A081908 -id:A081908 - cp's OEIS frontend

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

Henry Bottomley, Feb 13 2001

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

			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

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).

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.

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

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

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 *)

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

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

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

A081909 a(n) = 3^n(n^2 - n + 18)/18.

Original entry on oeis.org

1, 3, 10, 36, 135, 513, 1944, 7290, 26973, 98415, 354294, 1259712, 4428675, 15411789, 53144100, 181752822, 617003001, 2080591515, 6973568802, 23245229340, 77096677311, 254535261273, 836828256240, 2740612539186

Offset: 0

Paul Barry, Mar 31 2003

Binomial transform of A081908. 3rd binomial transform of (1,0,1,0,0,0,...). Case k=3 where a(n,k) = k^n*(n^2 - n + 2k^2)/(2k^2) with g.f. (1 - 2kx + (k^2+1)x^2)/(1-kx)^3.

a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used exactly twice. - Enrique Navarrete, Mar 29 2024

			a(2)=10 since the number of words of length 2 defined on {0,1,2,3} that don't use 0 or use it twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 00. - _Enrique Navarrete_, Mar 29 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..185
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A006234, A081910.

[3^n*(n^2-n+18)/18: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

Table[3^n(n^2-n+18)/18,{n,0,30}] (* or *) CoefficientList[Series[ (1-6x+10x^2)/(1-3x)^3,{x,0,30}],x]  (* Harvey P. Dale, Apr 26 2011 *)

a(n) = 3^n*(n^2 - n + 18)/18.
G.f.: (1 - 6x + 10x^2)/(1-3x)^3.
E.g.f.: exp(3*x)*(1+x^2/2). - Enrique Navarrete, Mar 29 2024

A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.

Original entry on oeis.org

1, 2, 5, 16, 28, 96, 160, 512, 896, 2560, 4864, 12288, 25600, 57344, 131072, 262144, 655360, 1179648, 3211264, 5242880, 15466496, 23068672, 73400320, 100663296, 343932928, 436207616, 1593835520, 1879048192, 7314866176, 8053063680, 33285996544, 34359738368

Offset: 0

Peter Luschny, Oct 01 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A327999, A328001.

Cf. A056040, A002699, A001477, A081908, A145018.

[IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // Marius A. Burtea, Feb 05 2020

swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);

A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* Harvey P. Dale, Jun 19 2022 *)

x='x + O('x^32);
Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))

Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ Colin Barker, Feb 05 2020

a(n) = Sum_{k=0..n} s(k)*s(n-k) where s(n) = A056040(n).
a(n) = [x^n] (4*x^2 - x - 1)^2 / (1 - 4*x^2)^3.
a(n) = 2^(n - 5)*(n*(n + 2) + 32) if n even else 2^(n - 1)*(n + 1).
a(2*n) = A327999(n).
a(2*n-1) = A002699(n), (with a(-1) = 0).
a(2^n-1) = 2^(2^n - 2 + n) for n >= 1.
2*a(2*n)/2^n = A081908(n+1).
4*a(2*n)/4^n = A145018(n+1).
2*a(2*n-1)/4^n = A001477(n).
From Stefano Spezia, Oct 19 2019: (Start)
a(n) = n! [x^n] (1/32)*exp(-2*x)*(8 + exp(4*x)*(8 + x)*(3 + 2*x) + x*(13 + 2*x)).
a(n) = 12*a(n-2) - 48*a(n-4) + 64*a(n-6) for n > 5. (End)

A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).

Original entry on oeis.org

1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

First difference yields A295288.
1 and 7 are the only odd terms.
a(n) gives the number of words of length n defined over the alphabet {a,b,c,d} such that letters from {a,b} are only used in pairs of at most one, and consist of (a,a), (a,b) and (b,a).

			a(4) = 88. The corresponding words are cccc, cccd, ccdc, ccdd, cdcc, cdcd, cddc, cddd, dccc, dccd, dcdc, dcdd, ddcc, ddcd, dddc, dddd, caac, caca, ccaa, caad, cada, caad, cabc, cacb, ccab, cabd, cadb, cabd, cbac, cbca, ccba, cbad, cbda, cbad, daac, daca, dcaa, daad, dada, daad, dabc, dacb, dcab, dabd, dadb, dabd, dbac, dbca, dcba, dbad, dbda, dbad, aacc, acac, acca, aacd, acad, acda, aadc, adac, adca, aadd, adad, adda, abcc, acbc, accb, abcd, acbd, acdb, abdc, adbc, adcb, abdd, adbd, addb, bacc, bcac, bcca, bacd, bcad, bcda, badc, bdac, bdca, badd, bdad, bdda.

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hermann Gruber, Jonathan Lee and Jeffrey Shallit, Enumerating regular expressions and their languages, arXiv preprint arXiv:1204.4982 [cs.FL], 2012.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.

List([0..30],n->(3*n^2-3*n+8)*2^(n-3)); # Muniru A Asiru, Mar 09 2018

[(3*n^2-3*n+8)*2^(n-3): n in [0..30]]; // Vincenzo Librandi, Mar 10 2018

A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
seq(A(n), n = 0 .. 70);

Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
LinearRecurrence[{6, -12, 8}, {1, 2, 7}, 30] (* Robert G. Wilson v, Mar 07 2018 *)

makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);

a(n) = (3*n^2-3*n+8)*2^(n-3); \\ Altug Alkan, Mar 09 2018

G.f.:  (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
a(n) = 3*A001788(n) + A000079(n).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), for n >= 3, with a(0) = 1, a(1) = 2 and a(2) = 7.
a(n) = A300184(n,2).

cp's OEIS Frontend

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

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A081909 a(n) = 3^n(n^2 - n + 18)/18.

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A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.

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A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).

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A081918 a(0) = 1; a(n) = n^(n-1)(3n-1)/2 (n>0).

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A328000 a(n) = Sum_{k=0..n}(k!(n - k)!)/(floor(k/2)!floor((n - k)/2)!)^2.

A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).