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A066982
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a(n) = Lucas(n+1) - (n+1).
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17
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1, 1, 3, 6, 12, 22, 39, 67, 113, 188, 310, 508, 829, 1349, 2191, 3554, 5760, 9330, 15107, 24455, 39581, 64056, 103658, 167736, 271417, 439177, 710619, 1149822, 1860468, 3010318, 4870815, 7881163, 12752009, 20633204, 33385246, 54018484, 87403765, 141422285
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OFFSET
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1,3
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COMMENTS
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Also the number of ways to split an (n + 1)-cycle into nonempty connected subgraphs with no singletons. For example, the a(1) = 1 through a(5) = 12 partitions are:
{{12}} {{123}} {{1234}} {{12345}} {{123456}}
{{12}{34}} {{12}{345}} {{12}{3456}}
{{14}{23}} {{123}{45}} {{123}{456}}
{{125}{34}} {{1234}{56}}
{{145}{23}} {{1236}{45}}
{{15}{234}} {{1256}{34}}
{{126}{345}}
{{1456}{23}}
{{156}{234}}
{{16}{2345}}
{{12}{34}{56}}
{{16}{23}{45}}
Also the number of non-singleton subsets of {1, ..., (n + 1)} with no cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(1) = 1 through a(5) = 12 subsets are:
{} {} {} {} {}
{1,3} {1,3} {1,3}
{2,4} {1,4} {1,4}
{2,4} {1,5}
{2,5} {2,4}
{3,5} {2,5}
{2,6}
{3,5}
{3,6}
{4,6}
{1,3,5}
{2,4,6}
(End)
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LINKS
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FORMULA
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a(1) = a(2) = 1, a(n + 2) = a(n + 1) + a(n) + n.
For n > 2, a(n) = floor(phi^(n+1) - (n+1)) + (1-(-1)^n)/2.
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x*(1-2*x+2*x^2)/((1-x)^2*(1-x-x^2)). (End)
a(n) is the sum of the n-th antidiagonal of A352744 (assuming offset 0). - Peter Luschny, Nov 16 2023
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MATHEMATICA
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a[1]=a[2]=1; a[n_]:= a[n] = a[n-1] +a[n-2] +n-2; Table[a[n], {n, 40}]
CoefficientList[Series[(1-2*x+2*x^2)/((1-x)^2*(1-x-x^2)), {x, 0, 40}], x] (* L. Edson Jeffery, Sep 28 2017 *)
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PROG
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(PARI) { for (n=1, 250, if (n>2, a=a1 + a2 + n - 2; a2=a1; a1=a, a=a1=1; a=a2=1); write("b066982.txt", n, " ", a) ) } \\ Harry J. Smith, Apr 14 2010
(PARI) vector(40, n, f=fibonacci; f(n+2)+f(n)-n-1) \\ G. C. Greubel, Jul 09 2019
(Magma) [Lucas(n+1)-n-1: n in [1..40]]; // G. C. Greubel, Jul 09 2019
(Sage) [lucas_number2(n+1, 1, -1) -n-1 for n in (1..40)] # G. C. Greubel, Jul 09 2019
(GAP) List([1..40], n-> Lucas(1, -1, n+1)[2] -n-1); # G. C. Greubel, Jul 09 2019
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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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