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A022097 Fibonacci sequence beginning 1, 7. 13
1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, 2872, 4647, 7519, 12166, 19685, 31851, 51536, 83387, 134923, 218310, 353233, 571543, 924776, 1496319, 2421095, 3917414, 6338509, 10255923, 16594432, 26850355, 43444787, 70295142, 113739929 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(7;n-1-k,k) with n>=1, a(-1)=6. These are the SW-NE diagonals in P(7;n,k), the (7,1) Pascal triangle A093564. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (perhaps the same as A001175). - R. J. Mathar, Aug 10 2012

LINKS

Table of n, a(n) for n=0..36.

Tanya Khovanova, Recursive Sequences

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

FORMULA

a(n) = a(n-1) + a(n-2) for n>=2, a(0)=1, a(1)=7, a(-1):=6.

G.f.: (1+6*x)/(1-x-x^2).

Row sums of triangle A131778 starting (1, 7, 8, 15, 23, 38,...). - Gary W. Adamson, Jul 14 2007

a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(-13 + sqrt(5)) + (1 + sqrt(5))^n*(13 + sqrt(5))))/sqrt(5). - Herbert Kociemba

a(n) = 6*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012

a(n) = 7*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015

a(n) = 8*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

MAPLE

with(combinat):  P:=proc(q) local n; for n from 0 to q do

print(7*fibonacci(n)+fibonacci(n-1)); od; end: P(10^2); # Paolo P. Lava, May 18 2015

MATHEMATICA

First /@ NestList[{Last@ #, Total@ #} &, {1, 7}, 36] (* or *)

CoefficientList[Series[(1 + 6 x)/(1 - x - x^2), {x, 0, 36}], x] (* Michael De Vlieger, Feb 20 2017 *)

PROG

(MAGMA) a0:=1; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

(PARI) a(n)=([0, 1; 1, 1]^n*[1; 7])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

a(n) = A101220(6, 0, n+1) = A109754(6, n+1) = A118654(3, n).

Cf. A000045, A131778.

Sequence in context: A231390 A231458 A070424 * A041100 A129658 A041693

Adjacent sequences:  A022094 A022095 A022096 * A022098 A022099 A022100

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 22 12:02 EDT 2017. Contains 289669 sequences.