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 A254371 Sum of cubes of the first n even numbers (A016743). 2
 0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3. Diophante, Une miniature avec trois entiers consécutifs (in French). Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5. a(n) = 2*n^2*(n + 1)^2. a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017 a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017 From Amiram Eldar, Aug 25 2022: (Start) Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2. Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End) MAPLE A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017 MATHEMATICA Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40] Accumulate[Range[0, 80, 2]^3] (* Harvey P. Dale, Jun 26 2017 *) PROG (PARI) a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015 (Magma) [2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015 (GAP) List([0..35], n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018 CROSSREFS Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes). Cf. A060300 (2*a(n)). First bisection of A105636; second bisection of A212892. Cf. A000217, A000415, A002378, A035287. Sequence in context: A064015 A044576 A104453 * A143945 A239095 A189954 Adjacent sequences: A254368 A254369 A254370 * A254372 A254373 A254374 KEYWORD nonn,easy AUTHOR Luciano Ancora, Mar 16 2015 STATUS approved

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Last modified November 30 01:31 EST 2022. Contains 358431 sequences. (Running on oeis4.)