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A063436
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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.
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2
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0, 15, 54, 117, 204, 315, 450, 609, 792, 999, 1230, 1485, 1764, 2067, 2394, 2745, 3120, 3519, 3942, 4389, 4860, 5355, 5874, 6417, 6984, 7575, 8190, 8829, 9492, 10179, 10890, 11625, 12384, 13167, 13974, 14805, 15660, 16539, 17442, 18369, 19320, 20295, 21294, 22317
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OFFSET
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0,2
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COMMENTS
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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
Or, sum of multiples of 2 and 3 from 0 to 6n. - Zak Seidov, Aug 06 2016
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 3*x*(5+3*x)/(1-x)^3. (End)
Sum_{n>=1} 1/a(n) = 4/3 - Pi/6 - log(2).
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)
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EXAMPLE
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The spiral begins:
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16--15--14
/ \
17 5---4 13
/ / \ \
18 6 0 3 12
/ / / / /
19 7 1---2 11 26
\ \ / /
20 8---9--10 25
\ /
21--22--23--24
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MATHEMATICA
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a[n_] := 3*n*(4*n + 1); Array[a, 40, 0] (* Amiram Eldar, Mar 27 2022 *)
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PROG
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(PARI) { for (n=0, 1000, write("b063436.txt", n, " ", n*(12*n + 3)) ) } \\ Harry J. Smith, Aug 21 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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