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A022388 Fibonacci sequence beginning 6, 13. 4
6, 13, 19, 32, 51, 83, 134, 217, 351, 568, 919, 1487, 2406, 3893, 6299, 10192, 16491, 26683, 43174, 69857, 113031, 182888, 295919, 478807, 774726, 1253533, 2028259, 3281792, 5310051, 8591843, 13901894, 22493737, 36395631, 58889368, 95284999, 154174367, 249459366, 403633733, 653093099 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The Pisano periods for this sequence are different from those for the Fibonacci numbers (A001175) for modulus 11 and 22. Furthermore, its Pisano periods are exactly the same as those of the Lucas sequence (A000032), given in A106291. - Klaus Purath, Apr 20 2019

a(n) is the alternating sum of 5 consecutive Lucas numbers (A000032). Also the sum of 4*k consecutive terms of A000285 divided by Fibonacci(2*k) (A000045), k = {1, 2, …}. All involved sequences extended to negative indices, following the rule a(n-1) = a(n+1) - a(n). - Klaus Purath, Jul 29 2019

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

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

FORMULA

G.f.: (6+7*x)/(1-x-x^2). - Philippe Deléham, Nov 20 2008

a(n) = 6*Fibonacci(n+2) + Fibonacci(n) = 6*Fibonacci(n-1) + 13*Fibonacci(n). - G. C. Greubel, Mar 02 2018

From Klaus Purath, Jul 29 2019: (Start)

L = Lucas (A000032), F = Fibonacci (A000045). All involved sequences extended to negative indices, following the rule a(n-1) = a(n+1) - a(n).

a(n+1) - a(n-4) = L(n)*11.

a(n) = L(n-1) + L(n+4).

a(n) = 3*L(n+1) + L(n+2) = L(n) + 4*L(n+1) = L(n+6) - 4*L(n+2).

a(n) = L(n+1) + 5*F(n+2) = L(n+5) - 5*F(n+1).

a(n) = (7*L(n+1) + 5*F(n+1))/2.

a(n) = (13*L(n+1) + L(n+5) - 5*F(n))/4.

a(n) = 7*F(n) + 6*F(n+1) = 7*F(n+2) - F(n+1).

a(n) = 8*F(n+2) - F(n+3) = 17*F(n+4) - 9*F(n+5).

The following six formulas apply for all sequences of the Fibonacci type.

a(n) = L(2*m)*a(n+2*m) - a(n+4*m).

a(n) = (F(m+2)*a(n+2) - a(m+n+2))/F(m).

a(n) = F(n-m-1)*a(m) + F(n-m)*a(m+1).

a(n)^2 + a(n+3)^2 = 2*(a(n+1)^2 + a(n+2)^2).

a(n)^2 + a(n+2)^2 + a(n+1)^2 + a(n+3)^2 = 3*(a(n)*a(n+2) + a(n+1)*a(n+3)).

3*a(n+2)*a(n+1)*a(n) = a(n+2)^3 - a(n+1)^3 - a(n)^3. (End)

E.g.f.: exp(-2*x/(1+sqrt(5)))*(-15-sqrt(5)+(45+19*sqrt(5))*exp(sqrt(5)*x))/(5+3*sqrt(5)). - Stefano Spezia, Aug 16 2019

MATHEMATICA

Table[6*Fibonacci[n+2] + Fibonacci[n], {n, 0, 40}] (* or *) LinearRecurrence[{1, 1}, {6, 13}, 40] (* G. C. Greubel, Mar 02 2018 *)

PROG

(PARI) vector(40, n, n--; 6*fibonacci(n+2) + fibonacci(n)) \\ G. C. Greubel, Mar 02 2018

(MAGMA) [6*Fibonacci(n+2) + Fibonacci(n): n in [0..40]]; // G. C. Greubel, Mar 02 2018

(Sage) [6*fibonacci(n+2) + fibonacci(n) for n in (0..40)] # G. C. Greubel, Jun 30 2019

(GAP) List([0..40], n-> 6*Fibonacci(n+2) + Fibonacci(n)) # G. C. Greubel, Jun 30 2019

CROSSREFS

Cf. A000032.

Sequence in context: A246306 A135274 A189378 * A041471 A041695 A109236

Adjacent sequences:  A022385 A022386 A022387 * A022389 A022390 A022391

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Terms a(36) onward added by G. C. Greubel, Mar 02 2018

STATUS

approved

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)