A224808 -id:A224808 - cp's OEIS frontend

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}.

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).

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2T(n-2,k-2) - T(n-3,k-1) - T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-2) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)delta(k,0) - delta(n,2)*(delta(k,1) + delta(k,2)), T(n

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 4, 0, 1, 1, 2, 6, 0, 3, 0, 1, 3, 9, 4, 9, 0, 1, 1, 4, 12, 10, 18, 0, 4, 0, 1, 5, 16, 21, 36, 10, 16, 0, 1, 1, 6, 21, 36, 60, 30, 40, 0, 5, 0, 1, 7, 27, 57, 100, 81, 100, 20, 25, 0, 1, 1, 8, 34, 84, 158, 168

Offset: 0

Michael A. Allen, Jun 04 2022

This is the m=2, t=4 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(2*j+r-3*k,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a (1,4)-bonacci polynomial defined by f(n,x)=f(n-1,x)+x*f(n-4,x)+delta(n,0) where f(n<0,x)=0.
T(n+6-3*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 2, 4, or 6.

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   0,   2,   0;
  1,   1,   4,   0,   1;
  1,   2,   6,   0,   3,   0;
  1,   3,   9,   4,   9,   0,   1;
  1,   4,  12,  10,  18,   0,   4,   0;
  1,   5,  16,  21,  36,  10,  16,   0,   1;
  1,   6,  21,  36,  60,  30,  40,   0,   5,   0;
  1,   7,  27,  57, 100,  81, 100,  20,  25,   0,   1;
  1,   8,  34,  84, 158, 168, 200,  70,  75,   0,   6,   0;
  1,   9,  42, 118, 243, 322, 400, 231, 225,  35,  36,   0,   1;
...

Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, On A Two-Parameter Family of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.9.8.

Row sums are A099163.
Sums over k of T(n-3*k,k) are A224808.
Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Mathematica
```
T[n_,k_]:=If[k<0 || n
				
```

T(n,0) = 1.
T(n,n) = delta(n mod 2,0).
T(n,1) = n-3 for n>2.
T(2*j-r,2*j-1) = 0 for j>0, r=-1,0,1.
T(2*(j-1)+p,2*(j-1)) = j^p for j>0 and p=0,1,2.
T(2*j+p,2*(j-1)) = j^2*((j+1)/2)^p for j>0 and p=1,2.
T(2*j+3,2*(j-1)) = (j*(j+1))^2*(j+2)/12 for j>0.
T(2*(j+p),2*j-p) = C(j+2,3)^p for j>0 and p=0,1,2.
G.f. of row sums: (1-2*x^2)/(1-x-3*x^2+2*x^3).
G.f. of sums of T(n-3*k,k) over k: (1-x^5-x^8)/(1-x-x^5+x^6-x^7-2*x^8+x^9-x^10+x^13+x^16).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=3*k+1 if k>=0.

A224811 Number of subsets of {1,2,...,n-8} without differences equal to 2, 4, 6 or 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 48, 64, 88, 121, 165, 225, 300, 400, 520, 676, 884, 1156, 1530, 2025, 2700, 3600, 4800, 6400, 8480, 11236, 14840, 19600, 25900, 34225, 45325, 60025, 79625, 105625, 140075, 185761, 246101, 326041, 431676, 571536, 756756, 1002001, 1327326, 1758276, 2329782, 3087049, 4090296, 5419584

Offset: 0

Vladimir Baltic, May 18 2013

nonn

Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=8, I={-2,0,8}.

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, -1, 1, 2, -1, 1, 0, 0, -2, 1, -2, 0, 0, -1, 0, 0, 0, 0, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694,A224808-A224810.

CoefficientList[Series[(1 - x^10 - x^5 - x^7 + x^15)/((1 - x)*(1 + x)*(x^2 - x + 1)*(x^3 + x^2 - 1)*(x^6 - x^2 - 1)*(x^12 + x^10 + x^8 + 2*x^6 + x^4 + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec((1-x^10-x^5-x^7+x^15)/((1-x)*(1+x)*(x^2-x+1)*( x^3+x^2-1)*(x^6-x^2-1)*(x^12+x^10+x^8+2*x^6+x^4+1) )) \\ G. C. Greubel, Oct 28 2017

a(n) = a(n-1) +a(n-5) -a(n-6) +a(n-7) -a(n-8) +a(n-9) +2*a(n-10) -a(n-11) +a(n-12) -2*a(n-15) +a(n-16) -2*a(n-17) -a(n-20) +a(n-25).
G.f.: (1-x^10-x^5-x^7+x^15) / ( (1-x) *(1+x) *(x^2-x+1) *(x^3+x^2-1) *(x^6-x^2-1) *(x^12+x^10+x^8+2*x^6+x^4+1) ).
a(2*k) = (A003520(k))^2,
a(2*k+1) = A003520(k) * A003520(k+1)

