A280154 a(n) = 5*Lucas(n).

10, 5, 15, 20, 35, 55, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6820, 11035, 17855, 28890, 46745, 75635, 122380, 198015, 320395, 518410, 838805, 1357215, 2196020, 3553235, 5749255, 9302490, 15051745, 24354235, 39405980, 63760215, 103166195, 166926410, 270092605, 437019015

Offset: 0

Author

Bruno Berselli, Dec 27 2016

Keywords

• nonn
• easy

Comments

Fibonacci sequence beginning 10, 5.

After 5, the sequence provides the 3rd column of the rectangular array in A213590.

After 5, all terms belong to A191921 because a(n) = Lucas(n+4) - 3*Lucas(n-1).

From G. C. Greubel, Dec 27 2016: (Start)

{a(n) mod 3} yields (1,2,0,2,2,1,0,1), repeated, and is given as A082115.

{a(n) mod 6} yields (4,5,3,2,5,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1) and is given as A082117. (End)

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Subsequence of A084176.

Cf. A022088: 5*Fibonacci(n).

Cf. A022359: Lucas(n+5) + Lucas(n-5).

Cf. A000032, A000045, A191921, A213590.

Cf. sequences with formula Fibonacci(n+k) + Fibonacci(n-k): A006355 (k=0, without the initial 1), A000032 (k=1), A022086 (k=2), A022112 (k=3, with an initial 4), A022090 (k=4), this sequence (k=5), A022352 (k=6).

Programs

Magma

[5*Lucas(n): n in [0..40]];

Maple

F := n -> combinat:-fibonacci(n):
seq(F(n+5) + F(n-5), n=0..38); # Peter Luschny, Dec 29 2016

Mathematica

Table[5 LucasL[n], {n, 0, 40}]

PARI

vector(40, n, n--; fibonacci(n+5)+fibonacci(n-5))

Sage

def A280154():
    x, y = 10, 5
    while True:
        yield x
        x, y = y, x + y
a = A280154(); print([next(a) for  in range(39)]) # _Peter Luschny, Dec 29 2016

Formula

G.f.: 5*(2 - x)/(1 - x - x^2).

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

a(n) = Fibonacci(n+5) + Fibonacci(n-5), with Fibonacci(-k) = -(-1)^k*Fibonacci(k) for the negative indices.