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0, 8, 32, 72, 128, 200, 288, 392, 512, 648, 800, 968, 1152, 1352, 1568, 1800, 2048, 2312, 2592, 2888, 3200, 3528, 3872, 4232, 4608, 5000, 5408, 5832, 6272, 6728, 7200, 7688, 8192, 8712, 9248, 9800, 10368, 10952, 11552, 12168, 12800
(list;
graph;
refs;
listen;
history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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Opposite numbers to the centered 16-gonal numbers (A069129) in the square spiral whose vertices are the triangular numbers (A000217).
8 times the squares. - Omar E. Pol, Dec 09 2008
a(n-1) is the molecular topological index of the n-wheel graph W_n. - Eric W. Weisstein, Jul 11 2011
An n X n pandiagonal magic square has a(n) orientations. - Kausthub Gudipati, Sep 15 2011
Area of a square with diagonal 4n. - Wesley Ivan Hurt, Jun 19 2014
Sum of all the parts in the partitions of 4n into exactly two parts. - Wesley Ivan Hurt, Jul 23 2014
For n>1, a(n) is the third least number k = x + y, with x>0 and y>0, such that there are n different pairs (x,y) for which x*y/k is an integer. - Paolo P. Lava, Jan 29 2018
Equivalently: integers k such that k$ / (k/2-1)! and k$ / (k/2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..800
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Eric Weisstein's World of Mathematics, Molecular Topological Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 8*A000290(n) = 4*A001105(n) = 2*A016742(n). - Omar E. Pol, Dec 13 2008
G.f.: -8*x*(1+x) / (x-1)^3. - R. J. Mathar, Nov 27 2015
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/48 (A245058).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/96.
Product_{n>=1} (1 + 1/a(n)) = sqrt(8)*sinh(Pi/sqrt(8))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(8)*sin(Pi/sqrt(8))/Pi. (End)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Wesley Ivan Hurt, Dec 03 2021
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MAPLE
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A139098:=n->8*n^2; seq(A139098(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014
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MATHEMATICA
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8 Range[0, 50]^2 (* Wesley Ivan Hurt, Jun 19 2014 *)
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PROG
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(MAGMA) [8*n^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n)=8*n^2 \\ Charles R Greathouse IV, Jun 17 2017
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CROSSREFS
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Cf. A000217, A000290, A016766, A033582, A069129, A001105, A016742, A245058.
Cf. A348692.
Subsequence of A008586 and of A349081.
Sequence in context: A290960 A009245 A018842 * A224543 A211633 A130809
Adjacent sequences: A139095 A139096 A139097 * A139099 A139100 A139101
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KEYWORD
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nonn,easy
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AUTHOR
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Omar E. Pol, Apr 25 2008
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STATUS
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approved
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