This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A081007 a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1). 3

%I

%S 0,4,33,232,1596,10945,75024,514228,3524577,24157816,165580140,

%T 1134903169,7778742048,53316291172,365435296161,2504730781960,

%U 17167680177564,117669030460993,806515533049392,5527939700884756,37889062373143905,259695496911122584

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

%C 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

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

%H Nathaniel Johnston, <a href="/A081007/b081007.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).

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

%F 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

%F G.f.: x*(4+x)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 24 2012

%F a(n) = Sum_{i=1..2n} binomial(2n+i, 2n-i). - _Wesley Ivan Hurt_, Oct 06 2013

%F 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

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

%t Table[Fibonacci[4n+1] -1, {n,0,30}] (* _Wesley Ivan Hurt_, Oct 06 2013 *)

%t LinearRecurrence[{8,-8,1},{0,4,33},30] (* _Harvey P. Dale_, Jul 31 2018 *)

%t Table[Fibonacci[2n]LucasL[2n+1], {n,0,30}] (* _Rigoberto Florez_, Apr 19 2019 *)

%o (MAGMA) [Fibonacci(4*n+1) -1: n in [0..30]]; // _Vincenzo Librandi_, Apr 15 2011

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

%o makelist(A081007(n),n,0,30); /* _Martin Ettl_, Nov 12 2012 */

%o (PARI) concat(0, Vec(x*(4+x)/((1-x)*(1-7*x+x^2)) + O(x^30))) \\ _Colin Barker_, Dec 20 2014

%o (PARI) vector(30, n, n--; fibonacci(4*n+1)-1) \\ _G. C. Greubel_, Jul 14 2019

%o (Sage) [fibonacci(4*n+1)-1 for n in (0..30)] # _G. C. Greubel_, Jul 14 2019

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

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

%Y Cf. A081018.

%K nonn,easy

%O 0,2

%A _R. K. Guy_, Mar 01 2003

%E More terms from _James A. Sellers_, Mar 03 2003

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 15:11 EST 2019. Contains 329753 sequences. (Running on oeis4.)