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A059100 a(n) = n^2 + 2. 57

%I

%S 2,3,6,11,18,27,38,51,66,83,102,123,146,171,198,227,258,291,326,363,

%T 402,443,486,531,578,627,678,731,786,843,902,963,1026,1091,1158,1227,

%U 1298,1371,1446,1523,1602,1683,1766,1851,1938,2027,2118,2211,2306,2403

%N a(n) = n^2 + 2.

%C Let s(n) = Sum_{k>=1} 1/n^(2^k). Then I conjecture that the maximum element in the continued fraction for s(n) is n^2 + 2. - _Benoit Cloitre_, Aug 15 2002

%C Binomial transformation yields A081908, with A081908(0)=1 dropped. - _R. J. Mathar_, Oct 05 2008

%C a(3*n) mod 9 = 2. - _Paul Curtz_, Nov 07 2012

%C 1/a(n) = R(n)/r with R(n) the n-th radius of the Pappus chain of the symmetric arbelos with semicircle radii r, r1 = r/2 = r2. See the MathWorld link for Pappus chain (there are two of them, a left and a right one. In this case these two chains are congruent). - _Wolfdieter Lang_, Mar 01 2013

%C a(n) is the number of election results for an election with n+2 candidates, say C1, C2, ..., and C(n+2), and with only two voters (each casting a single vote) that have C1 and C2 receiving the same number of votes. See link below. - _Dennis P. Walsh_, May 08 2013

%C This sequence gives the set of values such that for sequences b(k+1) = a(n)*b(k) - b(k-1), with initial values b(0) = 2, b(1) = a(n), all such sequences are invariant under this transformation: b(k) = (b(j+k) + b(j-k))/b(j), except where b(j) = 0, for all integer values of j and k, including negative values. Examples are: at n=0, b(k) = 2 for all k; at n=1, b(k) = A005248; at n=2, b(k) = 2*A001541; at n=3, b(k)= A057076; at n=4, b(k) = 2*A023039. This b(k) family are also the transformation results for all related b'(k) (i.e., those with different initial values) including non-integer values. Further, these b(k) are also the bisections of the transformations of sequences of the form G(k+1) = n * G(k) + G(k-1), and those bisections are invariant for all initial values of g(0) and g(1), including non-integer values. For n = 1 this g(k) family includes Fibonacci and Lucas, where the invariant bisection is b(k) = A005248. The applicable bisection for this transformation of g(k) is for the odd values of k, and applies for all n. Also see A000032 for a related family of sequences. - _Richard R. Forberg_, Nov 22 2014

%C Also the number of maximum matchings in the n-gear graph. - _Eric W. Weisstein_, Dec 31 2017

%H Harry J. Smith, <a href="/A059100/b059100.txt">Table of n, a(n) for n = 0..1000</a>

%H Hesam Dashti, <a href="http://arxiv.org/abs/1009.5053">A New Upper Bound on the Length of Shortest Permutation Strings; An Algorithm for Generating Permutation Strings</a>, arXiv:1009.5053 [math.CO], Sep 26, 2010. - _Jonathan Vos Post_, Sep 28 2010

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/2tie1vote.pdf">Notes on a tied election</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumIndependentEdgeSet.html">Maximum Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PappusChain.html">Pappus Chain</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (2-3x+3x^2)/(1-x)^3. - _R. J. Mathar_, Oct 05 2008

%F a(n) = ((n-2)^2 + 2*(n+1)^2)/3. - _Reinhard Zumkeller_, Feb 13 2009

%F a(n) = A000196(A156798(n) - A000290(n)). - _Reinhard Zumkeller_, Feb 16 2009

%F a(n) = 2*n+a(n-1)-1 with a(0)=2. - _Vincenzo Librandi_, Aug 07 2010

%F a(n+3) = (A166464(n+5)-A166464(n))/20. - _Paul Curtz_, Nov 07 2012

%F a(3*n+1) = 3*A056109(n). a(3*n+2) = 3*A056105(n+1). - _Paul Curtz_, Nov 07 2012

%e For n=2, a(2)=6 since there are 6 election results in a 4-candidate, 2-voter election that have candidates c1 and c2 tied. Letting <i,j> denote voter 1 voting for candidate i and voter 2 voting for candidate j, the six election results are <1,2>, <2,1>, <3,3>, <3,4>, <4,3>, and <4,4>. - _Dennis P. Walsh_, May 08 2013

%p with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); # _Zerinvary Lajos_, Mar 20 2008

%t Table[n^2 + 2, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 15 2008 *)

%t LinearRecurrence[{3, -3, 1}, {2, 3, 6}, 50] (* _Vincenzo Librandi_, Feb 15 2012 *)

%t Range[0, 20]^2 + 2 (* _Eric W. Weisstein_, Dec 31 2017 *)

%t CoefficientList[Series[(-2 + 3 x - 3 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 31 2017 *)

%o (Sage) [lucas_number1(3,n,-2) for n in xrange(0, 50)] # _Zerinvary Lajos_, May 16 2009

%o (PARI) { for (n = 0, 1000, write("b059100.txt", n, " ", n^2+2); ) } \\ _Harry J. Smith_, Jun 24 2009

%o (Haskell)

%o a059100 = (+ 2) . (^ 2)

%o a059100_list = scanl (+) (2) [1, 3 ..]

%o -- _Reinhard Zumkeller_, Feb 09 2015

%Y Cf. A000290, A002522, A056899. Apart from initial terms, same as A010000.

%Y Cf. A069987, A114964 (see comment).

%Y Cf. A008865. 2nd row/column of A295707.

%K nonn,easy

%O 0,1

%A _Henry Bottomley_, Feb 13 2001

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Last modified February 25 08:45 EST 2018. Contains 299647 sequences. (Running on oeis4.)