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A066982 a(n) = Lucas(n+1) - (n+1). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

From Gus Wiseman, Feb 12 2019: (Start)

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)

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..250

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

FORMULA

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.

From Colin Barker, Jun 30 2012: (Start)

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)

MATHEMATICA

a[1]=a[2]=1; a[n_]:= a[n] = a[n-1] +a[n-2] +n-2; Table[a[n], {n, 40}]

LinearRecurrence[{3, -2, -1, 1}, {1, 1, 3, 6}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

Table[LucasL[n+1]-n-1, {n, 40}] (* Vladimir Reshetnikov, Sep 15 2016 *)

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 *)

PROG

(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

CROSSREFS

Cf. A000032, A000045, A000126, A000325, A005251, A169985, A323953, A323954.

Sequence in context: A309266 A081056 A242448 * A246597 A179906 A236913

Adjacent sequences:  A066979 A066980 A066981 * A066983 A066984 A066985

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Jan 27 2002

EXTENSIONS

Corrected and extended by Harvey P. Dale, Feb 08 2002

STATUS

approved

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Last modified November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)