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 A217843 #46 May 24 2021 00:15:40
%S A217843 0,1,8,9,27,35,36,64,91,99,100,125,189,216,224,225,341,343,405,432,
%T A217843 440,441,512,559,684,729,748,775,783,784,855,1000,1071,1196,1241,1260,
%U A217843 1287,1295,1296,1331,1584,1728,1729,1800,1925,1989,2016,2024,2025,2197
%N A217843 Numbers which are the sum of one or more consecutive nonnegative cubes.
%C A217843 Contains A000578 (cubes), A005898 (two consecutive cubes), A027602 (three consecutive cubes), A027603 (four consecutive cubes) etc. - _R. J. Mathar_, Nov 04 2012
%C A217843 See A265845 for sums of consecutive positive cubes in more than one way. - _Reinhard Zumkeller_, Dec 17 2015
%C A217843 From _Lamine Ngom_, Apr 15 2021: (Start)
%C A217843 a(n) can always be expressed as the difference of the squares of two triangular numbers (A000217).
%C A217843 A168566 is the subsequence A000217(n)^2 - 1.
%C A217843 a(n) is also the product of two nonnegative integers whose sum and difference are both promic.
%C A217843 See example and formula sections for details. (End)
%H A217843 T. D. Noe, <a href="/A217843/b217843.txt">Table of n, a(n) for n = 1..1000</a>
%F A217843 a(n) >> n^2. Probably a(n) ~ kn^2 for some k but I cannot prove this. - _Charles R Greathouse IV_, Aug 07 2013
%F A217843 a(n) is of the form [x*(x+2*k+1)*(x*(x+2*k+1)+2*k*(k+1))]/4, sum of n consecutive cubes starting from (k+1)^3. - _Lamine Ngom_, Apr 15 2021
%e A217843 From _Lamine Ngom_, Apr 15 2021: (Start)
%e A217843 Arrange the positive terms in a triangle as follows:
%e A217843 n\k | 1 2 3 4 5 6 7
%e A217843 ----+-----------------------------------
%e A217843 0 | 1;
%e A217843 1 | 8, 9;
%e A217843 2 | 27, 35, 36;
%e A217843 3 | 64, 91, 99, 100;
%e A217843 4 | 125, 189, 216, 224, 225;
%e A217843 5 | 216, 341, 405, 432, 440, 441;
%e A217843 6 | 343, 559, 684, 748, 775, 783, 784;
%e A217843 Column 1: cubes = A000217(n+1)^2 - A000217(n)^2.
%e A217843 The difference of the squares of two consecutive triangular numbers (A000217) is a cube (A000578).
%e A217843 Column 2: sums of 2 consecutive cubes (A027602).
%e A217843 Column 3: sums of 3 consecutive cubes (A027603).
%e A217843 etc.
%e A217843 Column k: sums of k consecutive cubes.
%e A217843 Row n: A000217(n)^2 - A000217(m)^2, m < n.
%e A217843 T(n,n) = A000217(n)^2 (main diagonal).
%e A217843 T(n,n-1) = A000217(n)^2 - 1 (A168566) (2nd diagonal).
%e A217843 Now rectangularize this triangle as follows:
%e A217843 n\k | 1 2 3 4 5 6 ...
%e A217843 ----+--------------------------------------
%e A217843 0 | 1, 9, 36, 100, 225, 441, ...
%e A217843 1 | 8, 35, 99, 224, 440, 783, ...
%e A217843 2 | 27, 91, 216, 432, 775, 1287, ...
%e A217843 3 | 64, 189, 405, 748, 1260, 1989, ...
%e A217843 4 | 125, 341, 684, 1196, 1925, 2925, ...
%e A217843 5 | 216, 559, 1071, 1800, 2800, 4131, ...
%e A217843 6 | 343, 855, 1584, 2584, 3915, 5643, ...
%e A217843 The general form of terms is:
%e A217843 T(n,k) = [n^4 + A016825(k)*n^3 + A003154(k)*n^2 + A300758(k)*n]/4, sum of n consecutive cubes after k^3.
%e A217843 This expression can be factorized into [n*(n + A005408(k))*(n*(n + A005408(k)) + 4*A000217(k))]/4.
%e A217843 For k = 1, the sequence provides all cubes: T(n,1) = A000578(k).
%e A217843 For k = 2, T(n,2) = A005898(k), centered cube numbers, sum of two consecutive cubes.
%e A217843 For k = 3, T(n,3) = A027602(k), sum of three consecutive cubes.
%e A217843 For k = 4, T(n,4) = A027603(k), sum of four consecutive cubes.
%e A217843 For k = 5, T(n,5) = A027604(k), sum of five consecutive cubes.
%e A217843 T(n,n) = A116149(n), sum of n consecutive cubes after n^3 (main diagonal).
%e A217843 For n = 0, we obtain the subsequence T(0,k) = A000217(n)^2, product of two numbers whose difference is 0*1 (promic) and sum is promic too.
%e A217843 For n = 1, we obtain the subsequence T(1,k) = A168566(x), product of two numbers whose difference is 1*2 (promic) and sum is promic too.
%e A217843 For n = 2, we obtain the subsequence T(2,k) = product of two numbers whose difference is 2*3 (promic) and sum is promic too.
%e A217843 etc.
%e A217843 For n = x, we obtain the subsequence formed by products of two numbers whose difference is the promic x*(x+1) and sum is promic too.
%e A217843 Consequently, if m is in the sequence, then m can be expressed as the product of two nonnegative integers whose sum and difference are both promic. (End)
%t A217843 nMax = 3000; t = {0}; Do[k = n; s = 0; While[s = s + k^3; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/3)}]; t = Union[t]
%o A217843 (Haskell)
%o A217843 import Data.Set (singleton, deleteFindMin, insert, Set)
%o A217843 a217843 n = a217843_list !! (n-1)
%o A217843 a217843_list = f (singleton (0, (0,0))) (-1) where
%o A217843 f s z = if y /= z then y : f s'' y else f s'' y
%o A217843 where s'' = (insert (y', (i, j')) $
%o A217843 insert (y' - i ^ 3 , (i + 1, j')) s')
%o A217843 y' = y + j' ^ 3; j' = j + 1
%o A217843 ((y, (i, j)), s') = deleteFindMin s
%o A217843 -- _Reinhard Zumkeller_, Dec 17 2015, May 12 2015
%o A217843 (PARI) lista(nn) = {my(list = List([0])); for (i=1, nn, my(s = 0); forstep(j=i, 1, -1, s += j^3; if (s > nn^3, break); listput(list, s););); Set(list);} \\ _Michel Marcus_, Nov 13 2020
%Y A217843 Cf. A034705, A217844-A217850, A062682, A131643, A240137.
%Y A217843 Cf. A000578, A005898, A027602, A027603, A027604.
%Y A217843 Cf. A265845 (subsequence).
%Y A217843 Cf. A000217 (triangular numbers), A046092 (4*A000217).
%Y A217843 Cf. A168566 (A000217^2 - 1).
%Y A217843 Cf. A002378 (promics), A016825 (singly even numbers), A003154 (stars numbers).
%Y A217843 Cf. A000330 (square pyramidal numbers), A300758 (12*A000330).
%Y A217843 Cf. A005408 (odd numbers).
%K A217843 nonn
%O A217843 1,3
%A A217843 _T. D. Noe_, Oct 23 2012
%E A217843 Name edited by _N. J. A. Sloane_, May 24 2021