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 A028896 6 times triangular numbers: a(n) = 3*n*(n+1). 33
 0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Floor van Lamoen, Jul 21 2001: (Start) Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ... The spiral begins:                  85--84--83--82--81--80                  /                     \                86  56--55--54--53--52  79                /   /                 \   \              87  57  33--32--31--30  51  78              /   /   /             \   \   \            88  58  34  16--15--14  29  50  77            /   /   /   /         \   \   \   \          89  59  35  17   5---4  13  28  49  76          /   /   /   /   /     \   \   \   \   \     <==90==60==36==18===6===0   3  12  27  48  75            /   /   /   /   /   /   /   /   /   /          61  37  19   7   1---2  11  26  47  74            \   \   \   \         /   /   /   /            62  38  20   8---9--10  25  46  73              \   \   \             /   /   /              63  39  21--22--23--24  45  72                \   \                 /   /                64  40--41--42--43--44  71                  \                     /                  65--66--67--68--69--70 (End) If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007 a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008 Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011 Partial sums of A008588. - R. J. Mathar, Aug 28 2014 Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017 LINKS Ivan Panchenko, Table of n, a(n) for n = 0..1000 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets Enrique Navarrete, Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019. Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014. Eric Weisstein's World of Mathematics, Graph Cycle Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA O.g.f.: 6*x/(1 - x)^3. E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017 a(n) = 6*A000217(n). a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003 From Zerinvary Lajos, Mar 06 2007: (Start) a(n) = A049598(n)/2. a(n) = A124080(n) - A046092(n). a(n) = A033996(n) - A002378(n). (End) a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008 a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010 a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011 From Philippe Deléham, Mar 26 2013: (Start) a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18. a(n) = A174709(6*n + 5). (End) a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014 a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018 a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018 a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 04 2020 MAPLE [seq(6*binomial(n, 2), n=1..44)]; # Zerinvary Lajos, Nov 24 2006 MATHEMATICA 6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *) 6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *) LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *) PROG (MAGMA) [3*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014 (PARI) a(n)=3*n*(n+1) \\ Charles R Greathouse IV, Sep 24 2015 (PARI) first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018 (GAP) List([0..44], n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019 CROSSREFS Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080. Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles). Sequence in context: A172522 A152539 A069958 * A295026 A034857 A116367 Adjacent sequences:  A028893 A028894 A028895 * A028897 A028898 A028899 KEYWORD nonn,easy AUTHOR Joe Keane (jgk(AT)jgk.org), Dec 11 1999 STATUS approved

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Last modified April 22 14:20 EDT 2021. Contains 343177 sequences. (Running on oeis4.)