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

0,14

Comments

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}.

Links

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

Formula

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).

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A002620 A087811 A025699 * A224812 A194256 A194246

Adjacent sequences: A224810 A224811 A224812 * A224814 A224815 A224816

Keyword

nonn,easy

Author

Vladimir Baltic, May 18 2013

Status

approved