This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033581 #154 Feb 16 2025 08:32:36
%S A033581 0,6,24,54,96,150,216,294,384,486,600,726,864,1014,1176,1350,1536,
%T A033581 1734,1944,2166,2400,2646,2904,3174,3456,3750,4056,4374,4704,5046,
%U A033581 5400,5766,6144,6534,6936,7350,7776,8214,8664,9126,9600,10086,10584,11094,11616
%N A033581 a(n) = 6*n^2.
%C A033581 Number of edges of a complete 4-partite graph of order 4n, K_n,n,n,n. - _Roberto E. Martinez II_, Oct 18 2001
%C A033581 Number of edges of the complete bipartite graph of order 7n, K_n, 6n. - _Roberto E. Martinez II_, Jan 07 2002
%C A033581 Number of edges in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n). - _Roberto E. Martinez II_, Jan 07 2002
%C A033581 Total surface area of a cube of edge length n. See A000578 for cube volume. See A070169 and A071399 for surface area and volume of a regular tetrahedron and links for the other Platonic solids. - _Rick L. Shepherd_, Apr 24 2002
%C A033581 a(n) can represented as n concentric hexagons (see example). - _Omar E. Pol_, Aug 21 2011
%C A033581 Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A003154 in the same spiral. - _Omar E. Pol_, Sep 08 2011
%C A033581 Together with 1, numbers m such that floor(2*m/3) and floor(3*m/2) are both squares. Example: floor(2*150/3) = 100 and floor(3*150/2) = 225 are both squares, so 150 is in the sequence. - _Bruno Berselli_, Sep 15 2014
%C A033581 a(n+1) gives the number of vertices in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter has 18 vertices. The core hexagon has 6 vertices. a(2) = 18 + 6 = 24 is the total number of vertices. - _Ivan N. Ianakiev_, Mar 11 2015
%C A033581 a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n). - _Sergey Pavlov_, Mar 31 2017
%C A033581 More generally, if k >= 5 we have that the sequence whose formula is a(n) = (2*k - 4)*n^2 is also the sequence found by reading the line from 0, in the direction 0, (2*k - 4), ..., in the square spiral whose vertices are the generalized k-gonal numbers. In this case k = 5. - _Omar E. Pol_, May 13 2018
%C A033581 The sequence also gives the number of size=1 triangles within a match-made hexagon of size n. - _John King_, Mar 31 2019
%C A033581 For hexagons, the number of matches required is A045945; thus number of size=1 triangles is A033581; number of larger triangles is A307253 and total number of triangles is A045949. See A045943 for analogs for Triangles; see A045946 for analogs for Stars. - _John King_, Apr 04 2019
%H A033581 Nathaniel Johnston, <a href="/A033581/b033581.txt">Table of n, a(n) for n = 0..10000</a>
%H A033581 Bruno Berselli, <a href="/A033581/a033581.jpg">An interpretation of initial terms</a>.
%H A033581 Ivan N. Ianakiev, <a href="http://mislandia.weebly.com/blog/hexagon-like-honeycomb-built-from-regular-congruent-hexagons">Hexagon-like honeycomb built form regular congruent hexagons</a>.
%H A033581 Leo Tavares, <a href="/A033581/a033581_1.jpg">Illustration: Diamond Star Rays</a>
%H A033581 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlatonicSolid.html">Platonic Solid</a>.
%H A033581 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033581 a(n) = A000290(n)*6. - _Omar E. Pol_, Dec 11 2008
%F A033581 a(n) = A001105(n)*3 = A033428(n)*2. - _Omar E. Pol_, Dec 13 2008
%F A033581 a(n) = 12*n + a(n-1) - 6, with a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F A033581 G.f.: 6*x*(1+x)/(1-x)^3. - _Colin Barker_, Feb 14 2012
%F A033581 For n > 0: a(n) = A005897(n) - 2. - _Reinhard Zumkeller_, Apr 27 2014
%F A033581 a(n) = 3*floor(1/(1-cos(1/n))) = floor(1/(1-n*sin(1/n))) for n > 0. - _Clark Kimberling_, Oct 08 2014
%F A033581 a(n) = t(4*n) - 4*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(4*n) - 4*A000217(n). - _Bruno Berselli_, Aug 31 2017
%F A033581 From _Amiram Eldar_, Feb 03 2021: (Start)
%F A033581 Sum_{n>=1} 1/a(n) = Pi^2/36.
%F A033581 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 (A086729).
%F A033581 Product_{n>=1} (1 + 1/a(n)) = sqrt(6)*sinh(Pi/sqrt(6))/Pi.
%F A033581 Product_{n>=1} (1 - 1/a(n)) = sqrt(6)*sin(Pi/sqrt(6))/Pi. (End)
%F A033581 E.g.f.: 6*exp(x)*x*(1 + x). - _Stefano Spezia_, Aug 19 2022
%e A033581 From _Omar E. Pol_, Aug 21 2011: (Start)
%e A033581 Illustration of initial terms as concentric hexagons:
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%e A033581 (End)
%p A033581 seq(6*n^2,n=0..44); # _Nathaniel Johnston_, Jun 26 2011
%t A033581 6 Range[44]^2 (* _Michael De Vlieger_, Apr 02 2017 *)
%t A033581 LinearRecurrence[{3,-3,1},{0,6,24},50] (* _Harvey P. Dale_, Jul 03 2017 *)
%o A033581 (Haskell)
%o A033581 a033581 = (* 6) . (^ 2) -- _Reinhard Zumkeller_, Apr 27 2014
%o A033581 (PARI) vector(100,n,6*(n-1)^2) \\ _Derek Orr_, Mar 11 2015
%Y A033581 Bisection of A032528. Central column of triangle A001283.
%Y A033581 Cf. A000217, A000290, A001105, A009111, A033428, A033583, A085250, A086729, A152734, A152751.
%Y A033581 Cf. A000578, A001318, A003215, A005897, A045943, A045945, A045949, A070169, A071399.
%Y A033581 Cf. A017593 (first differences).
%K A033581 nonn,easy
%O A033581 0,2
%A A033581 _N. J. A. Sloane_
%E A033581 More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001