A080014 -id:A080014 - cp's OEIS frontend

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}.

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

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)).

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).

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);

A072853 Number of permutations satisfying i-2<=p(i)<=i+6, i=1..n.

Original entry on oeis.org

1, 2, 6, 18, 54, 162, 486, 1394, 3991, 11593, 33772, 98320, 286072, 831952, 2418664, 7030816, 20441944, 59441521, 172843609, 502580846, 1461344622, 4249102850, 12354982862, 35924300898, 104456501102, 303726483778, 883140022543

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, 8, 14, 26, 44, 56, -11, -19, -28, -28, 0, -8, -20, -20, 0, 5, 11, 10, 0, 0, 2, 2, 0, 0, -1, -1).

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

Recurrence: a(n)= a(n - 1) + 2*a(n - 2) + 4*a(n - 3) + 8*a(n - 4) + 14*a(n - 5) + 26*a(n - 6) + 44*a(n - 7) + 56*a(n - 8) - 11*a(n - 9) - 19*a(n - 10) - 28*a(n - 11) - 28*a(n - 12) - 8*a(n - 14) - 20*a(n - 15) - 20*a(n - 16) + 5*a(n - 18) + 11*a(n - 19) + 10*a(n - 20) + 2*a(n - 23) + 2*a(n - 24) - a(n - 27) - a(n - 28).

G.f.: - (x^20 + x^18 - 2*x^16 - 2*x^14 - 6*x^12 - 2*x^11 - 4*x^10 - 4*x^9 + 12*x^8 + 2*x^7 + 8*x^6 + 6*x^5 + 4*x^4 + 2*x^3 + x^2 - 1)/(x^28 + x^27 - 2*x^24 - 2*x^23 - 10*x^20 - 11*x^19 - 5*x^18 + 20*x^16 + 20*x^15 + 8*x^14 + 28*x^12 + 28*x^11 + 19*x^10 + 11*x^9 - 56*x^8 - 44*x^7 - 26*x^6 - 14*x^5 - 8*x^4 - 4*x^3 - 2*x^2 - x + 1).

A072854 Number of permutations satisfying i-3<=p(i)<=i+4, i=1..n.

Original entry on oeis.org

1, 2, 6, 24, 96, 330, 1066, 3451, 11581, 39264, 132784, 446460, 1497108, 5023696, 16878488, 56739141, 190697893, 640763258, 2152824662, 7233281108, 24304468132, 81666680202, 274410023170, 922040339607, 3098121457769

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 (0, 3, 10, 24, 58, 128, 226, 164, 66, 8, 50, -72, -374, -640, -630, -518, -390, -426, -466, -216, 94, 48, 22, 52, 38, 48, 22, -8, -2, 0, -2, -1, -2, -1).

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

Recurrence: a(n) = 3*a(n - 2) + 10*a(n - 3) + 24*a(n - 4) + 58*a(n - 5) + 128*a(n - 6) + 226*a(n - 7) + 164*a(n - 8) + 66*a(n - 9) + 8*a(n - 10) + 50*a(n - 11) - 72*a(n - 12) - 374*a(n - 13) - 640*a(n - 14) - 630*a(n - 15) - 518*a(n - 16) - 390*a(n - 17) - 426*a(n - 18) - 466*a(n - 19) - 216*a(n - 20) + 94*a(n - 21) + 48*a(n - 22) + 22*a(n - 23) + 52*a(n - 24) + 38*a(n - 25) + 48*a(n - 26) + 22*a(n - 27) - 8*a(n - 28) - 2*a(n - 29) - 2*a(n - 31) - a(n - 32) - 2*a(n - 33) - a(n - 34).

G.f.: - (x^27 + x^26 + x^25 - x^24 + 4*x^22 + 4*x^21 - 16*x^20 - 23*x^19 - 29*x^18 + x^17 - 3*x^16 - 20*x^15 - 8*x^14 + 44*x^13 + 56*x^12 + 79*x^11 + 67*x^10 + 63*x^9 + 69*x^8 + 76*x^7 + 36*x^6 + 24*x^5 + 16*x^4 + 7*x^3 + x^2 - x - 1)/(x^34 + 2*x^33 + x^32 + 2*x^31 + 2*x^29 + 8*x^28 - 22*x^27 - 48*x^26 - 38*x^25 - 52*x^24 - 22*x^23 - 48*x^22 - 94*x^21 + 216*x^20 + 466*x^19 + 426*x^18 + 390*x^17 + 518*x^16 + 630*x^15 + 640*x^14 + 374*x^13 + 72*x^12 - 50*x^11 - 8*x^10 - 66*x^9 - 164*x^8 - 226*x^7 - 128*x^6 - 58*x^5 - 24*x^4 - 10*x^3 - 3*x^2 + 1).

