cp's OEIS Frontend

A001710 Order of alternating group A_n, or number of even permutations of n letters.

%I A001710 M2933 N1179 #351 Feb 16 2025 08:32:24
%S A001710 1,1,1,3,12,60,360,2520,20160,181440,1814400,19958400,239500800,
%T A001710 3113510400,43589145600,653837184000,10461394944000,177843714048000,
%U A001710 3201186852864000,60822550204416000,1216451004088320000,25545471085854720000,562000363888803840000
%N A001710 Order of alternating group A_n, or number of even permutations of n letters.
%C A001710 For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
%C A001710 a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - _Chad Brewbaker_, Jan 31 2003
%C A001710 a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - _Chad Brewbaker_, Jan 31 2003
%C A001710 Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - _Emeric Deutsch_, Aug 28 2004
%C A001710 Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - _Simone Severini_, Oct 15 2004
%C A001710 The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - _Jon Perry_, Sep 20 2008
%C A001710 Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - _Gary W. Adamson_, Dec 25 2008
%C A001710 First column of A092582. - _Mats Granvik_, Feb 08 2009
%C A001710 The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - _Johannes W. Meijer_, Oct 20 2009
%C A001710 For n>1: a(n) = A173333(n,2). - _Reinhard Zumkeller_, Feb 19 2010
%C A001710 Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - _Gary W. Adamson_, Aug 02 2010
%C A001710 For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - _Geoffrey Critzer_, Feb 16 2011.
%C A001710 The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - _Johannes W. Meijer_, Apr 20 2011
%C A001710 Also [1, 1] together with the row sums of triangle A162608. - _Omar E. Pol_, Mar 09 2012
%C A001710 a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - _Wolfdieter Lang_, Jun 26 2012
%C A001710 For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - _Stanislav Sykora_, May 10 2014
%C A001710 In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - _Antti Karttunen_, Dec 19 2015
%C A001710 Also (by definition) the independence number of the n-transposition graph. - _Eric W. Weisstein_, May 21 2017
%C A001710 Number of permutations of n letters containing an even number of even cycles. - _Michael Somos_, Jul 11 2018
%C A001710 Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - _Gus Wiseman_, Oct 20 2018
%C A001710 For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - _Noah A Rosenberg_, Feb 11 2019
%C A001710 Also the clique covering number of the n-Bruhat graph. - _Eric W. Weisstein_, Apr 19 2019
%C A001710 a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - _Bridget Tenner_, Jan 16 2020
%C A001710 For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - _Ivan N. Ianakiev_, Mar 11 2020
%C A001710 For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - _Dennis P. Walsh_, May 28 2020
%D A001710 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
%D A001710 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001710 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001710 N. J. A. Sloane, <a href="/A001710/b001710.txt">Table of n, a(n) for n = 0..100</a>
%H A001710 Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, <a href="https://arxiv.org/abs/1812.02955">Mixed restricted Stirling numbers</a>, arXiv:1812.02955 [math.CO], 2018.
%H A001710 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H A001710 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H A001710 Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
%H A001710 Olivier Bodini, Antoine Genitrini, and Mehdi Naima, <a href="https://arxiv.org/abs/1808.08376">Ranked Schröder Trees</a>, arXiv:1808.08376 [cs.DS], 2018.
%H A001710 Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
%H A001710 Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
%H A001710 Camille Combe and Samuele Giraudo, <a href="https://arxiv.org/abs/2106.14552">Cliff operads: a hierarchy of operads on words</a>, arXiv:2106.14552 [math.CO], 2021.
%H A001710 Mareike Fischer, <a href="https://doi.org/10.1007/s00026-021-00539-2">Extremal Values of the Sackin Tree Balance Index</a>, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
%H A001710 Hannah Golab, <a href="http://danaernst.com/scholarship/GolabThesis.pdf">Pattern avoidance in Cayley permutations</a>, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
%H A001710 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=262">Encyclopedia of Combinatorial Structures 262</a>.
%H A001710 Milan Janjic, <a href="http://web.archive.org/web/20150919034515/http://www.pmfbl.org/janjic/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A001710 Shirali Kadyrov and Farukh Mashurov, <a href="https://arxiv.org/abs/1912.03214">Generalized continued fraction expansions for Pi and E</a>, arXiv:1912.03214 [math.NT], 2019.
%H A001710 Chanchal Kumar and Amit Roy, <a href="https://arxiv.org/abs/2003.1009">Integer Sequences and Monomial Ideals</a>, arXiv:2003.10098 [math.CO], 2020.
%H A001710 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H A001710 Xah Lee, <a href="http://xahlee.org/MathGraphicsGallery_dir/Combinatorics_dir/loopNPoints.html">Combinatorics: Loop in n points</a>.
%H A001710 D. S. Mitrinovic and M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
%H A001710 Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
%H A001710 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
%H A001710 S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, <a href="http://dx.doi.org/10.1016/S0024-3795(03)00382-3">Extremes of permanents of (0,1)-matrices</a>, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
%H A001710 A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%H A001710 B. E. Tenner, <a href="https://arxiv.org/abs/2001.05011">Interval structures in the Bruhat and weak orders</a>, arXiv:2001.05011 [math.CO], 2020.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BruhatGraph.html">Bruhat Graph</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircularPermutation.html">Circular Permutation</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering Number</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EvenPermutation.html">Even Permutation</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddPermutation.html">Odd Permutation</a>.
