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%I #53 Mar 07 2024 16:01:27
%S 0,1,2,5,10,19,34,59,100,167,276,453,740,1205,1958,3177,5150,8343,
%T 13510,21871,35400,57291,92712,150025,242760,392809,635594,1028429,
%U 1664050,2692507,4356586,7049123
%N G.f.: x*(1 - x + x^2)/((1-x)^2 * (1 - x - x^2)).
%C A floretion-generated sequence.
%C Floretion Algebra Multiplication Program, FAMP Code: 1vesrokseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)('i + i' + 'ji' + 'ki' + e) ] RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + p.
%C Partial sums of Leonardo numbers A001595. - _Jonathan Vos Post_, Jan 01 2011
%H G. C. Greubel, <a href="/A104161/b104161.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F Superseeker results (incomplete): a(2) - 2a(n+1) + a(n) = A006355(n+1) (Number of binary vectors of length n containing no singletons); a(n+1) - a(n) = A001595(n) (2-ranks of difference sets constructed from Segre hyperovals); a(n) + n + 1 = A001595(n+1).
%F A107909(a(n)) = A000975(n). - _Reinhard Zumkeller_, May 28 2005
%F From _Ross La Haye_, Aug 03 2005: (Start)
%F a(n) = 2*(Fibonacci(n+2) - 1) - n.
%F a(n) = Sum_{k=0..n} A101220(n-k, 0, k). (End)
%F From _Gary W. Adamson_, Apr 02 2006: (Start)
%F a(n) = a(n-1) + a(n-2) + n-1.
%F a(n) = row sums of A117501, starting (1, 2, 5, 10, ...). (End)
%F a(n) = Sum_{k=0..n} A109754(n-k,k). - _Ross La Haye_, Apr 12 2006
%F a(n) = (Sum_{k=0..n} (n-k)*Fibonacci(k-1) + Fibonacci(k)) - n. - _Ross La Haye_, May 31 2006
%F From _R. J. Mathar_, Apr 18 2008: (Start)
%F a(n) = -2 - n + (-A094214)^n*(1-A010499/5) + (1+A010499/5)/A094214^n.
%F a(n) = A006355(n+3) - n - 2. (End)
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5. - _Harvey P. Dale_, Sep 06 2012
%t a=0;b=1;Table[c=b+a+n; a=b; b=c, {n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2011 *)
%t CoefficientList[Series[x*(1-x+x^2)/((1-x)^2*(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,2,5},40] (* _Harvey P. Dale_, Sep 06 2012 *)
%o (PARI) my(x='x+O('x^40)); concat(0, Vec(x*(1-x+x^2)/((1-x)^2*(1-x-x^2)))) \\ _G. C. Greubel_, Sep 26 2017
%o (Magma) [2*Fibonacci(n+2) -(n+2): n in [0..40]]; // _G. C. Greubel_, Jul 09 2019
%o (SageMath) [2*fibonacci(n+2) -(n+2) for n in (0..40)] # _G. C. Greubel_, Jul 09 2019
%o (GAP) List([0..40], n-> 2*Fibonacci(n+2) -(n+2)); # _G. C. Greubel_, Jul 09 2019
%Y Cf. A000045, A006355, A001595, A117501, A145912.
%K easy,nonn
%O 0,3
%A _Creighton Dement_, Mar 10 2005
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