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

%I

%S 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,

%T 100,130,169,221,289,374,484,616,784,980,1225,1505,1849,2279,2809,

%U 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

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

%C 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}.

%H G. C. Greubel, <a href="/A224813/b224813.txt">Table of n, a(n) for n = 0..1000</a>

%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-13

%F 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).

%F 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) ).

%F a(2*k) = (A005709(k))^2, a(2*k+1) = A005709(k) * A005709(k+1).

%t 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 *)

%o (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

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

%K nonn,easy

%O 0,14

%A _Vladimir Baltic_, May 18 2013

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Last modified July 26 15:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)