A224815 Number of subsets of {1,2,...,n-8} without differences equal to 4 or 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 24, 36, 54, 81, 108, 144, 192, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 9477, 13689, 19773, 28561, 41743, 61009, 89167, 130321, 192052, 283024, 417088, 614656, 900032, 1317904, 1929788, 2825761

Offset: 0

Vladimir Baltic, May 18 2013

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=4, r=8, I={-4,0,8}.

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 (April, 2010), 119-13
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -2, 2, 0, 2, -6, 6, 0, 6, 1, -1, 0, -1, 13, -13, 0, -13, 15, -15, 0, -15, -6, 6, 0, 6, 3, -3, 0, -3, -2, 2, 0, 2, 8, -8, 0, -8, 3, -3, 0, -3, -1, 1, 0, 1, -1, 1, 0, 1).

Cf. A000930, A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224814.

CoefficientList[Series[(1 - x^3 + x^4 - x^5 - x^6 - 3*x^7 + 3*x^8 - 2*x^9 - x^10 - 5*x^11 - 3*x^12 - 2*x^13 + 3*x^15 - 3*x^16 - 3*x^18 + 3*x^19 - 3*x^20 + 3*x^21 + 3*x^23 + 6*x^24 - 3*x^25 - 2*x^26 - 4*x^27 - x^29 - x^30 - 2*x^31 - x^32 + x^33 + x^35 - x^36 + x^37 + x^39)/((1 - x - x^3)*(1 + x^4 + x^6)*(1 + x^4 - x^6)*(1 - x^4 - x^12)*(1 + x^4 + 6*x^8 - 3*x^12 + 2*x^20 + x^24)), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

a(n) = a(n-1)+a(n-3)-2*a(n-4)+2*a(n-5)+2*a(n-7)-6*a(n-8)+6*a(n-9)+6*a(n-11) +a(n-12)-a(n-13)-a(n-15)+13*a(n-16)-13*a(n-17)-13*a(n-19)+15*a(n-20)-15*a(n-21)-15*a(n-23)-6*a(n-24)+6*a(n-25)+6*a(n-27)+3*a(n-28)-3*a(n-29)-3*a(n-31)-2*a(n-32)+2*a(n-33)+2*a(n-35)+8*a(n-36)-8*a(n-37)-8*a(n-39)+3*a(n-40)-3*a(n-41)-3*a(n-43)-a(n-44)+a(n-45)+a(n-47)-a(n-48)+a(n-49)+a(n-51).
G.f.: ( 1-x^3+x^4-x^5-x^6-3*x^7+3*x^8-2*x^9-x^10-5*x^11-3*x^12-2*x^13 +3*x^15-3*x^16-3*x^18+3*x^19-3*x^20+3*x^21+3*x^23+6*x^24-3*x^25-2*x^26-4*x^27-x^29-x^30-2*x^31-x^32+x^33+x^35-x^36+x^37+x^39 ) / ((1-x-x^3)*(1+x^4+x^6)*(1+x^4-x^6)*(1-x^4-x^12)*(1+x^4+6*x^8-3*x^12+2*x^20+x^24)).
a(4*k) = (A000930(k))^4,
a(4*k+1) = (A000930(k))^3 * A000930(k+1),
a(4*k+2) = (A000930(k))^2 * (A000930(k+1))^2,
a(4*k+3) = A000930(k) * (A000930(k+1))^3.

A224810 Subsets of {1,2,...,n-6} without differences equal to 3 or 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 96, 144, 216, 324, 486, 729, 1053, 1521, 2197, 3211, 4693, 6859, 10108, 14896, 21952, 32144, 47068, 68921, 100860, 147600, 216000, 316800, 464640, 681472, 998976

Offset: 0

Vladimir Baltic, May 16 2013

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=3, r=6, I={-2,-1,1,2,3,4,5}.

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 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 2, 0, -2, 4, 0, 2, 2, 0, 4, -2, 0, 2, -4, 0, -2, -2, 0, -1, -1, 0, -1).

