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).
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
Product_{n>=1} (1 - 1/a(n)) = (1 + 1/sqrt(5))/2 (A242671). - Amiram Eldar, Nov 28 2024
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
KEYWORD
nonn,easy
AUTHOR
R. K. Guy, Mar 01 2003
EXTENSIONS
More terms from James A. Sellers, Mar 03 2003
STATUS
approved