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A101946
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6*2^n - 3*n - 5.
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9
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1, 4, 13, 34, 79, 172, 361, 742, 1507, 3040, 6109, 12250, 24535, 49108, 98257, 196558, 393163, 786376, 1572805, 3145666, 6291391, 12582844, 25165753, 50331574, 100663219, 201326512, 402653101, 805306282, 1610612647, 3221225380, 6442450849, 12884901790
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OFFSET
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0,2
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COMMENTS
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Sequence generated from a 3 X 3 matrix, companion to A101945.
Characteristic polynomial of M = x^3 - 4x^2 + 5x - 2.
Sequence can also be generated by the same method as A061777 with slightly different rules. Refer to A061777, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e. "vertex to side" expansion version. Let us assign the label "1" to the triangle at the origin; at n-th generation add a triangle at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping triangles will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The triangles count is A005488. See illustration. - Kival Ngaokrajang, Sep 26 2014
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LINKS
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Table of n, a(n) for n=0..31.
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
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FORMULA
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a(0)=1, a(1)=4, a(2)=13 and for n>2, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 2 0 / 1 2 1]. M^n * [1 1 1] = [1 A033484(n) a(n)].
a(0) = 1, for n >= 1, a(n) = 3*A000225(n) + a(n-1). - Kival Ngaokrajang, Sep 26 2014
G.f.: -(2*x^2+1) / ((x-1)^2*(2*x-1)). - Colin Barker, Sep 26 2014
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EXAMPLE
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a(4) = 79 = 4*34 - 5*13 + 2*4 = 4*a(3) - 5*a(2) + 2*a(1).
a(4) = right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 46 a(4)], where 46 = A033484(4).
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MATHEMATICA
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a[0] = 1; a[1] = 4; a[2] = 13; a[n_] := a[n] = 4a[n - 1] - 5a[n - 2] + 2a[n - 3]; Table[ a[n], {n, 0, 30}] (* Or *)
a[n_] := (MatrixPower[{{1, 0, 0}, {2, 2, 0}, {1, 2, 1}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 12 2005 *)
Table[6*2^n-3n-5, {n, 0, 40}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 4, 13}, 40] (* Harvey P. Dale, Jun 03 2017 *)
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PROG
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(PARI) a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))\\ Kival Ngaokrajang, Sep 26 2014
(PARI) Vec(-(2*x^2+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014
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CROSSREFS
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Cf. A101945, A033484.
Cf. A061777, A000225, A005488. - Kival Ngaokrajang, Sep 26 2014
Sequence in context: A322599 A135859 A161531 * A029860 A262200 A213578
Adjacent sequences: A101943 A101944 A101945 * A101947 A101948 A101949
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KEYWORD
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nonn,easy
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AUTHOR
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Gary W. Adamson, Dec 22 2004
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EXTENSIONS
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New definition from Ralf Stephan, May 17 2007
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STATUS
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approved
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