Roberto Tauraso - cp's OEIS frontend

A336734 The number of tight 5 X n pavings.

0, 1, 57, 1071, 12279, 106738, 781458, 5111986, 30980370, 178047831, 985621119, 5311715977, 28075774881, 146309927344, 754544640000, 3861338821620, 19646614600164, 99532074868285, 502608221035605, 2531829420822835, 12730273358124315, 63919766245452606

Offset: 0

Roberto Tauraso, Aug 02 2020

This is row (or column) m=5 of the array T in A285357.

D. E. Knuth (Proposer), Problem 12005, Amer. Math. Monthly 124 (No. 8, Oct. 2017), page 755. For the solution see op. cit., 126 (No. 7, 2019), 660-664.
Roberto Tauraso, Problem 12005, Proposed solution.
Index entries for linear recurrences with constant coefficients, signature (31,-432,3580,-19666,75558,-208736,419600,-613605,644771,-473432,230220,-66528,8640).

Cf. A000295 (m=2), A285357, A285361 (m=3), A336732 (m=4).

a(n) = (5^(n+7)+(2*n-66)*4^(n+6)+(16*n^2-1432*n+13164)*3^(n+3) +(303*n-1505)*2^(n+10)+576*n^4+13248*n^3+129936*n^2+646972*n+1377903)/576.

G.f.: (x +26*x^2 -264*x^3 +122*x^4 +4367*x^5 -11668*x^6 +3000*x^7 +11168*x^8 +160*x^9) / ((1-x)^5*(1-2*x)^2*(1-3*x)^3*(1-4*x)^2*(1-5*x)).

A336732 The number of tight 4 X n pavings.

Original entry on oeis.org

0, 1, 26, 282, 2072, 12279, 63858, 305464, 1382648, 6029325, 25628762, 107026662, 441439944, 1804904755, 7334032754, 29669499492, 119647095176, 481400350185, 1933747745850, 7758556171570, 31102292517560, 124605486285231, 498987240470066, 1997573938402512

Offset: 0

Roberto Tauraso, Aug 02 2020

This is row (or column) m=4 of the array T in A285357.

D. E. Knuth (Proposer), Problem 12005, Amer. Math. Monthly 124 (No. 8, Oct. 2017), page 755. For the solution see op. cit., 126 (No. 7, 2019), 660-664.
Roberto Tauraso, Problem 12005, Proposed solution.
Index entries for linear recurrences with constant coefficients, signature (18,-139,604,-1627,2818,-3141,2176,-852,144).

Cf. A000295 (m=2), A285357, A285361 (m=3), A336734 (m=5).

seq((4^(n+5)+(n-42)*3^(n+4)-9*(2*n-27)*2^(n+5)-36*n^3-486*n^2-2577*n-5398)/36,n=0..20);

num=(x+8*x^2-47*x^3+6*x^4+104*x^5); den=((1-x)^4*(1-2*x)^2*(1-3*x)^2*(1-4*x)); CoefficientList[Series[num/den,{x,0,20}],x]

a(n) = (4^(n+5)+(n-42)*3^(n+4)-9*(2*n-27)*2^(n+5)-36*n^3-486*n^2-2577*n-5398)/36.

G.f.: (x+8*x^2-47*x^3+6*x^4+104*x^5)/((1-x)^4*(1-2*x)^2*(1-3*x)^2*(1-4*x)).

A134988 Number of formal expressions obtained by applying iterated binary brackets to n indexed symbols x_1, ..., x_n such that: 1) each symbol appears exactly once; 2) the smallest index inside a bracket appears on the left hand side and the largest index appears on the right hand side; 3) the outer bracket is the only bracket whose set of indices is a sequence of consecutive integers.

Original entry on oeis.org

1, 0, 1, 4, 22, 144, 1089, 9308, 88562, 927584, 10603178, 131368648, 1753970380, 25112732512, 383925637137, 6243618722124, 107644162715098, 1961478594977856, 37671587406585006, 760654555198989240, 16110333600696417780, 357148428086308848480, 8271374327887650503130

Offset: 2

Paolo Salvatore and Roberto Tauraso, Feb 05 2008, Feb 22 2008

nonn

a(n) is the number of generators in arity n of the operad Lie, when considered as a free non-symmetric operad.

Francis Brown and Jonas Bergström, Inversion of series and the cohomology of the moduli spaces m_(0,n)^δ, arXiv:0910.0120 [math.AG], 2009.
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
P. Salvatore and R. Tauraso, The Operad Lie is Free, arXiv:0802.3010 [math.QA], 2008.

Cf. A075834.

terms = 23; F[x_] = Sum[n! x^n, {n, 0, terms+1}]; CoefficientList[(x - InverseSeries[Series[x F[x], {x, 0, terms+1}], x])/x^2, x] (* Jean-François Alcover, Feb 17 2019 *)

N=66;  x='x+O('x^N);
F = sum(n=0,N,x^n*n!);
gf= x - serreverse(x*F);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

a(2) = 1, a(n) = Sum_{k=2..n-2} ((k+1)*a(k+1) + a(k))*a(n-k), n > 2;
G.f.: x - series_reversion(x*F(x)), where F(x) is the g.f. of the factorials (A000142).
a(n) = (1/e)*(1 - 3/n - 5/(2n^2) + O(1/n^3)).