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

CoefficientList[Series[(1 + x^3 - x^4 - x^5 + x^6 - 2*x^7 - x^8 - x^9 - 2*x^10 - x^12 - x^13 - x^15)/((1 - x)*(1 + x + x^2)*(1 - x - x^3)*(1 + 3*x^3 + 7*x^6 + 9*x^9 + 7*x^12 + 3*x^15 + x^18)), {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2017 *)

x='x+O('x^50); Vec((1 + x^3 - x^4 - x^5 + x^6 - 2*x^7 - x^8 - x^9 - 2*x^10 - x^12 - x^13 - x^15)/((1 - x)*(1 + x + x^2)*(1 - x - x^3)*(1 + 3*x^3 + 7*x^6 + 9*x^9 + 7*x^12 + 3*x^15 + x^18))) \\ G. C. Greubel, Apr 30 2017

a(3*k) = (A000930(k))^3.
a(3*k+1) = (A000930(k))^2 * A000930(k+1).
a(3*k+2) = A000930(k) * (A000930(k+1))^2.
a(n) = a(n-1) -a(n-3) +2*a(n-4) -2*a(n-6) +4*a(n-7) +2*a(n-9) +2*a(n-10) +4*a(n-12) -2*a(n-13) +2*a(n-15) -4*a(n-16) -2*a(n-18) -2*a(n-19) -a(n-21) -a(n-22) -a(n-24)
G.f.: (1+x^3-x^4-x^5+x^6-2*x^7-x^8-x^9-2*x^10-x^12-x^13-x^15) / ((1-x)*(1+x+x^2)*(1-x-x^3)*(1+3*x^3+7*x^6+9*x^9+7*x^12+3*x^15+x^18))

A224814 Number of subsets of {1,2,...,n-9} without differences equal to 3, 6 or 9.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 80, 100, 125, 175, 245, 343, 490, 700, 1000, 1400, 1960, 2744, 3724, 5054, 6859, 9386, 12844, 17576, 24336, 33696, 46656, 64800, 90000, 125000, 172500, 238050, 328509, 452295, 622725, 857375, 1182275, 1630295, 2248091, 3106141, 4291691, 5929741, 8190250, 11312500, 15625000, 21562500

Offset: 0

Vladimir Baltic, May 18 2013

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=3, r=9, I={-3,0,9}.

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 (April, 2010), 119-13
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, 0, -1, 1, 0, -3, 4, 0, 5, -2, 0, 3, -8, 0, 1, -4, 0, 3, -4, 0, -3, 0, 0, -5, 8, 0, 7, -2, 0, -9, 2, 0, 5, 4, 0, 1, -6, 0, -3, 2, 0, 2, 1, 0, -1, -1, 0, 0, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

CoefficientList[Series[(1 - x^4 - x^5 - x^7 - x^8 + 2*x^9 - x^10 - 3*x^12 - x^13 - 2*x^15 + 3*x^16 + 3*x^17 + 2*x^18 - x^20 - 4*x^21 + x^23 + 3*x^24 + 3*x^25 + x^27 - 4*x^28 - x^29 - 2*x^30 + x^31 + 2*x^33 +x^34 - x^36 - x^37 + x^40)/((1 - x - x^4)*(1 - x^9 - x^12)*(1 + x^6 + 4*x^9 - 4*x^12 - 2*x^15 + 4*x^18 - 3*x^21 - 3*x^24 + 7*x^27 - 6*x^30 + 3*x^33 - x^36)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec((1 -x^4 -x^5 -x^7 -x^8 +2*x^9 -x^10 -3*x^12 -x^13 -2*x^15 +3*x^16 +3*x^17 +2*x^18 -x^20 -4*x^21 +x^23 +3*x^24 +3*x^25 +x^27 -4*x^28 -x^29 -2*x^30 +x^31 +2*x^33 +x^34 -x^36 -x^37 +x^40 )/((1-x-x^4)*(1-x^9-x^12)*(1 +x^6 +4*x^9 -4*x^12 -2*x^15 +4*x^18 -3*x^21 -3*x^24 +7*x^27 -6*x^30 +3*x^33 -x^36))) \\ G. C. Greubel, Oct 28 2017

