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Number of subsets of {1,2,...,n-10} without differences equal to 2, 4, 6, 8 or 10.
3

%I #16 Sep 04 2024 09:58:16

%S 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,

%T 192,256,336,441,567,729,918,1156,1462,1849,2365,3025,3905,5041,6532,

%U 8464,10948,14161,18207,23409,29988,38416,49196,63001,80822,103684,133308,171396,220662,284089,365638,470596,605052,777924,999306,1283689,1648515

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

%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=10, I={-2,0,10}.

%H G. C. Greubel, <a href="/A224812/b224812.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

%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, 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).

%F 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).

%F 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) ).

%F a(2*k) = (A005708(k))^2, a(2*k+1) = A005708(k) * A005708(k+1).

%t 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 *)

%o (PARI) 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

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

%K nonn,easy

%O 0,12

%A _Vladimir Baltic_, May 18 2013