A002528 -id:A002528 - cp's OEIS frontend

A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.

1, 2, 6, 18, 46, 115, 301, 792, 2068, 5380, 14020, 36581, 95413, 248786, 648714, 1691686, 4411530, 11503991, 29998953, 78228640, 203998184, 531969064, 1387222648, 3617479225, 9433351129, 24599481138, 64148406350, 167280683834

Offset: 1

Vladimir Baltic, Jul 21 2002

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 (1,2,3,5,6,-1,-1,0,-1,-1).

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

LinearRecurrence[{1,2,3,5,6,-1,-1,0,-1,-1},{1,2,6,18,46,115,301,792,2068,5380},30] (* Harvey P. Dale, Aug 15 2014 *)

a(n)=([0,1,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,1; -1,-1,0,-1,-1,6,5,3,2,1]^(n-1)*[1;2;6;18;46;115;301;792;2068;5380])[1,1] \\ Charles R Greathouse IV, Jul 28 2015

Recurrence: a(n) = a(n-1)+2*a(n-2)+3*a(n-3)+5*a(n-4)+6*a(n-5)-a(n-6)-a(n-7)-a(n-9)-a(n-10).

G.f.: (1-x^5-x^3-x^2)/(x^10+x^9+x^7+x^6-6*x^5-5*x^4-3*x^3-2*x^2-x+1). [Corrected by Georg Fischer, May 15 2019]

A079955 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={0,2,3}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11, 15, 19, 26, 34, 46, 60, 80, 105, 140, 185, 246, 325, 431, 570, 756, 1001, 1327, 1757, 2328, 3083, 4085, 5411, 7169, 9496, 12580, 16664, 22076, 29244, 38741, 51320, 67985, 90060, 119305, 158045, 209366, 277350, 367411

Offset: 0

Vladimir Baltic, Feb 19 2003

Number of compositions (ordered partitions) of n into elements of the set {2,5,6}.

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 (0,1,0,0,1,1).

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

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-x^5-x^6) )); // G. C. Greubel, Dec 11 2019

seq(coeff(series(1/(1-x^2-x^5-x^6), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 11 2019

LinearRecurrence[{0, 1, 0, 0, 1, 1}, {1, 0, 1, 0, 1, 1}, 51] (* Jean-François Alcover, Dec 11 2019 *)

a(n) = ([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,0,0,1,0]^n*[1;0;1;0;1;1])[1,1] \\ Charles R Greathouse IV, Jul 28 2015

def A079955_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^2-x^5-x^6) ).list()
A079955_list(50) # G. C. Greubel, Dec 11 2019

a(n) = a(n-2) + a(n-5) + a(n-6).

G.f.: 1/(1 - x^2 - x^5 - x^6).

A002526 Number of permutations of length n within distance 3 of a fixed permutation.

Original entry on oeis.org

1, 1, 2, 6, 24, 78, 230, 675, 2069, 6404, 19708, 60216, 183988, 563172, 1725349, 5284109, 16177694, 49526506, 151635752, 464286962, 1421566698, 4352505527, 13326304313, 40802053896, 124926806216, 382497958000, 1171122069784, 3585709284968, 10978628154457

Offset: 0

N. J. A. Sloane

For positive n, a(n) equals the permanent of the n X n matrix with 1's along the seven central diagonals, and 0's everywhere else. - John M. Campbell, Jul 09 2011

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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. H. Hardin, Table of n, a(n) for n=0..400, Jul 11 2010
V. Baltic, On the number of certain types of strongly restricted permutations, Appl. An. Disc. Math. 4 (2010), 119-135; DOI:10.2298/AADM1000008B.
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. (Table 3, top row).
O. Krafft and M. Schaefer, On the number of permutations within a given distance, Fib. Quart. 40 (5) (2002) 429-434.
R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

The 14 sequences in Kløve's Table 3 are A002526, A002527, A002529, A188379, A188491, A188492, A188493, A188494, A002528, A188495, A188496, A188497, A188498, A002526.
Cf. A002524.
Column k=3 of A306209.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) )); // G. C. Greubel, Jan 22 2022

CoefficientList[Series[(1-x-2x^2-2x^4+x^7+x^8)/(1-2x-2x^2-10x^4-8x^5+ 2x^6+ 16x^7+10x^8+2x^9-4x^10-2x^11-2x^13-x^14),{x,0,50}],x] (* or *) LinearRecurrence[{2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1},{1,1,2,6,24,78, 230, 675,2069,6404,19708,60216,183988,563172},51] (* Harvey P. Dale, Jun 22 2011 *)

Vec((1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8+2*x^9-4*x^10-2*x^11-2*x^13-x^14)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