a(n) = a(n-1) +a(n-4) -a(n-6) +a(n-7) -3*a(n-9) +4*a(n-10) +5*a(n-12) -2*a(n-13) +3*a(n-15) -8*a(n-16) +a(n-18) -4*a(n-19) +3*a(n-21) -4*a(n-22) -3*a(n-24) -5*a(n-27) +8*a(n-28) +7*a(n-30) -2*a(n-31) -9*a(n-33) +2*a(n-34) +5*a(n-36) +4*a(n-37) +a(n-39) -6*a(n-40) -3*a(n-42) +2*a(n-43) +2*a(n-45) +a(n-46) -a(n-48) -a(n-49) +a(n-52).
G.f.: (1 -x^4 -x^5 -x^7 -x^8 +2*x^9 -x^10 -3*x^12 -x^13 -2*x^15 +3*x^16 +3*x^17 +2*x^18 -x^20 -4*x^21 +x^23 +3*x^24 +3*x^25 +x^27 -4*x^28 -x^29 -2*x^30 +x^31 +2*x^33 +x^34 -x^36 -x^37 +x^40 )/((1-x-x^4)*(1-x^9-x^12)*(1 +x^6 +4*x^9 -4*x^12 -2*x^15 +4*x^18 -3*x^21 -3*x^24 +7*x^27 -6*x^30 +3*x^33 -x^36)).
a(3*k) = (A003269(k))^3,
a(3*k+1) = (A003269(k))^2 * A003269(k+1),
a(3*k+2) = A003269(k) * (A003269(k+1))^2.

A224812 Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 63, 81, 108, 144, 192, 256, 336, 441, 567, 729, 918, 1156, 1462, 1849, 2365, 3025, 3905, 5041, 6532, 8464, 10948, 14161, 18207, 23409, 29988, 38416, 49196, 63001, 80822, 103684, 133308, 171396, 220662, 284089, 365638, 470596, 605052, 777924, 999306, 1283689, 1648515

Offset: 0

Vladimir Baltic, May 18 2013

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=10, I={-2,0,10}.

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 (April, 2010), 119-13
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, 3, -2, 2, -1, 1, 0, 0, -2, 1, -2, 0, 0, -3, 1, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

CoefficientList[Series[-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec(-(x + 1)*(x^23 - x^22 + x^21 - x^20 + x^19 - x^13 + x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 2*x^8 - 2*x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x^6 + x - 1)*(x^30 + x^24 - 2*x^20 - 2*x^18 - x^14 - 2*x^12 + x^10 + x^8 + x^6 + 1))) \\ G. C. Greubel, Oct 28 2017

a(n) = a(n-1) +a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11) +3*a(n-12) -2*a(n-13) +2*a(n-14) -a(n-15) +a(n-16) -2*a(n-19) +a(n-20) -2*a(n-21) -3*a(n-24) +a(n-25) -2*a(n-26) +a(n-31) +a(n-36).
G.f.: -(x+1) *(x^23 -x^22 +x^21 -x^20 +x^19 -x^13 +x^12 -3*x^11 +3*x^10 -3*x^9 +2*x^8 -2*x^7 +x^6 -x^5 +x^4 -x^3 +x^2 -x +1)/ ((x^6 +x -1) *(x^30 +x^24 -2*x^20 -2*x^18 -x^14 -2*x^12 +x^10 +x^8 +x^6+1) ).
a(2*k) = (A005708(k))^2, a(2*k+1) = A005708(k) * A005708(k+1).

A224813 Number of subsets of {1,2,...,n-12} without differences equal to 2, 4, 6, 8, 10 or 12.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 80, 100, 130, 169, 221, 289, 374, 484, 616, 784, 980, 1225, 1505, 1849, 2279, 2809, 3498, 4356, 5478, 6889, 8715, 11025, 13965, 17689, 22344, 28224, 35448, 44521, 55704, 69696, 87120, 108900, 136290, 170569, 213934, 268324, 337218, 423801, 533169, 670761, 843570

Offset: 0

Vladimir Baltic, May 18 2013

a(n) is the number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=12, I={-2,0,12}.

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 (April, 2010), 119-13
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 3, -2, 2, -1, 1, 0, 0, -3, 2, -4, 2, -3, 0, 0, -3, 1, -2, 0, 0, 0, 0, 3, -1, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1).

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694, A224808-A224815.

