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A200048
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Number of real singularities on a family of degree-3n algebraic surfaces.
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2
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4, 59, 220, 581, 1162, 2105, 3370, 5171, 7384, 10319, 13744, 18089, 22990, 29021, 35662, 43655, 52300, 62531, 73444, 86189, 99634, 115169, 131410, 150011, 169312, 191255, 213880, 239441, 265654, 295109, 325174, 358799, 392980, 431051, 469612, 512405, 555610
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OFFSET
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1,1
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COMMENTS
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The sequence gives the number of real nodes of a family of algebraic surfaces with degrees 3n. They have been introduced by means of a kind of duality in the basic geometric constructions corresponding to the generation of substitution tilings.
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LINKS
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FORMULA
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a(n) = (1/12)*(135*n^3-126*n^2 + 45*n-6) if n is odd; a(n) = (1/12)*(135*n^3- 117*n^2 + 54*n - 12) if n is even.
G.f.: x*(4 + 55*x + 149*x^2 + 196*x^3 + 110*x^4 + 25*x^5 + x^6) / ((1 - x)^4*(1 + x)^3).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7.
(End)
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PROG
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(PARI) a(n) = if (n % 2, (1/12)*(135*n^3-126*n^2 + 45*n-6), (1/12)*(135*n^3- 117*n^2 + 54*n - 12)); \\ Michel Marcus, Aug 17 2013
(PARI) Vec(x*(4 + 55*x + 149*x^2 + 196*x^3 + 110*x^4 + 25*x^5 + x^6) / ((1 - x)^4*(1 + x)^3) + O(x^40)) \\ Colin Barker, Nov 04 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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EXTENSIONS
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STATUS
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approved
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