A002524 -id:A002524 - cp's OEIS frontend

A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 5, 1, 1, 1, 2, 6, 14, 8, 1, 1, 1, 2, 6, 24, 31, 13, 1, 1, 1, 2, 6, 24, 78, 73, 21, 1, 1, 1, 2, 6, 24, 120, 230, 172, 34, 1, 1, 1, 2, 6, 24, 120, 504, 675, 400, 55, 1, 1, 1, 2, 6, 24, 120, 720, 1902, 2069, 932, 89, 1, 1, 1, 2, 6, 24, 120, 720, 3720, 6902, 6404, 2177, 144, 1

Offset: 0

Alois P. Heinz, Jan 29 2019

A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].

			A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.
A(5,2) = 31: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14253, 14325, 14523, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 34125.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  2,   2,    2,    2,     2,     2,     2,     2, ...
  1,  3,   6,    6,    6,     6,     6,     6,     6, ...
  1,  5,  14,   24,   24,    24,    24,    24,    24, ...
  1,  8,  31,   78,  120,   120,   120,   120,   120, ...
  1, 13,  73,  230,  504,   720,   720,   720,   720, ...
  1, 21, 172,  675, 1902,  3720,  5040,  5040,  5040, ...
  1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104.

Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.
Rows n=1-2 give: A000012, A040000.
Main diagonal gives A000142.
A(2n,n) gives A048163(n+1).
A(2n+1,n) gives A092552(n+1).
A(n,floor(n/2)) gives A306267.
A(n+2,n) gives A001564.
Cf. A130152.

A[0, _] = 1;
A[n_ /; n > 0, k_] := A[n, k] = Permanent[Table[If[Abs[i - j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1 }] // Flatten (* Jean-François Alcover, Oct 18 2021, after Alois P. Heinz in A130152 *)

A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.

A079977 Fibonacci numbers interspersed with zeros.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, 13, 0, 21, 0, 34, 0, 55, 0, 89, 0, 144, 0, 233, 0, 377, 0, 610, 0, 987, 0, 1597, 0, 2584, 0, 4181, 0, 6765, 0, 10946, 0, 17711, 0, 28657, 0, 46368, 0, 75025, 0, 121393, 0, 196418, 0, 317811, 0, 514229, 0, 832040, 0, 1346269

Offset: 0

Vladimir Baltic, Feb 17 2003

Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={0,2}.
Number of compositions of n into elements of the set {2,4}.
a(n-2) is the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 3. - Ethan Patrick White, Jun 24 2020

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
Ethan P. White, Richard K. Guy, and Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A002524, A002525, A002526, A002527, A002528, A002529, A072827.
Cf. A072850, A072851, A072852, A072853, A072854, A072855, A072856, A099574.
Cf. A079955 - A080014.

A079977:= func< n | (1+(-1)^n)*Fibonacci(Floor((n+2)/2))/2 >;
[A079977(n): n in [0..50]]; // G. C. Greubel, Jul 25 2022

Riffle[Fibonacci[Range[50]],0] (* Harvey P. Dale, Dec 20 2015 *)

a(n)=if(n%2,0,fibonacci(n/2+1)) \\ Charles R Greathouse IV, Jun 11 2015

def A079977(n): return ((n+1)%2)*fibonacci((n+2)//2)
[A079977(n) for n in (0..50)] # G. C. Greubel, Jul 25 2022

a(n) = A000045(k+1) if n=2k, a(n)=0 otherwise.
a(n) = a(n-2) + a(n-4).
G.f.: 1/(1 - x^2 - x^4).

Editorial note: normally the alternate zeros are omitted from sequences like this. This entry is an exception. - N. J. A. Sloane

A079962 Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 14, 22, 36, 58, 94, 153, 247, 399, 646, 1045, 1691, 2737, 4428, 7164, 11592, 18756, 30348, 49105, 79453, 128557, 208010, 336567, 544577, 881145, 1425722, 2306866, 3732588, 6039454, 9772042, 15811497, 25583539, 41395035

Offset: 0

Vladimir Baltic, Feb 19 2003

Number of compositions (ordered partitions) of n into elements of the set {1,3,5,6}. - Mark Dols, Aug 20 2010

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,1).

Cf. A002524, A002525, A002526, A002527, A002528, A002529, A072827, A072850, A072851, A072852, A072853, A072854, A072855, A072856, A079955 - A080014.

