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A130748
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Place n points on each of the three sides of a triangle, 3n points in all; a(n) = number of nondegenerate triangles that can be constructed using these points (plus the 3 original vertices) as vertices.
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3
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17, 72, 190, 395, 711, 1162, 1772, 2565, 3565, 4796, 6282, 8047, 10115, 12510, 15256, 18377, 21897, 25840, 30230, 35091, 40447, 46322, 52740, 59725, 67301, 75492, 84322, 93815, 103995, 114886
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OFFSET
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1,1
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COMMENTS
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Define b(1)=1 and b(n)=a(n-1) for n>1. Then (b(n)) is the principal diagonal of the convolution array A213833. [Clark Kimberling, Jul 04 2012]
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LINKS
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Table of n, a(n) for n=1..30.
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = C(3*(n+1), 3) - 3*C(n+2, 3) where n>0.
a(n) = (n+1)*A002414(n+1) - n*A002414(n). [Bruno Berselli, Dec 11 2012]
a(n) = (8n^3 + 15n^2 + 9n + 2)/2 \\ Charles R Greathouse IV, Feb 14 2013
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EXAMPLE
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5 points are put on each side of a triangle (n = 5); we then have 18 vertices to construct with: 5 * 3 + 3 originals. The number of total arrangements = combi(18,3) : combi[3(n+1),3]. But these include degenerates along the 3 sides: 7 points on each side, so combi(7,3) on each side : 3 * combi[n+2, 3] combi[18,3] - 3 * combi[7,3] = 816 - 105 = 711.
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PROG
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(PARI) a(n)=4*n^3+n*(15*n+9)/2+1 \\ Charles R Greathouse IV, Feb 14 2013
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CROSSREFS
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Cf. A002414, A213833, A220084 (for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).
Sequence in context: A214530 A087514 A119625 * A131692 A157864 A059704
Adjacent sequences: A130745 A130746 A130747 * A130749 A130750 A130751
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KEYWORD
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nonn,easy
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AUTHOR
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Denis Borris (daborris(AT)rogers.com), Jul 12 2007
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STATUS
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approved
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