This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094584 #37 Jan 05 2025 19:51:37
%S A094584 1,5,14,34,74,152,299,571,1066,1956,3540,6336,11237,19777,34582,60134,
%T A094584 104062,179320,307855,526775,898706,1529160,2595624,4396224,7431049,
%U A094584 12537917,21118814,35517226,59646386,100034456,167562035,280348531,468543802,782277612
%N A094584 Dot product of (1,2,...,n) and first n distinct Fibonacci numbers.
%C A094584 a(n) is the cost of all non-leaf nodes in the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. In a Fibonacci tree the cost of a left (right) edge is defined to be 1 (2). The cost of a node of a Fibonacci tree is defined to be the sum of the costs of the edges that form the path from the root to this node. - _Emeric Deutsch_, Jun 14 2010
%D A094584 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
%D A094584 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. [From _Emeric Deutsch_, Jun 14 2010]
%H A094584 Muniru A Asiru, <a href="/A094584/b094584.txt">Table of n, a(n) for n = 1..2000</a>
%H A094584 Y. Horibe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From _Emeric Deutsch_, Jun 14 2010]
%H A094584 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F A094584 a(n) = F(2) + 2*F(3) + 3*F(4) + ... + n*F(n+1) = (n+1)*F(n+3) - F(n+5) + 3.
%F A094584 G.f.: x*(1+2*x)/((1-x)*(1-x-x^2)^2). - _Colin Barker_, Nov 11 2012
%F A094584 From _Wesley Ivan Hurt_, Mar 10 2015: (Start)
%F A094584 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).
%F A094584 a(n) = Sum_{i=1..n+2} (n-i+1) * F(n-i+2).
%F A094584 a(n) = (30*(-1-sqrt(5))^n + (-15+7*sqrt(5))*2^n - (15+7*sqrt(5))*(-3-sqrt(5))^n + 2n*((5-2*sqrt(5))*2^n + (5+2*sqrt(5))*(-3-sqrt(5))^n)) / (10*(-1-sqrt(5))^n). (End)
%e A094584 a(4) = (1,2,3,4)*(1,2,3,5) = 1+4+9+20 = 34.
%p A094584 with(combinat): A094584:=n->(n+1)*fibonacci(n+3)-fibonacci(n+5)+3: seq(A094584(n), n=1..50); # _Wesley Ivan Hurt_, Mar 10 2015
%t A094584 Table[Range[n].Fibonacci[Range[2,n+1]],{n,40}] (* _Harvey P. Dale_, Aug 21 2011 *)
%o A094584 (Magma) I:=[1,5,14,34,74]; [n le 5 select I[n] else 3*Self(n-1)-Self(n-2)-3*Self(n-3)+Self(n-4)+Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Mar 11 2015
%o A094584 (Magma) [n*Fibonacci(n+3)-Fibonacci(n+4)+3: n in [1..40]]; // _G. C. Greubel_, Apr 28 2019
%o A094584 (GAP) List([1..40],n->(n+1)*Fibonacci(n+3)-Fibonacci(n+5)+3); # _Muniru A Asiru_, Apr 27 2019
%o A094584 (PARI) {a(n) = n*fibonacci(n+3) - fibonacci(n+4) +3}; \\ _G. C. Greubel_, Apr 28 2019
%o A094584 (Sage) [n*fibonacci(n+3) - fibonacci(n+4) +3 for n in (1..40)] # _G. C. Greubel_, Apr 28 2019
%Y A094584 Cf. A000045, A094585.
%Y A094584 Partial sums of A023607.
%K A094584 nonn,easy
%O A094584 1,2
%A A094584 _Clark Kimberling_, May 13 2004