%H A001710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TranspositionGraph.html">Transposition Graph</a>.
%H A001710 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 7.
%H A001710 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.
%H A001710 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H A001710 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>.
%H A001710 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A001710 a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
%F A001710 D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - _Chad Brewbaker_, Jan 31 2003 [Corrected by _N. J. A. Sloane_, Jul 25 2008]
%F A001710 a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - _Amarnath Murthy_, Oct 29 2002
%F A001710 Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - _Michael Somos_, Mar 04 2004
%F A001710 First Eulerian transform of A000027. See A000142 for definition of FET. - _Ross La Haye_, Feb 14 2005
%F A001710 From _Paul Barry_, Apr 18 2005: (Start)
%F A001710 a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
%F A001710 T(n,k) = abs(A008276(n, k)). (End)
%F A001710 E.g.f.: (2 - x^2)/(2 - 2*x).
%F A001710 E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
%F A001710 E.g.f.: 1 + sinh(log(1/(1-x))). - _Geoffrey Critzer_, Dec 12 2010
%F A001710 a(n+1) = (-1)^n * A136656(n,1), n>=1.
%F A001710 a(n) = n!/2 for n>=2 (proof from the e.g.f). - _Wolfdieter Lang_, Apr 30 2010
%F A001710 a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - _Gary Detlefs_, May 21 2010
%F A001710 a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - _Gary Detlefs_, May 24 2010
%F A001710 O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - _Paul D. Hanna_, Sep 13 2011
%F A001710 a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - _Peter Luschny_, Nov 07 2011
%F A001710 a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - _Mircea Merca_, Apr 07 2012
%F A001710 a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - _Wolfdieter Lang_, Jun 26 2012
%F A001710 G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Dec 26 2012.
%F A001710 G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013
%F A001710 G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - _Sergei N. Gladkovskii_, May 15 2013
%F A001710 G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013
%F A001710 G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013
%F A001710 G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 26 2013
%F A001710 From _Antti Karttunen_, Dec 19 2015: (Start)
%F A001710 a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
%F A001710 For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
%F A001710 Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - _Anton Zakharov_, Aug 07 2016
%F A001710 a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - _Ilya Gutkovskiy_, Aug 07 2016
%F A001710 a(n) = A006595(n-1)*n/A000124(n) for n>=2. - _Anton Zakharov_, Aug 23 2016
%F A001710 a(n) = A001563(n-1) - A001286(n-1) for n>=2. - _Anton Zakharov_, Sep 23 2016
%F A001710 From _Peter Bala_, May 24 2017: (Start)
%F A001710 The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
%F A001710 G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
%F A001710 A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
%F A001710 H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - _Tom Copeland_, Dec 27 2019
%F A001710 From _Amiram Eldar_, Jan 08 2023: (Start)
%F A001710 Sum_{n>=0} 1/a(n) = 2*(e-1).
%F A001710 Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)
%e A001710 G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...
%p A001710 seq(mul(k, k=3..n), n=0..20); # _Zerinvary Lajos_, Sep 14 2007
%t A001710 a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
%t A001710 a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* _Robert G. Wilson v_, Apr 16 2011 *)
%t A001710 a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* _Michael Somos_, May 22 2014 *)
%t A001710 a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* _Michael Somos_, May 22 2014 *)
%t A001710 Numerator[Range[0, 20]!/2] (* _Eric W. Weisstein_, May 21 2017 *)
%t A001710 Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* _Eric W. Weisstein_, May 21 2017 *)
%o A001710 (Magma) [1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // _Arkadiusz Wesolowski_, May 17 2014
%o A001710 (PARI) {a(n) = if( n<2, n>=0, n!/2)};
%o A001710 (PARI) a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ _Paul D. Hanna_
%o A001710 (PARI) A001710=n->n!\2+(n<2) \\ _M. F. Hasler_, Dec 01 2013
%o A001710 (Scheme) ;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
%o A001710 (definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
%o A001710 ;; _Antti Karttunen_, Dec 19 2015
%o A001710 (Python)
%o A001710 from math import factorial
%o A001710 def A001710(n): return factorial(n)>>1 if n > 1 else 1 # _Chai Wah Wu_, Feb 14 2023
%o A001710 (SageMath)
%o A001710 def A001710(n): return (factorial(n) +int(n<2))//2
%o A001710 [A001710(n) for n in range(31)] # _G. C. Greubel_, Sep 28 2024
%Y A001710 Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
%Y A001710 a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
%Y A001710 Bisections are A002674 and A085990 (essentially).
%Y A001710 Row 3 of A265609 (essentially).
%Y A001710 Row sums of A307429.
%K A001710 nonn,easy,nice
%O A001710 0,4
%A A001710 _N. J. A. Sloane_
%E A001710 More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
%E A001710 Further terms from _Simone Severini_, Oct 15 2004