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

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

Author

Tomasz Dziekanski, Feb 18 2024

Keywords

• easy
• nonn

Examples

			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
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-1).

Crossrefs

Cf. A006356, A006054, A306334.

Programs

Mathematica

LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* Paolo Xausa, Feb 22 2024 *)

PARI

Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ Joerg Arndt, Feb 18 2024

Python

a = [1, 3, 2, 6, 5, 14]
for i in range(30):
    a.append(2*a[-2]+a[-4]-a[-6])
print(a)

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