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A186707
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Partial sums of A007202 (crystal ball sequence for hexagonal close-packing).
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3
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1, 14, 71, 224, 547, 1134, 2101, 3584, 5741, 8750, 12811, 18144, 24991, 33614, 44297, 57344, 73081, 91854, 114031, 140000, 170171, 204974, 244861, 290304, 341797, 399854, 465011, 537824, 618871, 708750, 808081, 917504, 1037681, 1169294, 1313047, 1469664
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OFFSET
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0,2
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COMMENTS
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Subsequence of primes begins 71, 547, 5741, 114031, 244861, 465011, 808081, 1037681. Subsequence of powers includes 537824 = 2^5 * 7^5.
The sequence is a quasipolynomial, so under the Bunyakovsky conjecture there are infinitely many primes in this sequence. - Charles R Greathouse IV, Aug 21 2011
Let s(0) = 0 and s(n) = A186707(n-1) for n > 0. Then s(n) is the number of 4-tuples (w,x,y,z) having all terms in {1, ..., n} and |w - x| < w + |y - z|. - Clark Kimberling, May 24 2012
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LINKS
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FORMULA
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a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) = 7*n*(n^3/8 + n^2/2 + 3*n/4 + 1/2) + (15 + (-1)^n)/16.
G.f.: ( -1 - 10*x - 20*x^2 - 10*x^3 - x^4 ) / ( (1 + x)*(x - 1)^5 ). (End)
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MATHEMATICA
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CoefficientList[Series[ (-1-10 x-20 x^2-10 x^3-x^4)/((x-1)^5 (1+x)), {x, 0, 40}], x] (* Harvey P. Dale, Apr 04 2011 *)
Table[7*n*(n^3 + 4*n^2 + 6*n + 4)/8 + (15 + (-1)^n)/16, {n, 0, 40}] (* T. D. Noe, Apr 04 2011 *)
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PROG
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(Magma)[7*n*(n^3+4*n^2+6*n+4)/8+(15+(-1)^n)/16: n in [0..40] ]; // Vincenzo Librandi, Aug 22 2011
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CROSSREFS
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KEYWORD
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nonn,easy,less
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AUTHOR
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STATUS
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approved
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