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A063436 Write 1, 2, 3, 4, ... counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical upward direction.

%I A063436 #50 Oct 31 2024 18:39:00
%S A063436 0,15,54,117,204,315,450,609,792,999,1230,1485,1764,2067,2394,2745,
%T A063436 3120,3519,3942,4389,4860,5355,5874,6417,6984,7575,8190,8829,9492,
%U A063436 10179,10890,11625,12384,13167,13974,14805,15660,16539,17442,18369,19320,20295,21294,22317
%N A063436 Write 1, 2, 3, 4, ... counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical upward direction.
%C A063436 Related to parity of Beatty sequences for exp(-(1/2)/n). Let f(k,n)=-sum(i=1,n,sum(j=1,i,(-1)^floor(j*exp(-(1/2)/n)))), then a(n)=Max{f(k,n) : 1<=k<=4*a(n)-2} and for 0<=i<=4*a(n)-3, f(i,n)=f(4*a(n)-2-i,n). - _Benoit Cloitre_, May 26 2004
%C A063436 Or, sum of multiples of 2 and 3 from 0 to 6*n. - _Zak Seidov_, Aug 06 2016
%H A063436 Harry J. Smith, <a href="/A063436/b063436.txt">Table of n, a(n) for n = 0..1000</a>
%H A063436 Leo Tavares, <a href="/A063436/a063436.jpg">Illustration: Stellar Triangles</a>.
%H A063436 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A063436 a(n) = 3*n*(4*n+1) = 3*A007742(n).
%F A063436 a(n) = 24*n + a(n-1) - 9 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F A063436 From _Colin Barker_, Jul 07 2012: (Start)
%F A063436 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A063436 G.f.: 3*x*(5 + 3*x)/(1-x)^3. (End)
%F A063436 a(n) = A272399(n+1) - A003154(n+1). - _Leo Tavares_, Mar 24 2022
%F A063436 From _Amiram Eldar_, Mar 27 2022: (Start)
%F A063436 Sum_{n>=1} 1/a(n) = 4/3 - Pi/6 - log(2).
%F A063436 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 + sqrt(2)*log(sqrt(2)+1)/3 - 4/3. (End)
%F A063436 E.g.f.: 3*x*(5 + 4*x)*exp(x). - _Elmo R. Oliveira_, Oct 31 2024
%e A063436 The spiral begins:
%e A063436 .
%e A063436         16--15--14
%e A063436         /         \
%e A063436       17   5---4  13
%e A063436       /   /     \   \
%e A063436     18   6   0   3  12
%e A063436     /   /   /   /   /
%e A063436   19   7   1---2  11  26
%e A063436     \   \         /   /
%e A063436     20   8---9--10  25
%e A063436       \             /
%e A063436       21--22--23--24
%t A063436 a[n_] := 3*n*(4*n + 1); Array[a, 40, 0] (* _Amiram Eldar_, Mar 27 2022 *)
%o A063436 (PARI) { for (n=0, 1000, write("b063436.txt", n, " ", n*(12*n + 3)) ) } \\ _Harry J. Smith_, Aug 21 2009
%Y A063436 Cf. A000567, A003154, A007742, A062783, A272399.
%K A063436 nonn,easy
%O A063436 0,2
%A A063436 _Floor van Lamoen_, Jul 21 2001