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 A135453 a(n) = 12*n^2. 17
 0, 12, 48, 108, 192, 300, 432, 588, 768, 972, 1200, 1452, 1728, 2028, 2352, 2700, 3072, 3468, 3888, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9408, 10092, 10800, 11532, 12288, 13068, 13872, 14700, 15552, 16428, 17328, 18252, 19200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Areas of perfect 4:3 rectangles (for n>0). Sequence found by reading the line from 0, in the direction 0, 12,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011 For n>2, a(n) is the fifth 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 (x,y,z) = (-a(n),1 + n*a(n),1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022 LINKS Ivan Panchenko, Table of n, a(n) for n = 0..10000 Leo Tavares, Illustration. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = A000290(n)*12 = A001105(n)*6 = A033428(n)*4 = A016742(n)*3 = A033581(n)*2. - Omar E. Pol, Dec 13 2008 a(n) = a(n-1)+24*n-12, n>0. - Vincenzo Librandi, Nov 24 2010 From Amiram Eldar, Feb 03 2021: (Start) Sum_{n>=1} 1/a(n) = Pi^2/72 (A086729). Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144. Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi. Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End) EXAMPLE 192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192. MAPLE seq(12*h^2, n=0..100); # Muniru A Asiru, Jan 29 2018 MATHEMATICA Table[12*n^2, {n, 0, 60}] (* Stefan Steinerberger, Dec 17 2007 *) LinearRecurrence[{3, -3, 1}, {0, 12, 48}, 50] (* Harvey P. Dale, Jan 19 2020 *) PROG (PARI) a(n)=12*n^2 \\ Charles R Greathouse IV, Jun 17 2017 (GAP) List([0..100], n->12*n^2); # Muniru A Asiru, Jan 29 2018 CROSSREFS Cf. A000290, A001105, A016742, A033428, A033581, A086729. Sequence in context: A213493 A009958 A256695 * A165280 A280058 A173548 Adjacent sequences:  A135450 A135451 A135452 * A135454 A135455 A135456 KEYWORD nonn,easy AUTHOR Ben Paul Thurston, Dec 14 2007 EXTENSIONS More terms from Stefan Steinerberger, Dec 17 2007 Minor edits from Omar E. Pol, Dec 15 2008 STATUS approved

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Last modified September 26 17:47 EDT 2022. Contains 357002 sequences. (Running on oeis4.)