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%I #44 Feb 28 2024 16:22:46
%S 1,1,4,7,13,22,37,61,100,163,265,430,697,1129,1828,2959,4789,7750,
%T 12541,20293,32836,53131,85969,139102,225073,364177,589252,953431,
%U 1542685,2496118,4038805,6534925,10573732,17108659,27682393,44791054,72473449
%N a(n) = a(n-1) + a(n-2) + 2 where a(0) = a(1) = 1.
%C This is the sequence A(1,1;1,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - _Wolfdieter Lang_, Oct 17 2010
%H T. D. Noe, <a href="/A111314/b111314.txt">Table of n, a(n) for n=0..500</a>
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 13.
%H Wolfdieter Lang, <a href="/A111314/a111314.pdf">Notes on certain inhomogeneous three term recurrences</a>.
%H Elif Tan and Ho-Hon Leung, <a href="https://doi.org/10.5281/zenodo.7569221">On Leonardo p-Numbers</a>, Integers (2023) Vol. 23, #A7.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F a(n) = 2*Fib(n+1)-Fib(n+2)+Fib(n+3)-2. - _Robert G. Wilson v_, Nov 10 2005
%F G.f.: (2x^2-x+1)/((x-1)(x^2+x-1)). - _T. D. Noe_, Oct 19 2007
%F a(n) = F(n-1)+F(n+3)-2, where F(n) is the n-th Fibonacci number. - _Zerinvary Lajos_, Jan 31 2008
%F a(n) = 3*Fib(n+1)-2. - _Olivier Pirson_, Jun 30 2015
%p with(combinat): seq(fibonacci(n-1)+fibonacci(n+3)-2, n=0..35); # _Zerinvary Lajos_, Jan 31 2008
%t a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + 2; Table[ a[n], {n, 0, 36}] (* _Robert G. Wilson v_ *)
%t RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+2},a,{n,40}] (* _Harvey P. Dale_, Mar 27 2022 *)
%o (Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,1,1, lambda n: 2); [next(it) for i in range(1,38)] # _Zerinvary Lajos_, Jul 09 2008
%Y Cf. A000045, A000071.
%K easy,nonn
%O 0,3
%A _Parthasarathy Nambi_, Nov 03 2005
%E More terms from _Robert G. Wilson v_, Nov 07 2005
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