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%I #56 Jan 01 2024 11:07:40
%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 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
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