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A060300
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a(n) = (2n(n+1))^2.
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10
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0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600
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listen;
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OFFSET
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0,2
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COMMENTS
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Arises from middle column of following triangle: 4^2, 12^2, 24^2,...:
....................... 3^2 + 4^2 = 5^2
............... 10^2 + 11^2 + 12^2 = 13^2 + 14^2
........ 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
. 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2, etc.
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REFERENCES
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C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..1000
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FORMULA
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G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. [Colin Barker, Apr 22 2012]
a(n) = 4*A035287(n+1) = 4*A002378(n)^2. - Michel Marcus, May 24 2016
a(n) = 16 * A000537(n) = 16 * (n*(n+1)/2)^2 = 16 * A000217(n)^2 = A046092(n)^2. - Bruce J. Nicholson, Jun 05 2017
a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - César Aguilera, Jun 27 2020
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MATHEMATICA
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CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Nov 18 2016 *)
Table[(2n(n+1))^2, {n, 0, 30}] (* Harvey P. Dale, Jan 19 2019 *)
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PROG
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(PARI) { for (n=0, 1000, write("b060300.txt", n, " ", (2*n*(n + 1))^2); ) } \\ Harry J. Smith, Jul 03 2009
(MAGMA) [(2*n*(n+1))^2: n in [0..30]]; // Vincenzo Librandi, Nov 18 2016
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CROSSREFS
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Cf. A000217, A000537, A002378, A035287, A046092.
Sequence in context: A017114 A331741 A092820 * A128985 A341369 A004409
Adjacent sequences: A060297 A060298 A060299 * A060301 A060302 A060303
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KEYWORD
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nonn,easy
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AUTHOR
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Jason Earls, Mar 25 2001
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EXTENSIONS
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Corrected the definition from 2n(n+1)^2 to (2n(n+1))^2. - Harry J. Smith, Jul 03 2009
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STATUS
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approved
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