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A131269 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=6.

%I A131269 #45 Dec 09 2024 06:00:45
%S A131269 1,2,3,6,11,20,35,60,101,168,277,454,741,1206,1959,3178,5151,8344,
%T A131269 13511,21872,35401,57292,92713,150026,242761,392810,635595,1028430,
%U A131269 1664051,2692508,4356587,7049124,11405741,18454896,29860669,48315598,78176301,126491934
%N A131269 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=6.
%C A131269 Row sums of triangles A131268 and A131270.
%C A131269 a(n)/a(n-1) tends to phi (A001622).
%H A131269 Bruno Berselli, <a href="/A131269/b131269.txt">Table of n, a(n) for n = 0..1000</a>
%H A131269 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A131269 a(n) = a(n-2) + a(n-1) + n - 2 with n>1, a(0)=1, a(1)=2. - _Alex Ratushnyak_, May 02 2012
%F A131269 From _Bruno Berselli_, May 03 2012: (Start)
%F A131269 G.f.: (1-x-x^2+2*x^3)/((1-x-x^2)*(1-x)^2). - _Bruno Berselli_, May 03 2012
%F A131269 a(n) = A001595(n+1) - n = A006355(n+3) - n - 1 = ((1+sqrt(5))^(n+2) - (1-sqrt(5))^(n+2))/(2^(n+1)*sqrt(5))-n-1. (End)
%e A131269 a(4) = 11 = sum of row 4 terms of triangle A131268: (1 + 1 + 5 + 3 + 1), or the reversed terms of triangle A131270, row 4.
%t A131269 LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 6}, 41] (* _Bruno Berselli_, May 03 2012 *)
%t A131269 Table[2*Fibonacci[n+2]-n-1, {n,0,40}] (* _G. C. Greubel_, Jul 09 2019 *)
%o A131269 (Python)
%o A131269 prpr = 1
%o A131269 prev = 2
%o A131269 for n in range(2,99):
%o A131269     current = prpr + prev + n - 2
%o A131269     print(prpr, end=',')
%o A131269     prpr = prev
%o A131269     prev = current  # _Alex Ratushnyak_, May 02 2012
%o A131269 (PARI) Vec((1-x-x^2+2*x^3)/((1-x-x^2)*(1-x)^2)+O(x^40)) \\ _Bruno Berselli_, May 03 2012
%o A131269 (PARI) vector(40, n, n--; 2*fibonacci(n+2)-n-1) \\ _G. C. Greubel_, Jul 09 2019
%o A131269 (Magma) /* By the first comment: */ [&+[2*Binomial(n-Floor((k+1)/2), Floor(k/2))-1: k in [0..n]]: n in [0..40]]; /* _Bruno Berselli_, May 03 2012 */
%o A131269 (Magma) [2*Fibonacci(n+2)-n-1: n in [0..40]]; // _G. C. Greubel_, Jul 09 2019
%o A131269 (Maxima) makelist(expand(((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2) )/(2^(n+1)*sqrt(5))-n-1), n, 0, 40); /* _Bruno Berselli_, May 03 2012 */
%o A131269 (Sage) [2*fibonacci(n+2)-n-1 for n in (0..40)] # _G. C. Greubel_, Jul 09 2019
%o A131269 (GAP) List([0..40], n-> 2*Fibonacci(n+2)-n-1); # _G. C. Greubel_, Jul 09 2019
%Y A131269 Cf. A000045, A001622, A065941, A131268, A131270.
%Y A131269 Cf. A001595 (first differences).
%K A131269 nonn,easy
%O A131269 0,2
%A A131269 _Gary W. Adamson_, Jun 23 2007
%E A131269 Better definition and more terms from _Bruno Berselli_, May 03 2012