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

2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502, 2603

Offset: 0

Author

Henry Bottomley, Feb 13 2001

Keywords

• nonn
• easy

Comments

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

Binomial transformation yields A081908, with A081908(0)=1 dropped. - R. J. Mathar, Oct 05 2008

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

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

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

Also the number of maximum matchings in the n-gear graph. - Eric W. Weisstein, Dec 31 2017

Also the Wiener index of the n-dipyramidal graph. - Eric W. Weisstein, Jun 14 2018

Numbers of the form n^2+2 have no factors that are congruent to 7 (mod 8). - Gordon E. Michaels, Sep 12 2019

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {n, 2n}]. - Magus K. Chu, Sep 10 2022

Examples

			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
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 52, 56.
• Hesam Dashti, A New Upper Bound on the Length of Shortest Permutation Strings; An Algorithm for Generating Permutation Strings, arXiv:1009.5053 [math.CO], 2010. - _Jonathan Vos Post_, Sep 28 2010
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Alexander Soifer, Coffee Hour and the Conway-Soifer Cover-Up, In: How Does One Cut a Triangle? (2009), pp. 147-156. See also here
• Dennis P. Walsh, Notes on a tied election.
• Eric Weisstein's World of Mathematics, Dipyramidal Graph.
• Eric Weisstein's World of Mathematics, Gear Graph.
• Eric Weisstein's World of Mathematics, Matching.
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
• Eric Weisstein's World of Mathematics, Near-Square Prime.
• Eric Weisstein's World of Mathematics, Pappus Chain.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000032, A000196, A000290, A001541, A002522, A005248, A008865, A056105, A056109, A056899, A057076, A069987, A081908, A114964, A156798, A166464.

Apart from initial terms, same as A010000.

2nd row/column of A295707.

Programs

Haskell

a059100 = (+ 2) . (^ 2)
a059100_list = scanl (+) (2) [1, 3 ..]
-- Reinhard Zumkeller, Feb 09 2015

Maple

with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); # Zerinvary Lajos, Mar 20 2008

Mathematica

Table[n^2 + 2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)
LinearRecurrence[{3, -3, 1}, {2, 3, 6}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
Range[0, 20]^2 + 2 (* Eric W. Weisstein, Dec 31 2017 *)
CoefficientList[Series[(-2 + 3 x - 3 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 31 2017 *)

PARI

a(n) = { n^2+2 } \\ Harry J. Smith, Jun 24 2009

Sage

[lucas_number1(3,n,-2) for n in range(0, 50)] # Zerinvary Lajos, May 16 2009

Formula

G.f.: (2 - 3*x + 3*x^2)/(1 - x)^3. - R. J. Mathar, Oct 05 2008

a(n) = ((n - 2)^2 + 2*(n + 1)^2)/3. - Reinhard Zumkeller, Feb 13 2009

a(n) = A000196(A156798(n) - A000290(n)). - Reinhard Zumkeller, Feb 16 2009

a(n) = 2*n + a(n-1) - 1 with a(0) = 2. - Vincenzo Librandi, Aug 07 2010

a(n+3) = (A166464(n+5) - A166464(n))/20. - Paul Curtz, Nov 07 2012

From Paul Curtz, Nov 07 2012: (Start)

a(3*n) mod 9 = 2.

a(3*n+1) = 3*A056109(n).

a(3*n+2) = 3*A056105(n+1). (End)

Sum_{n >= 1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018

From Amiram Eldar, Jan 29 2021: (Start)

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*(csch(sqrt(2)*Pi)))/4.

Product_{n>=0} (1 + 1/a(n)) = sqrt(3/2)*csch(sqrt(2)*Pi)*sinh(sqrt(3)*Pi).

Product_{n>=0} (1 - 1/a(n)) = csch(sqrt(2)*Pi)*sinh(Pi)/sqrt(2). (End)

E.g.f.: exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024