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A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.

%I A370377 #84 Mar 07 2024 10:39:39
%S A370377 1,3,2,6,5,14,11,31,25,70,56,157,126,353,283,793,636,1782,1429,4004,
%T A370377 3211,8997,7215,20216,16212,45425,36428,102069,81853,229347,183922,
%U A370377 515338,413269,1157954,928607,2601899,2086561,5846414,4688460,13136773,10534874
%N A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.
%H A370377 Paolo Xausa, <a href="/A370377/b370377.txt">Table of n, a(n) for n = 0..1000</a>
%H A370377 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1,0,-1).
%F A370377 a(n) = 2*A306334(n) - A006356(n).
%F A370377 Also:
%F A370377 a(0) = 1;
%F A370377 a(2) = 2;
%F A370377 a(n) = A006356((n+1)/2) if n is odd;
%F A370377 a(n) = A006356(n/2) - A006356((n-4)/2) if n is even.
%F A370377 G.f.: (1+3*x-x^5)/(1-2*x^2-x^4+x^6). - _Joerg Arndt_, Feb 18 2024
%e A370377 For n = 1: a(1) = A006356(1) = 3
%e A370377  CH3-CH3, CH2=CH2, CH≡CH
%e A370377 For n = 3: a(3) = A006356(2) = 6
%e A370377  CH3-CH2-CH2-CH3, CH3-CH=CH-CH3, CH3-C≡C-CH3, CH2=CH-CH=CH2, CH≡C-C≡CH, CH2=C=C=CH2
%e A370377 For n = 4: a(4) = A006356(2) - A006356(0) = 6 - 1 = 5
%e A370377  CH3-CH2-CH2-CH2-CH3, CH3-CH=C=CH-CH3, CH2=CH-CH2-CH=CH2, CH≡C-CH2-C≡CH, CH2=C=C=C=CH2
%t A370377 LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* _Paolo Xausa_, Feb 22 2024 *)
%o A370377 (Python)
%o A370377 a = [1, 3, 2, 6, 5, 14]
%o A370377 for i in range(30):
%o A370377     a.append(2*a[-2]+a[-4]-a[-6])
%o A370377 print(a)
%o A370377 (PARI) Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ _Joerg Arndt_, Feb 18 2024
%Y A370377 Cf. A006356, A006054, A306334.
%K A370377 easy,nonn
%O A370377 0,2
%A A370377 _Tomasz Dziekanski_, Feb 18 2024