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A370377
a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.
1
1, 3, 2, 6, 5, 14, 11, 31, 25, 70, 56, 157, 126, 353, 283, 793, 636, 1782, 1429, 4004, 3211, 8997, 7215, 20216, 16212, 45425, 36428, 102069, 81853, 229347, 183922, 515338, 413269, 1157954, 928607, 2601899, 2086561, 5846414, 4688460, 13136773, 10534874
OFFSET
0,2
FORMULA
a(n) = 2*A306334(n) - A006356(n).
Also:
a(0) = 1;
a(2) = 2;
a(n) = A006356((n+1)/2) if n is odd;
a(n) = A006356(n/2) - A006356((n-4)/2) if n is even.
G.f.: (1+3*x-x^5)/(1-2*x^2-x^4+x^6). - Joerg Arndt, Feb 18 2024
EXAMPLE
For n = 1: a(1) = A006356(1) = 3
CH3-CH3, CH2=CH2, CH≡CH
For n = 3: a(3) = A006356(2) = 6
CH3-CH2-CH2-CH3, CH3-CH=CH-CH3, CH3-C≡C-CH3, CH2=CH-CH=CH2, CH≡C-C≡CH, CH2=C=C=CH2
For n = 4: a(4) = A006356(2) - A006356(0) = 6 - 1 = 5
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
MATHEMATICA
LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* Paolo Xausa, Feb 22 2024 *)
PROG
(Python)
a = [1, 3, 2, 6, 5, 14]
for i in range(30):
a.append(2*a[-2]+a[-4]-a[-6])
print(a)
(PARI) Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ Joerg Arndt, Feb 18 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Tomasz Dziekanski, Feb 18 2024
STATUS
approved