A393414 - OEIS

%I #62 Mar 02 2026 18:20:07

%S 29,21,50,71,121,192,313,505,818,1323,2141,3464,5605,9069,14674,23743,

%T 38417,62160,100577,162737,263314,426051,689365,1115416,1804781,

%U 2920197,4724978,7645175,12370153,20015328,32385481,52400809,84786290,137187099,221973389,359160488

%N Fibonacci sequence beginning 29, 21.

%C For n >= 3, a(n) is half the number of matchings in the graph consisting of an n-cycle connected to a 7-cycle by a bridge.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F a(n) = a(n-1) + a(n-2).

%F a(n) = 29*Fibonacci(n+1) - 8*Fibonacci(n).

%F G.f.: (8*x - 29) / (x^2 + x - 1).

%F E.g.f.: exp(x/2)*(145*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2))/5. - _Enrique Navarrete_, Feb 28 2026

%e For n = 3, a(n) = 71, which is half the number of matchings in the graph of a 7-cycle joined by a bridge to a 3-cycle:

%e     o---o---o---o     o---o

%e     |           |     | /

%e     o-----o-----o-----o

%p a:= n-> (<<0|1>, <1|1>>^n. <<29, 21>>)[1,1]:

%p seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 23 2026

%t a[n_]:= 29*Fibonacci[n+1] -8*Fibonacci[n]; Table[a[n],{n,0,40}]

%t Table[SeriesCoefficient[(29 -8*x)/(1 - x - x^2), {x, 0, n}], {n,0, 40}]

%o (Python)

%o from gmpy2 import fib2

%o def A393414(n): return (lambda x:int(29*x[0]-8*x[1]))(fib2(n+1)) # _Chai Wah Wu_, Feb 28 2026

%Y Cf. A000045, A069921, A022121.

%K nonn,easy

%O 0,1

%A _Birhanu Gebrehanna Habtemariam_, Feb 14 2026