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A278130
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a(n) = 492*2^n - 222.
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1
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270, 762, 1746, 3714, 7650, 15522, 31266, 62754, 125730, 251682, 503586, 1007394, 2015010, 4030242, 8060706, 16121634, 32243490, 64487202, 128974626, 257949474, 515899170, 1031798562, 2063597346, 4127194914, 8254390050, 16508780322, 33017560866, 66035121954, 132070244130
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OFFSET
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0,1
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COMMENTS
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a(n) is the first Zagreb index of the phenylacetylene dendrimer NSB[n] defined pictorially in the Yarahmadi references. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
The M-polynomial of the dendrimer NSB[n] is M(NSB[n],x,y) = 9*2^n*x*y^4 + (24*2^n - 12)*x^2*y^2 + (48*2^n - 24)*x^2*y^3 + (15*2^n - 9)*x^3*y^3 + 3*2^n*x^3*y^4.
In general, this type of formula b(n) = k*2^n - k (where n>=0 and h, k are given constants) has:
O.g.f.: (h - k - (h - 2*k)*x)/((1 - x)*(1 - 2*x));
E.g.f.: (-k + h*exp(x))*exp(x);
Linear recurrence: b(n) = 3*b(n-1) - 2*b(n-2);
Signature of the recurrence: (3,-2). (End)
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LINKS
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FORMULA
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O.g.f.: 6*(45 - 8*x)/((1 - x)*(1 - 2*x)).
E.g.f.: 6*(-37 + 82*exp(x))*exp(x).
a(n) = 3*a(n-1) - 2*a(n-2).
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MAPLE
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seq(492*2^n-222, n=0..35)
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MATHEMATICA
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LinearRecurrence[{3, -2}, {270, 762}, 30] (* Harvey P. Dale, Jan 27 2024 *)
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PROG
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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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