|
%I #129 Jun 19 2023 14:38:53
%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 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
%C Also the Wiener index of the n-dipyramidal graph. - _Eric W. Weisstein_, Jun 14 2018
%C Numbers of the form n^2+2 have no factors that are congruent to 7 (mod 8). - _Gordon E. Michaels_, Sep 12 2019
%C For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {n, 2n}]. - _Magus K. Chu_, Sep 10 2022
%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], 2010. - _Jonathan Vos Post_, Sep 28 2010
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Alexander Soifer, <a href="https://doi.org/10.1007/978-0-387-74652-4_15">Coffee Hour and the Conway-Soifer Cover-Up</a>, In: How Does One Cut a Triangle? (2009), pp. 147-156. See also <a href="https://twitter.com/latzplacian/status/1267134639040352257">here</a>
%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/DipyramidalGraph.html">Dipyramidal Graph</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</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 - 3*x + 3*x^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 From _Paul Curtz_, Nov 07 2012: (Start)
%F a(3*n) mod 9 = 2.
%F a(3*n+1) = 3*A056109(n).
%F a(3*n+2) = 3*A056105(n+1). (End)
%F Sum_{n >= 1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - _Vaclav Kotesovec_, May 01 2018
%F From _Amiram Eldar_, Jan 29 2021: (Start)
%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*(csch(sqrt(2)*Pi))/4.
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(3/2)*csch(sqrt(2)*Pi)*sinh(sqrt(3)*Pi).
%F Product_{n>=0} (1 - 1/a(n)) = csch(sqrt(2)*Pi)*sinh(Pi)/sqrt(2). (End)
%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 range(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. A000032, A000196, A000290, A001541, A002522, A005248, A008865, A056105, A056109, A056899, A057076, A069987, A114964, A156798, A166464.
%Y Apart from initial terms, same as A010000.
%Y 2nd row/column of A295707.
%K nonn,easy
%O 0,1
%A _Henry Bottomley_, Feb 13 2001
|