[Round(Fibonacci(n+3)/4): n in [0..40]]; // G. C. Greubel, Jan 21 2022

with(combinat,fibonacci): seq(round(fibonacci(n+3)/4),n=0..38) # Mircea Merca, Jan 04 2011

LinearRecurrence[{1,0,1,0,1,1}, {1,1,1,2,3,5}, 41] (* G. C. Greubel, Jan 21 2022 *)

a(n)=fibonacci(n+3)\/4 \\ Charles R Greathouse IV, Oct 07 2015

[(1/4)*(fibonacci(n+3) + chebyshev_U(n,1/2) + chebyshev_U(2*n,1/2)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6).
G.f.: 1/((1+x+x^2)*(1-x+x^2)*(1-x-x^2)).
a(n+1)/a(n) -> golden ratio A001622. - Roger L. Bagula, Mar 13 2006
a(n) + a(n+2) + a(n+4) = Fibonacci(n+5). - Mark Dols, Aug 20 2010
a(n) = round(Fibonacci(n+3)/4). - Mircea Merca, Jan 04 2011
a(n+6) - a(n) = A000045(n+6). - Paul Curtz, Jun 29 2013
a(n) + a(n+1) + a(n+2) = A024490(n+6). - R. J. Mathar, Jun 30 2013
a(n) - a(n-1) + a(n-2) = A094686(n). - R. J. Mathar, Jun 30 2013
4*a(n) = A057078(n) + A010892(n) + A000045(n+3). - R. J. Mathar, Nov 02 2016

A154654 Number of permutations of length n within distance 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 3720, 17304, 76110, 329462, 1441923, 6487445, 29555588, 135025756, 615260976, 2791161792, 12618600768, 57008446080, 257708989200, 1166042944564, 5279435858788, 23908888017477, 108262665958797, 490132089640318, 2218641353956314

Offset: 0

Torleiv Kløve, Jan 13 2009

a(n) equals the permanent of the n X n matrix with 1's along the eleven central diagonals and 0's everywhere else. - John M. Campbell, Jul 10 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..400 from R. H. Hardin)
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Cf. A000045, A002524, A002526, A072856.

Column k=5 of A306209.

G.f. is a rational function f(x)/g(x) where f has degree 132 and g has degree 142.

a(0)=1 prepended by Alois P. Heinz, Jan 28 2019

A224808 Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i not in the set I, i=1..n, with k=2, r=6, I={-1,1,2,3,4,5}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 35, 49, 70, 100, 140, 196, 266, 361, 494, 676, 936, 1296, 1800, 2500, 3450, 4761, 6555, 9025, 12445, 17161, 23711, 32761, 45250, 62500, 86250, 119025, 164220, 226576, 312732, 431649, 595899, 822649, 1135564, 1567504, 2163456, 2985984

Offset: 0

Vladimir Baltic, Apr 18 2013

a(n) is the number of subsets of {1,2,...,n-6} without differences equal to 2, 4 or 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1,1,2,-1,1,0,0,-1,0,0, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694.

CoefficientList[Series[(1 - x^5 - x^8)/(1 - x - x^5 + x^6 - x^7 - 2*x^8 + x^9 - x^10 + x^13 + x^16), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)
LinearRecurrence[{1,0,0,0,1,-1,1,2,-1,1,0,0,-1,0,0,-1},{1,1,1,1,1,1,1,2,4,6,9,12,16,20,25,35},60] (* Harvey P. Dale, Dec 02 2024 *)

x='x+O('x^66); Vec((1-x^5-x^8)/(1-x-x^5+x^6-x^7-2*x^8+x^9-x^10+x^13+x^16) ) \\ Joerg Arndt, Apr 19 2013

a(n) = a(n-1) + a(n-5) - a(n-6) + a(n-7) + 2*a(n-8) - a(n-9) + a(n-10) - a(n-13) + a(n-16).
G.f.: (1-x^5-x^8)/(1-x-x^5+x^6-x^7-2*x^8+x^9-x^10+x^13+x^16).
a(2*k-2) = (A003269(k))^2,
a(2*k-1) = A003269(k) * A003269(k+1)

A006500 Restricted combinations.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 27, 45, 75, 125, 200, 320, 512, 832, 1352, 2197, 3549, 5733, 9261, 14994, 24276, 39304, 63580, 102850, 166375, 269225, 435655, 704969, 1140624, 1845504, 2985984, 4831488, 7817616, 12649337, 20466953, 33116057, 53582633

Offset: 0

N. J. A. Sloane

a(n)=( A000045(k+2) )^3 if n=3k, a(n)=( A000045(k+2) )^3 * A000045(k+3) if n=3k+1, a(n)= A000045(k+2) * ( A000045(k+3) )^2 if n=3k+2. Number of all subsets of the set {1,2,...,n} which do not contain two elements whose difference is 3. a(n) is number of compositions of n+3 into elements of the set {1,2,4,5,6}, but with condition that 2 succeed only 2 or 4. Number of all permutations of {1,2,...,n+3} satisfying p(i)-i in {-3,0,3}. - Vladimir Baltic, Feb 17 2003

			For example, a_4=12 and 12 subsets are: emptyset, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {2,3}, {2,4}, {3,4}, {1,2,3}, {2,3,4}. Corresponding compositions of 7=4+3 are: 1+1+1+1+1+1+1+1, 4+1+1+1, 1+4+1+1, 1+1+4+1, 1+1+1+4, 5+1+1, 4+2+1, 1+5+1, 1+4+2, 1+1+5, 6+1 and 1+6.

M. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461 doi:10.1016/j.amc.2009.12.069, Table 1 k=3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 16.
G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1,1,-1,-1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

A006500:=-(2*z**6+z**7-z**4+z**5-3*z**3-z**2-z-1)/(z**6-z**3-1)/(z**2+z-1); # Conjectured by Simon Plouffe in his 1992 dissertation.

Table[Fibonacci[Floor[n/3] + 3]^Mod[n, 3] * Fibonacci[Floor[n/3] + 2]^(3 - Mod[n, 3]), {n, 0, 40}]  (* David Nacin, Feb 29 2012 *)
Table[Product[Fibonacci[Floor[(n + i)/3] + 2], {i, 0, 2}], {n, 0, 30}] (* David Nacin, Mar 07 2012 *)
LinearRecurrence[{1, 1, -1, 1, 1, 1, -1, -1}, {1, 2, 4, 8, 12, 18, 27, 45}, 40] (* David Nacin, Mar 07 2012 *)

def a(n, adict={0:1, 1:2, 2:4, 3:8, 4:12, 5:18, 6:27, 7:45}):
    if n in adict:
        return adict[n]
    adict[n]=a(n-1)+a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)-a(n-7)-a(n-8)
    return adict[n] # David Nacin, Mar 07 2012

Recurrence: a(n) = a(n-1)+a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)-a(n-7)-a(n-8) G.f.: -(x^7+2*x^6+x^5-x^4-3*x^3-x^2-x-1)/(x^8+x^7-x^6-x^5-x^4+x^3-x^2-x+1). - Vladimir Baltic, Feb 17 2003

a(n) = F(floor(n/3) + 3)^(n mod 3)*F(floor(n/3) + 2)^(3 - (n mod 3)) where F(n) is the n-th Fibonacci number. - David Nacin, Feb 29 2012

A224809 Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i not in the set I, i=1..n, with k=2, r=4, I={-1,1,2,3}.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 24, 36, 54, 81, 117, 169, 247, 361, 532, 784, 1148, 1681, 2460, 3600, 5280, 7744, 11352, 16641, 24381, 35721, 52353, 76729, 112462, 164836, 241570, 354025, 518840, 760384, 1114416, 1633284

Offset: 0

Vladimir Baltic, May 16 2013

nonn

Number of subsets of {1,2,...,n-4} without differences equal to 2 or 4.

Gheorghe Coserea, Table of n, a(n) for n = 0..4096
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,1,0,0,-1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808.

CoefficientList[Series[-(x-1)*(1+x+x^2)/((x^3+x-1)*(x^6-x^4-1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

N = 42; x = 'x + O('x^N);
Vec(Ser(-(x-1)*(1+x+x^2)/((x^3+x-1)*(x^6-x^4-1))))  \\ Gheorghe Coserea, Nov 11 2016

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-9).
G.f.: -(x-1)*(1+x+x^2) / ( (x^3+x-1)*(x^6-x^4-1) ).
a(2*k) = (A000930(k))^2, a(2*k+1) = A000930(k) * A000930(k+1).

A079997 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={0}.

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 57, 140, 376, 1016, 2692, 7020, 18369, 48344, 127465, 335510, 882081, 2319136, 6100393, 16049440, 42220168, 111053856, 292109320, 768373144, 2021186393, 5316647448, 13985104873, 36786882378, 96765680857, 254536684328

Offset: 0

Vladimir Baltic, Feb 17 2003

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1, 3, 0, 6, 10, 0, -12, -10, -2, 0, 0, -1, 1, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.
Column k=3 of A259776.
Cf. A260081.

LinearRecurrence[{1,3,0,6,10,0,-12,-10,-2,0,0,-1,1,1},{1,0,1,2,9,24,57,140,376,1016,2692,7020,18369,48344},40] (* Harvey P. Dale, Nov 27 2013 *)

a(n) = a(n-1)+3*a(n-2)+6*a(n-4)+10*a(n-5)-12*a(n-7)-10*a(n-8)-2*a(n-9)-a(n-12)+a(n-13)+a(n-14)

G.f.: -(x^8+x^7-x^5-2*x^4+x^3-2*x^2-x+1)/(x^14 +x^13 -x^12 -2*x^9 -10*x^8 -12*x^7 +10*x^5 +6*x^4 +3*x^2 +x-1).

