Vladimir Baltic - cp's OEIS frontend

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

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).

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.

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.

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).

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)

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))

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).

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)

A217694 Number of n-variations of the set {1,2,...,n+1} satisfying p(i)-i in {-2,0,2}, i=1..n (an n-variation of the set N_{n+s} = {1,2,...,n+s} is any 1-to-1 mapping p from the set N_n = {1,2,...,n} into N_{n+s} = {1,2,...,n+s}).

Original entry on oeis.org

1, 1, 2, 4, 8, 12, 21, 35, 60, 96, 160, 260, 429, 693, 1134, 1836, 2992, 4840, 7865, 12727, 20648, 33408, 54144, 87608, 141897, 229593, 371722, 601460, 973560, 1575252, 2549421, 4125051, 6675460, 10801120, 17478176, 28280284, 45761045, 74042925, 119808150

Offset: 0

Vladimir Baltic, Oct 11 2012

V. Baltic, Applications of the finite state automata for counting restricted permutations and variations, Yugoslav Journal of Operations Research, 22 (2012), Number 2, 183-198. - _N. J. A. Sloane_, Jan 02 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,0,2,-2,-1,-1,-1).

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

LinearRecurrence[{1,1,0,2,-2,-1,-1,-1},{1,1,2,4,8,12,21,35},40] (* Harvey P. Dale, Feb 29 2020 *)

Recurrence: a(n)=a(n-1)+a(n-2)+2*a(n-4)-2*a(n-5)-a(n-6)-a(n-7)-a(n-8).

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

A086113 Number of 3 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.

Original entry on oeis.org

6, 36, 102, 216, 390, 636, 966, 1392, 1926, 2580, 3366, 4296, 5382, 6636, 8070, 9696, 11526, 13572, 15846, 18360, 21126, 24156, 27462, 31056, 34950, 39156, 43686, 48552, 53766, 59340, 65286, 71616, 78342, 85476, 93030, 101016, 109446, 118332

Offset: 1

Vladimir Baltic, Vladeta Jovovic, Jul 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Don Coppersmith, Ponder This: IBM Research Monthly Puzzles, March 2004 challenge
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A032260, A016742, A086114, A086115.

I:=[6, 36, 102, 216]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 24 2012

CoefficientList[Series[6*(1+2x-x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 24 2012 *)

a(n) = 2*n*(n^2 + 3*n - 1) = 2*n*A014209(n). More generally, number of m X n (0, 1) matrices such that each row and each column is increasing or decreasing is 2*n*(2*binomial(n+m-1, n)-m) = 2*m*(2*binomial(m+n-1, m)-n).
G.f.: 6*x*(1 + 2*x - x^2)/(1-x)^4. - Vincenzo Librandi, Jun 24 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 24 2012

cp's OEIS Frontend

User: Vladimir Baltic

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

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

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

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

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

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

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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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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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