A081007 a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1).

0, 4, 33, 232, 1596, 10945, 75024, 514228, 3524577, 24157816, 165580140, 1134903169, 7778742048, 53316291172, 365435296161, 2504730781960, 17167680177564, 117669030460993, 806515533049392, 5527939700884756, 37889062373143905, 259695496911122584

Offset: 0

Author

R. K. Guy, Mar 01 2003

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. A081018, A242671.

Programs

GAP

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

Magma

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

Maple

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

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

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

Extensions

More terms from James Sellers, Mar 03 2003