[( (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Jan 22 2022

G.f.: (1-x-2*x^2-2*x^4+x^7+x^8)/(1-2*x-2*x^2-10*x^4-8*x^5+2*x^6+16*x^7+10*x^8 +2*x^9-4*x^10-2*x^11-2*x^13-x^14).

a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=78, a(6)=230, a(7)=675, a(8)=2069, a(9)=6404, a(10)=19708, a(11)=60216, a(12)=183988, a(13)=563172, a(n) = 2*a(n-1) +2*a(n-2) +10*a(n-4) +8*a(n-5) -2*a(n-6) -16*a(n-7) -10*a(n-8) -2*a(n-9) +4*a(n-10) +2*a(n-11) +2*a(n-13) +a(n-14). - Harvey P. Dale, Jun 22 2011

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

A188491 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 3, and p(4) >= 2.

Original entry on oeis.org

0, 1, 2, 6, 14, 48, 152, 476, 1425, 4340, 13288, 40852, 125124, 382888, 1171612, 3587505, 10985790, 33638142, 102988410, 315318756, 965432832, 2955964296, 9050522241, 27710613432, 84843476928, 259771465608, 795361704776, 2435217884992

Offset: 0

N. J. A. Sloane, Apr 01 2011

nonn

a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of a single zero in the (1,4)-entry), ones on its three subdiagonals (with the exception of a single zero in the (4,1)-entry), and is zero elsewhere.

This is row 5 of Kløve's Table 3.

Nathaniel Johnston, Table of n, a(n) for n = 0..100
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.

a:= n-> (Matrix(13, (i, j)-> `if`(i=j-1, 1, `if`(i=13, [-1, -3, -3, -5, -9, -7, 3, 19, 21, 13, 3, 3, 1][j], 0)))^n. <<0, 0, 1, (0$6), 1, 2, 6, 14>>)[9, 1]: seq(a(n), n=0..30);  # Alois P. Heinz, Apr 08 2011

a[n_] := ((Table[Which[i == j-1, 1, i == 13, {-1, -3, -3, -5, -9, -7, 3, 19, 21, 13, 3, 3, 1}[[j]], True, 0], {i, 1, 13}, {j, 1, 13}] // MatrixPower[#, n]&).{0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 6, 14})[[9]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

a(n) = A002526(n-1) + A002528(n-1) + A188494(n-1). - Nathaniel Johnston, Apr 08 2011

G.f.: -x*(x^3+x^2-1)*(x^3+2*x^2+x+1) / (x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x+1).

Name and comments edited by Nathaniel Johnston, Apr 08 2011

A188494 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i and p(1) <= 2.

Original entry on oeis.org

0, 1, 2, 4, 12, 42, 138, 414, 1235, 3764, 11604, 35664, 109132, 333652, 1021220, 3127709, 9578526, 29326904, 89785684, 274896606, 841682902, 2577075290, 7890425175, 24158602552, 73968049928, 226473538032, 693411153800, 2123068036904, 6500352097064

Offset: 0

N. J. A. Sloane, Apr 01 2011

a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals, ones on its three subdiagonals (with the exception of zeros in the (3,1) and (4,1)-entries), and is zero elsewhere.

This is row 8 of Kløve's Table 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000(first 93 terms from Nathaniel Johnston)
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.
Index entries for linear recurrences with constant coefficients, signature (1,3,3,13,21,19,3,-7,-9,-5,-3,-3,-1).

with(LinearAlgebra):
A188494:= n-> `if`(n=0, 0, Permanent(Matrix(n, (i, j)->
              `if`(abs(j-i)<4 and [i, j]<>[3, 1] and [i, j]<>[4, 1], 1, 0)))):
seq(A188494(n), n=0..20);

LinearRecurrence[{1,3,3,13,21,19,3,-7,-9,-5,-3,-3,-1},{0,1,2,4,12,42,138,414,1235,3764,11604,35664,109132},30] (* Harvey P. Dale, Dec 27 2015 *)

concat(0, Vec(x*(x^6 +x^5 -x^4 -x^3 -x^2 +x +1) / (x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1) + O(x^100))) \\ Colin Barker, Dec 13 2014

From Nathaniel Johnston, Apr 10 2011: (Start)
a(n) = A188491(n+1) - A002528(n) - A002526(n).
a(n) = A002526(n-1) + A002527(n-1).
(End)
G.f.: x*(x^6 +x^5 -x^4 -x^3 -x^2 +x +1) / (x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1). - Colin Barker, Dec 13 2014

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 10 2011

cp's OEIS Frontend

A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A079955 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={0,2,3}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A002526 Number of permutations of length n within distance 3 of a fixed permutation.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A079977 Fibonacci numbers interspersed with zeros.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

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}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A188491 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 3, and p(4) >= 2.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Formula