A393414 Fibonacci sequence beginning 29, 21.

29, 21, 50, 71, 121, 192, 313, 505, 818, 1323, 2141, 3464, 5605, 9069, 14674, 23743, 38417, 62160, 100577, 162737, 263314, 426051, 689365, 1115416, 1804781, 2920197, 4724978, 7645175, 12370153, 20015328, 32385481, 52400809, 84786290, 137187099, 221973389, 359160488

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (1,1).

Formula

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

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

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

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

Example

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:
    o---o---o---o     o---o
    |           |     | /
    o-----o-----o-----o

Maple

a:= n-> (<<0|1>, <1|1>>^n. <<29, 21>>)[1, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 23 2026

Mathematica

a[n_]:= 29*Fibonacci[n+1] -8*Fibonacci[n]; Table[a[n], {n, 0, 40}]
Table[SeriesCoefficient[(29 -8*x)/(1 - x - x^2), {x, 0, n}], {n, 0, 40}]

Prog

(Python)
from gmpy2 import fib2
def A393414(n): return (lambda x:int(29*x[0]-8*x[1]))(fib2(n+1)) # Chai Wah Wu, Feb 28 2026

Crossrefs

Cf. A000045, A069921, A022121.

Sequence in context: A088400 A040814 A307129 * A303615 A291492 A256441

Adjacent sequences: A393411 A393412 A393413 * A393415 A393416 A393417

Keyword

nonn,easy

Author

Birhanu Gebrehanna Habtemariam, Feb 14 2026

Status

approved