A072855 Number of permutations satisfying i-3<=p(i)<=i+5, i=1..n.

Original entry on oeis.org

1, 2, 6, 24, 96, 384, 1374, 4718, 16275, 57749, 206756, 739780, 2637348, 9378840, 33318804, 118439044, 421340612, 1499388117, 5335199213, 18980987054, 67522942850, 240204885524, 854523535096, 3040023558788, 10815153542594

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, 3, 8, 20, 46, 114, 242, 354, -250, -490, -660, -496, -24, -1242, -2430, -2270, -566, 2241, 5071, 4259, -632, 1392, 6396, 5596, -132, 1316, -6220, -11116, 736, 344, -5128, -3684, 1148, -388, 980, 1665, 239, -199, 688, 540, -106, 50, -78, -102, -58, 22, -44, -40, 0, -2, 2, 2, 2, -1, 1, 1).

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

G.f.: -(1- 2*x^2 - 7*x^3 - 16*x^4 - 28*x^5 - 32*x^6 - 58*x^7 - 156*x^8 + 67*x^9 + 76*x^10 + 68*x^11 + 145*x^12 + 12*x^13 + 156*x^14 + 180*x^15 + 704*x^16 + 344*x^17 - 454*x^18 - 276*x^19 - 480*x^20 + 158*x^21 - 260*x^22 - 116*x^23 - 780*x^24 - 756*x^25 + 168*x^26 + 206*x^27 + 900*x^28 - 340*x^29 + 126*x^30 + 132*x^31 + 276*x^32 + 28*x^33 + 16*x^34 + 24*x^35 - 107*x^36 + 36*x^37 - 14*x^38 - 7*x^39 - 28*x^40 - 4*x^42 - 2*x^43 + 4*x^44 - x^45 + x^48) / (-1 + x + 3*x^2 + 8*x^3 + 20*x^4 + 46*x^5 + 114*x^6 + 242*x^7 + 354*x^8 - 250*x^9 - 490*x^10 - 660*x^11 - 496*x^12 - 24*x^13 - 1242*x^14 - 2430*x^15 - 2270*x^16 - 566*x^17 + 2241*x^18 + 5071*x^19 + 4259*x^20 - 632*x^21 + 1392*x^22 + 6396*x^23 + 5596*x^24 - 132*x^25 + 1316*x^26 - 6220*x^27 - 11116*x^28 + 736*x^29 + 344*x^30 - 5128*x^31 - 3684*x^32 + 1148*x^33 - 388*x^34 + 980*x^35 + 1665*x^36 + 239*x^37 - 199*x^38 + 688*x^39 + 540*x^40 - 106*x^41 + 50*x^42 - 78*x^43 - 102*x^44 - 58*x^45 + 22*x^46 - 44*x^47 - 40*x^48 - 2*x^50 + 2*x^51 + 2*x^52 + 2*x^53 - x^54 + x^55 + x^56). - Vaclav Kotesovec, Dec 01 2012

A080013 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={0,1}.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 3, 3, 4, 6, 9, 12, 16, 24, 33, 46, 64, 91, 127, 177, 249, 349, 489, 684, 960, 1345, 1884, 2640, 3700, 5185, 7264, 10180, 14265, 19989, 28009, 39249, 54999, 77067, 107992, 151326, 212049, 297136, 416368, 583444, 817561, 1145622, 1605324, 2249491, 3152139, 4416993

Offset: 0

Vladimir Baltic, Jan 24 2003

nonn

Also the number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={0,-1}.

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.

Michael De Vlieger, Table of n, a(n) for n = 0..6829
Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, arXiv:2107.02589 [math.CO], 2021.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
P. L. Krapivsky and J. M. Luck, Jamming and metastability in one dimension: from the kinetically constrained Ising chain to the Riviera model, arXiv:2211.12815 [cond-mat.stat-mech], 2022.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

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

LinearRecurrence[{0,1,1,1,0,-1},{1,0,0,1,1,1},60] (* Harvey P. Dale, Aug 08 2019 *)

Recurrence: a(n) = a(n-2)+a(n-3)+a(n-4)-a(n-6).

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

cp's OEIS Frontend

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}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A006500 Restricted combinations.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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}).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

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}.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A072853 Number of permutations satisfying i-2<=p(i)<=i+6, i=1..n.

Views

Author

Keywords

Links

Crossrefs

Formula

A072854 Number of permutations satisfying i-3<=p(i)<=i+4, i=1..n.

Views

Author

Keywords

Links

Crossrefs

Formula

A072855 Number of permutations satisfying i-3<=p(i)<=i+5, i=1..n.