B42 B43 B53
%e A211540 C32 B52 B62
%e A211540 C42 C43
%e A211540 D32 C52
%e A211540 D42
%e A211540 E32
%e A211540 The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
%e A211540 321 421 431 432 532 542 543 643 653 654 754
%e A211540 521 531 541 632 642 652 743 753 763
%e A211540 621 631 641 651 742 752 762 853
%e A211540 721 731 732 751 761 843 862
%e A211540 821 741 832 842 852 871
%e A211540 831 841 851 861 943
%e A211540 921 931 932 942 952
%e A211540 A21 941 951 961
%e A211540 A31 A32 A42
%e A211540 B21 A41 A51
%e A211540 B31 B32
%e A211540 C21 B41
%e A211540 C31
%e A211540 D21
%e A211540 (End)
%p A211540 f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
%p A211540 seq(f(i),i=0..100); # _Robert Israel_, Dec 31 2015
%t A211540 t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
%t A211540 c[n_] := Count[t[n], 0]
%t A211540 t = Table[c[n], {n, 0, 80}] (* A211540 *)
%t A211540 FindLinearRecurrence[t]
%t A211540 LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* _Vincenzo Librandi_, Dec 31 2015 *)
%t A211540 Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* _Gus Wiseman_, Oct 05 2020 *)
%o A211540 (PARI) {a(n) = round( (n-2)^2 / 12 )}; / * _Michael Somos_, Feb 02 2015 */
%o A211540 (Magma) I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // _Vincenzo Librandi_, Dec 31 2015
%o A211540 (PARI) concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ _Altug Alkan_, Jan 10 2016
%Y A211540 Cf. A001399, A069905, A211422.
%Y A211540 All of the following pertain to 3-part strict partitions.
%Y A211540 - A000009 counts these partitions of any length, with non-strict version A000041.
%Y A211540 - A007304 gives the Heinz numbers, with non-strict version A014612.
%Y A211540 - A101271 counts the relatively prime case, with non-strict version A023023.
%Y A211540 - A220377 counts the pairwise coprime case, with non-strict version A307719.
%Y A211540 - A337605 counts the pairwise non-coprime case, with non-strict version A337599.
%Y A211540 Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.
%K A211540 nonn,easy
%O A211540 0,8
%A A211540 _Clark Kimberling_, Apr 15 2012