A188958 T(n,k)=Number of nXk array permutations with each element moved no more than a city block distance of two.

Original entry on oeis.org

1, 2, 2, 6, 24, 6, 14, 362, 362, 14, 31, 3969, 29172, 3969, 31, 73, 45288, 1722256, 1722256, 45288, 73, 172, 532224, 104735558, 504690017, 104735558, 532224, 172, 400, 6187928, 6513224285, 151700751624, 151700751624, 6513224285, 6187928, 400

Offset: 1

R. H. Hardin Apr 14 2011

Table starts
....1.........2..............6................14.................31
....2........24............362..............3969..............45288
....6.......362..........29172...........1722256..........104735558
...14......3969........1722256.........504690017.......151700751624
...31.....45288......104735558......151700751624....229188213745576
...73....532224.....6513224285....47085945014364.359214228037483580
..172...6187928...402549615544.14529766291673016
..400..71851073.24857280209008
..932.835579642
.2177

			Some solutions for 5X3
..0..3..1....0..3..1....0..3..1....0..3..1....0..3..1....0..3..1....0..3..1
..5..2.11....5..2..4....5..2..4....5..6..2....5..2..7....5..6..8....5..2.11
..4..7..8....7..9.10....6..8.14....9.13..8....4.11..8....4.13..2....4..8..7
.10.13.14....6..8.11...11..7..9....7..4.14....6.12.14...11.14..7....6.12.10
..6..9.12...12.14.13...10.13.12...12.10.11...10..9.13...10..9.12...14..9.13

R. H. Hardin, Table of n, a(n) for n = 1..59

Column 1 is A002524

A072852 Number of permutations satisfying i-2<=p(i)<=i+5, i=1..n.

Original entry on oeis.org

1, 2, 6, 18, 54, 162, 454, 1267, 3613, 10344, 29572, 84436, 240868, 686884, 1959636, 5592181, 15957717, 45533682, 129922090, 370708166, 1057755082, 3018154342, 8611878218, 24572725639, 70114579881, 200061418144, 570845362600

Offset: 1

Vladimir Baltic, Jul 25 2002

nonn

R. H. Hardin, Table of n, a(n) for n=1..400
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1, 2, 4, 7, 13, 22, 28, -4, -6, -6, 0, -4, -10, -10, 0, 1, 1, 0, 0, 1, 1).

Cf. A002524..A002529, A072827, A072850..A072856, A079955..A080014.

CoefficientList[Series[-(x^14+x^12+x^10+x^8-6x^7-x^6-4x^5-3x^4-2x^3-x^2+1)/(x^21+x^20+ x^17+x^16- 10x^14-10x^13-4x^12-6x^10- 6x^9-4x^8+28x^7+22x^6+ 13x^5+7x^4+4x^3+ 2x^2+x-1),{x,0,30}],x] (* or *) LinearRecurrence[{1,2,4,7,13,22,28,-4,-6,-6,0,-4,-10,-10,0,1,1,0,0,1,1},{1,1,2,6,18,54,162,454,1267,3613,10344,29572,84436,240868,686884,1959636,5592181,15957717,45533682,129922090,370708166},30] (* Harvey P. Dale, Jul 28 2024 *)

Recurrence: a(n) = a(n - 1) + 2*a(n - 2) + 4*a(n - 3) + 7*a(n - 4) + 13*a(n - 5) + 22*a(n - 6) + 28*a(n - 7) - 4*a(n - 8) - 6*a(n - 9) - 6*a(n - 10) - 4*a(n - 12) - 10*a(n - 13) - 10*a(n - 14) + a(n - 16) + a(n - 17) + a(n - 20) + a(n - 21). G.f.: - (x^14 + x^12 + x^10 + x^8 - 6*x^7 - x^6 - 4*x^5 - 3*x^4 - 2*x^3 - x^2 + 1)/(x^21 + x^20 + x^17 + x^16 - 10*x^14 - 10*x^13 - 4*x^12 - 6*x^10 - 6*x^9 - 4*x^8 + 28*x^7 + 22*x^6 + 13*x^5 + 7*x^4 + 4*x^3 + 2*x^2 + x - 1);

cp's OEIS Frontend

A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A079977 Fibonacci numbers interspersed with zeros.

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A079962 Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.

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A154654 Number of permutations of length n within distance 5.

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A224808 Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i not in the set I, i=1..n, with k=2, r=6, I={-1,1,2,3,4,5}.

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A006500 Restricted combinations.

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A224809 Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i not in the set I, i=1..n, with k=2, r=4, I={-1,1,2,3}.

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