CoefficientList[Series[-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 1)), {x, 0, 1000}], x] (* G. C. Greubel, Oct 28 2017 *)

x='x+O('x^50); Vec(-(-1 + x^7 + x^9 + x^11 + 2*x^14 + x^16 - 2*x^21 - 2*x^23 - x^28 + x^35)/((x^7 + x - 1)*(x^42 - x^36 - 2*x^30 - 3*x^28 + 2*x^24 + 2*x^22 + x^18 + 2*x^16 + 3*x^14 - x^12 - x^10 - x^8 - 1))) \\ G. C. Greubel, Oct 28 2017

a(n) = a(n-1) +a(n-7) -a(n-8) +a(n-9) -a(n-10) +a(n-11) -a(n-12) +a(n-13) +3*a(n-14) -2*a(n-15) +2*a(n-16) -a(n-17) +a(n-18) -3*a(n-21) +2*a(n-22) -4*a(n-23) +2*a(n-24) -3*a(n-25) -3*a(n-28) +a(n-29) -2*a(n-30) +3*a(n-35) -a(n-36) +3*a(n-37) +a(n-42) -a(n-49).
G.f.: -(-1 +x^7 +x^9 +x^11 +2*x^14 +x^16 -2*x^21 -2*x^23 -x^28 +x^35)/( (x^7+x-1) *(x^42 -x^36 -2*x^30 -3*x^28 +2*x^24 +2*x^22 +x^18 +2*x^16 +3*x^14 -x^12 -x^10 -x^8 -1) ).
a(2*k) = (A005709(k))^2, a(2*k+1) = A005709(k) * A005709(k+1).

A387020 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,5} for all i=1,...,n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130, 175, 231, 305, 400, 540, 729, 999, 1363, 1855, 2510, 3370, 4531, 6070, 8180, 11026, 14921, 20197, 27322, 36940, 49820, 67204, 90528, 122091, 164686, 222344, 300316, 405574, 547768, 739291, 997794, 1346130

Offset: 0

Michael A. Allen, Aug 13 2025

			a(7) = 2: 1234567, 6712345.
a(8) = 3: 12345678, 17823456, 67123458.

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

Vladimir Baltić, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,3,-2,2,-1,1,0,0,-3,1,-2,0,0,0,0,1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,..,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 2*x^7 - x^9 + x^14)/(1 - x - 3*x^7 + 2*x^8 - 2*x^9 + x^10 - x^11 + 3*x^14 - x^15 + 2*x^16 - x^21),{x,0,51}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 3, -2, 2, -1, 1, 0, 0, -3, 1, -2, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130}, 52]

a(n) = a(n-1) + 3*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-10) + a(n-11) - 3*a(n-14) + a(n-15) - 2*a(n-16) + a(n-21) for n >= 21.

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

A387021 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,7} for all i=1,...,n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773, 3670, 4861, 6388, 8344, 10848, 14019, 18166, 23479, 30556, 39762, 52049, 68125, 89345, 117034, 153078, 199979, 260572, 339546, 441669, 575341

Offset: 0

Michael A. Allen, Aug 13 2025

			a(9)=2: 123456789, 891234567.

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

V. Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,4,-3,3,-2,2,-1,1,0,0,-6,3,-6,2,-3,0,0,0,0,4,-1,3,0,0,0,0,0,0,-1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,...,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36),{x,0,55}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 4, -3, 3, -2, 2, -1, 1, 0, 0, -6, 3, -6, 2, -3, 0, 0, 0, 0, 4, -1, 3, 0, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773}, 56]

a(n) = a(n-1) + 4*a(n-9) - 3*a(n-10) + 3*a(n-11) - 2*a(n-12) + 2*a(n-13) - a(n-14) + a(n-15) - 6*a(n-18) + 3*a(n-19) - 6*a(n-20) + 2*a(n-21) - 3*a(n-22) + 4*a(n-27) - a(n-28) + 3*a(n-29) - a(n-36) for n >= 36.

G.f.: (1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36).

cp's OEIS Frontend

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}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2*T(n-2,k-2) - T(n-3,k-1) - T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-2) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)*delta(k,0) - delta(n,2)*(delta(k,1) + delta(k,2)), T(n

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A224811 Number of subsets of {1,2,...,n-8} without differences equal to 2, 4, 6 or 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224815 Number of subsets of {1,2,...,n-8} without differences equal to 4 or 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A224810 Subsets of {1,2,...,n-6} without differences equal to 3 or 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224814 Number of subsets of {1,2,...,n-9} without differences equal to 3, 6 or 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224812 Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224813 Number of subsets of {1,2,...,n-12} without differences equal to 2, 4, 6, 8, 10 or 12.

Views

Author

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2T(n-2,k-2) - T(n-3,k-1) - T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-2) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)delta(k,0) - delta(n,2)*(delta(k,1) + delta(k,2)), T(n