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 A253680 Numbers c(n) whose square are equal to the sum of an odd number M of consecutive cubed integers b^3 + (b+1)^3 + ... + (b+M-1)^3 = c(n)^2, starting at b(n) (A253679). 7
 204, 2940, 16296, 57960, 159060, 368004, 754320, 1412496, 2465820, 4070220, 6418104, 9742200, 14319396, 20474580, 28584480, 39081504, 52457580, 69267996, 90135240, 115752840, 146889204, 184391460, 229189296, 282298800, 344826300, 417972204, 503034840 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers c(n) such that b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2 has nontrivial solutions over the integers for M being an odd positive integer. To every odd positive integer M corresponds a sum of M consecutive cubed integers starting at b^3 having at least one nontrivial solution. For n>=1, M(n)=(2n+1) (A005408), b(n) = M^3 -(3M-1)/2 = (2n+1)^3 - (3n+1) (A253679) and c(n) = M*(M^2-1)*(2M^2-1)/2 = 2n*(n+1)*(2n+1)*(8n*(n+1)+1) (A253680). The trivial solutions with M < 1 and b < 2 are not considered here. Stroeker stated that all odd values of M yield a solution to  b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2. This was further demonstrated by Pletser. LINKS Vladimir Pletser, Table of n, a(n) for n = 1..50000 Vladimir Pletser, File Triplets (M,b,c) for M=(2n+1) Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015. Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015 R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307. Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1). FORMULA c(n) = 2n(n+1)*(2n+1)*(8n*(n+1)+1). G.f.: 12*x*(x+1)*(17*x^2+126*x+17) / (x-1)^6. - Colin Barker, Jan 09 2015 EXAMPLE For n=1, M(n)=3, b(n)=23, c(n)=204. See "File Triplets (M,b,c) for M=(2n+1)" link. MAPLE restart: for n from 1 to 50000 do c:=2*n*(n+1)*(2*n+1)*(8*n*(n+1)+1): print (c); end do: MATHEMATICA f[n_] := 2 n (n + 1) (2 n + 1) (8 n (n + 1) + 1); Array[f, 36] (* Michael De Vlieger, Jan 10 2015 *) PROG (PARI) Vec(12*x*(x+1)*(17*x^2+126*x+17)/(x-1)^6 + O(x^100)) \\ Colin Barker, Jan 09 2015 (MAGMA) [2*n*(n+1)*(2*n+1)*(8*n*(n+1)+1): n in [1..30]]; // Vincenzo Librandi, Feb 19 2015 CROSSREFS Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253681. Sequence in context: A234796 A234789 A099105 * A209790 A348603 A194192 Adjacent sequences:  A253677 A253678 A253679 * A253681 A253682 A253683 KEYWORD nonn,easy AUTHOR Vladimir Pletser, Jan 08 2015 STATUS approved

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)