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A145018
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a(n) = (n^2 - n + 8)/2.
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10
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4, 5, 7, 10, 14, 19, 25, 32, 40, 49, 59, 70, 82, 95, 109, 124, 140, 157, 175, 194, 214, 235, 257, 280, 304, 329, 355, 382, 410, 439, 469, 500, 532, 565, 599, 634, 670, 707, 745, 784, 824, 865, 907, 950, 994, 1039, 1085, 1132, 1180, 1229, 1279, 1330, 1382, 1435
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OFFSET
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1,1
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COMMENTS
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The previous name was "a(1) = 4; then add 1 to the first number, then 2, then 3 and so on".
a(n) is the minimal number of vertices for a polyhedron with at least one vertex of degree k and at least one k-gonal face for each k=3..n+2. - Riccardo Maffucci, Aug 03 2021
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LINKS
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FORMULA
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G.f.: x*(4 -7*x +4*x^2)/(1-x)^3.
a(n) = a(n-1) + n - 1.
Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(31)*Pi/2)/sqrt(31). - Amiram Eldar, Dec 13 2022
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MAPLE
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MATHEMATICA
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Nest[Append[#, #[[-1]] + Length@ #] &, {4}, 66] (* or *)
Rest@ CoefficientList[Series[x (4 - 7 x + 4 x^2)/(1 - x)^3, {x, 0, 67}], x] (* Michael De Vlieger, Jan 23 2019 *)
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PROG
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(Sage)[4+binomial(n, 2) for n in range(1, 68)] # Zerinvary Lajos, Mar 12 2009
(PARI) x='x+O('x^50); Vec(x*(4 -7*x +4*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 18 2017
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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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Jayanth (mergujayanth(AT)yahoo.com), Sep 29 2008
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EXTENSIONS
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STATUS
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approved
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