A123304 Number of edge covers for the circular ladder (n-prism graph) C_n X K_2.

Original entry on oeis.org

4, 5, 43, 263, 1699, 10895, 69943, 448943, 2881699, 18497135, 118730023, 762108143, 4891844659, 31399932335, 201550911703, 1293721577903, 8304182337859, 53303156937455, 342144045482503, 2196165379031663, 14096818096762579, 90485116626705455, 580808823292457143

Offset: 0

Roberto Tauraso, Sep 24 2006

An edge covering for a graph is a set of edges so that every vertex is adjacent to at least one edge of this set.

The number of edge coverings for the circle C_n for n>0 is the n-th Lucas number.

R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Prism Graph
Index entries for linear recurrences with constant coefficients, signature (5,9,1,-2).

a[0] = 4; a[1] = 5; a[2] = 43; a[3] = 263; a[n_] := a[n] = 5a[n - 1] + 9a[n - 2] + a[n - 3] - 2a[n - 4]; Table[a[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 26 2006 *)
CoefficientList[ Series[(4 - 15x - 18x^2 - x^3)/((1 + x)*(1 - 6x - 3x^2 + 2x^3)), {x, 0, 19}], x] (* Robert G. Wilson v, Sep 26 2006 *)
Table[(-1)^n + RootSum[2 - 3 # - 6 #^2 + #^3 &, #^n &], {n, 0, 20}] (* Eric W. Weisstein, Mar 29 2017 *)
LinearRecurrence[{5, 9, 1, -2}, {5, 43, 263, 1699}, {0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)

x='x+O('x^99); Vec((4-15*x-18*x^2-x^3)/((1+x)*(1-6*x-3*x^2+2*x^3))) \\ Altug Alkan, Aug 10 2017

a(n) = 5*a(n-1) +9*a(n-2) +a(n-3) -2*a(n-4).

G.f.: (4-15*x-18*x^2-x^3) / ((1+x)*(1-6*x-3*x^2+2*x^3)).

A115418 Define a k-th-power loop of length m>1 to be a circular permutation of the numbers 1 to m such that the sum of any two consecutive numbers is a perfect k-th-power; these numbers are the lengths of the possible k-th-power loops.

Original entry on oeis.org

2, 32, 473, 9641

Offset: 1

Roberto Tauraso, Jan 22 2006

nonn

Cf. A071984, A112663.

A110128 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)| not equal to 2 for all 0

Original entry on oeis.org

1, 1, 2, 4, 16, 44, 200, 1288, 9512, 78652, 744360, 7867148, 91310696, 1154292796, 15784573160, 232050062524, 3648471927912, 61080818510972, 1084657970877416, 20361216987032284, 402839381030339816, 8377409956454452732

Offset: 0

Roberto Tauraso, A. Nicolosi and G. Minenkov, Jul 13 2005

When n is even: 1) Number of ways that n persons seated at a rectangular table with n/2 seats along the two opposite sides can be rearranged in such a way that neighbors are no more neighbors after the rearrangement. 2) Number of ways to arrange n kings on an n X n board, with 1 in each row and column, which are non-attacking with respect to the main four quadrants.
a(n) is also number of ways to place n nonattacking pieces rook + alfil on an n X n chessboard (Alfil is a leaper [2,2]) [From Vaclav Kotesovec, Jun 16 2010]
Note that the conjectured recurrence was based on the 600-term b-file, not the other way round. - N. J. A. Sloane, Dec 07 2022

Rintaro Matsuo, Table of n, a(n) for n = 0..600 (terms up to a(35) from Vaclav Kotesovec)
Manuel Kauers, Guessed recurrence operator of order 24 and degree 64
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS, vol. 6 (2006), paper A11. arXiv:math/0507293.

Cf. A089222, A002464, A117574, A189281.

Column k=2 of A333706.

A formula is given in the Tauraso reference.
Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 4/n + 8/n^2)/e^2.
a(n) ~ exp(-2) * n! * (1 + 4/n + 8/n^2 + 68/(3*n^3) + 242/(3*n^4) + 1692/(5*n^5) + 72802/(45*n^6) + 2725708/(315*n^7) + 16083826/(315*n^8) + 186091480/(567*n^9) + 32213578294/(14175*n^10) + ...), based on the recurrence by Manuel Kauers. - Vaclav Kotesovec, Dec 05 2022

Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jan 01 2008

Terms a(33)-a(35) from Vaclav Kotesovec, Apr 20 2012

cp's OEIS Frontend

User: Roberto Tauraso

A336734 The number of tight 5 X n pavings.

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A336732 The number of tight 4 X n pavings.

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A123304 Number of edge covers for the circular ladder (n-prism graph) C_n X K_2.

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A115418 Define a k-th-power loop of length m>1 to be a circular permutation of the numbers 1 to m such that the sum of any two consecutive numbers is a perfect k-th-power; these numbers are the lengths of the possible k-th-power loops.

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A110128 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)| not equal to 2 for all 0

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