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A022096 Fibonacci sequence beginning 1 6. 12
1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101, 27670, 44771, 72441, 117212, 189653, 306865, 496518, 803383, 1299901, 2103284, 3403185, 5506469, 8909654, 14416123, 23325777, 37741900, 61067677, 98809577, 159877254 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1)=sum(P(6;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=5. These are the sums of the SW-NE diagonals in P(6;n,k), the (6,1) Pascal triangle A093563. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of SW-NE diagonals in (1,5)-Pascal triangle A096940.

Subsequence of primes: 7, 13, 53, 139, 953, 44771, 189653, 1494692464747, ... - R. J. Mathar, Aug 09 2012

REFERENCES

J. L. Ramírez, G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).

LINKS

Table of n, a(n) for n=0..37.

Tanya Khovanova, Recursive Sequences

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368, 2012

Index to sequences with linear recurrences with constant coefficients, signature (1,1)

FORMULA

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

a(-1)=5.

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

Row sums of triangle A131777. - Gary W. Adamson, Jul 14 2007

a(n) = 5*Fibonacci(n+2)-4*Fibonacci(n+1)  [From Gary Detlefs Dec 21 2010]

a(n) = (2^(-1-n) ((1-Sqrt[5])^n (-11+Sqrt[5])+(1+Sqrt[5])^n (11+Sqrt[5])))/Sqrt[5] - Herbert Kociemba

MATHEMATICA

a={}; b=1; c=6; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 9, 1}]; a (* Vladimir Orlovsky, Jul 22 2008 *)

PROG

(MAGMA) a0:=1; a1:=6; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

CROSSREFS

a(n) = A101220(5, 0, n+1).

a(n) = A109754(5, n+1).

Cf. A131777.

Sequence in context: A127020 A154662 A070398 * A041175 A041074 A041749

Adjacent sequences:  A022093 A022094 A022095 * A022097 A022098 A022099

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Spelling correction by Jason G. Wurtzel, Aug 22 2010

STATUS

approved

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Last modified April 16 11:56 EDT 2014. Contains 240591 sequences.