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A197424 Number of subsets of {1, 2, ..., 4*n + 2} which do not contain two numbers whose difference is 4.

%I A197424 #47 Jul 03 2025 07:26:07
%S A197424 4,36,225,1600,10816,74529,509796,3496900,23961025,164249856,
%T A197424 1125736704,7716041281,52886200900,362488284900,2484529385121,
%U A197424 17029223715904,116720020119616,800010960336225,5483356589096100,37583485459535236,257601040852192129
%N A197424 Number of subsets of {1, 2, ..., 4*n + 2} which do not contain two numbers whose difference is 4.
%C A197424 This sequence is an instance of a general result given in Math. Mag. Problem 1854 (see Links).
%C A197424 From _Feryal Alayont_, May 20 2023: (Start)
%C A197424 a(n) is the number of edge covers of a caterpillar graph with spine P_(4n+5), one pendant attached at vertex n+2 counting from the left end of the spine, a second one at 2n+3 and a third at 3n+4. The caterpillar graph for n=1 is as follows:
%C A197424         *      *      *
%C A197424         |      |      |
%C A197424   *--*--*--v1--*--v2--*--*--*
%C A197424 Each pendant edge must be included in an edge cover leaving only the middle six edges flexible. Every vertex except v1 and v2 is incident with at least one of the pendant edges. Therefore, if we label the middle six edges in the spine with numbers 3, 1, 5, 2, 6, 4 (starting from the left), the edges have to be chosen so that both 1,5 and 2,6 cannot be missing. This corresponds to choosing subsets of {1, 2, ..., 6} which do not contain two numbers whose difference is 4. (End)
%D A197424 F. Alayont and E. Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs; submitted.
%H A197424 Colin Barker, <a href="/A197424/b197424.txt">Table of n, a(n) for n = 0..1000</a>
%H A197424 Mathramz Problem Solving Group, <a href="http://www.mathramz.com/mrpsg/Sol/MathMag1854.pdf">Solution for Problem 1854 in Mathematics Magazine Vol. 83, No.4, October 2010</a>, February 28, 2011
%H A197424 Marian Tetiva, <a href="http://www.jstor.org/stable/10.4169/mathmaga.83.4.0304a">Problem 1854</a>, Mathematics Magazine, 84 (2011) 300.
%H A197424 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F A197424 a(n) = F(n+2)^2*F(n+3)^2 = A001654(n+2)^2, where F(n) denotes the n-th Fibonacci number A000045(n).
%F A197424 G.f.: ( -4-16*x+15*x^2+5*x^3-x^4 ) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - _R. J. Mathar_, Oct 15 2011
%F A197424 Empirical: a(n) = A189145(2n+3). - _R. J. Mathar_, Oct 15 2011
%F A197424 For L=Lucas, a(n) = (1/25)*(L(2*(2*n+5)) - 2*(-1)^n*L(2*n+5) - 1), an instance of (F(n+p)*F(n+q))^2 = (1/25)*(L(2*(2*n+p+q)) - 2*(-1)^(n+q)*L(p-q)*L(2*n+p+q) + L(2*(p-q)) + 4*(-1)^(p-q)) which follows from squaring a specialization of identity 17b in the Vajda reference at A000045, F(n+p)*F(n+q) = (1/5)*(L(2*n+p+q) - (-1)*(n+q)*L(p-q)), then applying Vajda 17c, L(n)^2 = L(2*n) + 2*(-1)^n, to the expansion. - _Ehren Metcalfe_, Mar 26 2016
%F A197424 a(n) = A060635(n+2)/2. - _Alois P. Heinz_, Jul 03 2025
%t A197424 Table[(1/25) (LucasL[2 (2 n + 5)] - 2 (-1)^n LucasL[2 n + 5] - 1), {n, 0, 20}] (* _Michael De Vlieger_, Mar 27 2016 *)
%o A197424 (PARI) Vec((4+16*x-15*x^2-5*x^3+x^4) / ((1-x)*(1-7*x+x^2)*(1+3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Mar 26 2016
%o A197424 (PARI) a(n) = (fibonacci(n+2)*fibonacci(n+3))^2; \\ _Altug Alkan_, Mar 26 2016
%Y A197424 Cf. A000045, A001654, A060635, A066258.
%K A197424 nonn,easy
%O A197424 0,1
%A A197424 _John W. Layman_, Oct 14 2011