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A111314 a(n) = a(n-1) + a(n-2) + 2 where a(0) = a(1) = 1.

%I A111314 #54 Sep 16 2024 12:00:28
%S A111314 1,1,4,7,13,22,37,61,100,163,265,430,697,1129,1828,2959,4789,7750,
%T A111314 12541,20293,32836,53131,85969,139102,225073,364177,589252,953431,
%U A111314 1542685,2496118,4038805,6534925,10573732,17108659,27682393,44791054,72473449,117264505,189737956
%N A111314 a(n) = a(n-1) + a(n-2) + 2 where a(0) = a(1) = 1.
%C A111314 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 A111314 T. D. Noe, <a href="/A111314/b111314.txt">Table of n, a(n) for n=0..500</a>
%H A111314 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 A111314 Wolfdieter Lang, <a href="/A111314/a111314.pdf">Notes on certain inhomogeneous three term recurrences</a>.
%H A111314 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 A111314 Feng-Zhen Zhao, <a href="https://math.colgate.edu/~integers/y82/y82.pdf">The log-behavior of some sequences related to the generalized Leonardo numbers</a>, Integers (2024) Vol. 24, Art. No. A82.
%H A111314 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A111314 a(n) = 2*F(n+1)-F(n+2)+F(n+3)-2, where F(n) is the n-th Fibonacci number. - _Robert G. Wilson v_, Nov 10 2005
%F A111314 G.f.: (2*x^2-x+1)/((x-1)*(x^2+x-1)). - _T. D. Noe_, Oct 19 2007
%F A111314 a(n) = F(n-1)+F(n+3)-2. - _Zerinvary Lajos_, Jan 31 2008
%F A111314 a(n) = 3*F(n+1)-2. - _Olivier Pirson_, Jun 30 2015
%F A111314 E.g.f.: 3*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2*exp(x). - _Stefano Spezia_, Jul 21 2024
%p A111314 with(combinat): seq(fibonacci(n-1)+fibonacci(n+3)-2, n=0..35); # _Zerinvary Lajos_, Jan 31 2008
%t A111314 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 A111314 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 A111314 (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 A111314 Cf. A000045, A000071.
%K A111314 easy,nonn
%O A111314 0,3
%A A111314 _Parthasarathy Nambi_, Nov 03 2005
%E A111314 More terms from _Robert G. Wilson v_, Nov 07 2005