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A081007 a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1). 3
0, 4, 33, 232, 1596, 10945, 75024, 514228, 3524577, 24157816, 165580140, 1134903169, 7778742048, 53316291172, 365435296161, 2504730781960, 17167680177564, 117669030460993, 806515533049392, 5527939700884756, 37889062373143905, 259695496911122584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also the index of the first of two consecutive triangular numbers whose sum is equal to the sum of two consecutive heptagonal numbers. - Colin Barker, Dec 20 2014

REFERENCES

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

FORMULA

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

a(n) = -1 + (1/2)*((7/2 - (3/2)*sqrt(5))^n + (7/2 + (3/2)*sqrt(5))^n) + (1/10)*sqrt(5)*((7/2 + (3/2)*sqrt(5))^n - (7/2 - (3/2)*sqrt(5))^n), with n >= 0. - Paolo P. Lava, Dec 01 2008

G.f.: x*(4+x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

a(n) = Sum_{i=1..2n} binomial(2n+i, 2n-i). - Wesley Ivan Hurt, Oct 06 2013

a(n) = Sum_{i=0..2n-1} F(i)*L(i+2), F(i) = A000045(i) and L(i) = A000032(i). - Rigoberto Florez, Apr 19 2019

MAPLE

with(combinat) for n from 0 to 30 do printf(`%d, `, fibonacci(4*n+1)-1) od # James A. Sellers, Mar 03 2003

MATHEMATICA

Table[Fibonacci[4n+1] -1, {n, 0, 30}] (* Wesley Ivan Hurt, Oct 06 2013 *)

LinearRecurrence[{8, -8, 1}, {0, 4, 33}, 30] (* Harvey P. Dale, Jul 31 2018 *)

Table[Fibonacci[2n]LucasL[2n+1], {n, 0, 30}] (* Rigoberto Florez, Apr 19 2019 *)

PROG

(MAGMA) [Fibonacci(4*n+1) -1: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

(Maxima) A081007(n):=fib(4*n+1)-1$

makelist(A081007(n), n, 0, 30); /* Martin Ettl, Nov 12 2012 */

(PARI) concat(0, Vec(x*(4+x)/((1-x)*(1-7*x+x^2)) + O(x^30))) \\ Colin Barker, Dec 20 2014

(PARI) vector(30, n, n--; fibonacci(4*n+1)-1) \\ G. C. Greubel, Jul 14 2019

(Sage) [fibonacci(4*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 14 2019

(GAP) List([0..30], n-> Fibonacci(4*n+1)-1); # G. C. Greubel, Jul 14 2019

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Cf. A081018.

Sequence in context: A097705 A131509 A221030 * A213168 A203212 A088317

Adjacent sequences:  A081004 A081005 A081006 * A081008 A081009 A081010

KEYWORD

nonn,easy

AUTHOR

R. K. Guy, Mar 01 2003

EXTENSIONS

More terms from James A. Sellers, Mar 03 2003

STATUS

approved

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Last modified November 16 22:20 EST 2019. Contains 329208 sequences. (Running on oeis4.)