A094585 -id:A094585 - cp's OEIS frontend

A193999 Mirror of the triangle A094585.

1, 3, 2, 6, 5, 3, 11, 10, 8, 5, 19, 18, 16, 13, 8, 32, 31, 29, 26, 21, 13, 53, 52, 50, 47, 42, 34, 21, 87, 86, 84, 81, 76, 68, 55, 34, 142, 141, 139, 136, 131, 123, 110, 89, 55, 231, 230, 228, 225, 220, 212, 199, 178, 144, 89, 375, 374, 372, 369, 364, 356, 343

Offset: 1

Clark Kimberling, Aug 11 2011

nonn

A193999 is obtained by reversing the rows of the triangle A094585.

			First six rows:
   1;
   3,  2;
   6,  5,  3;
  11, 10,  8,  5;
  19, 18, 16, 13,  8;
  32, 31, 29, 26, 21, 13;

Muniru A Asiru, Table of n, a(n) for n = 1..11325

Cf. A094585.

Flat(List([1..11],n->Reversed(List([1..n],k->Fibonacci(n+3)-Fibonacci(n-k+3))))); # Muniru A Asiru, Apr 28 2019

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A094585 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193999 *)
(* alternate program *)
Table[Fibonacci[n+3]-Fibonacci[k+2], {n,1,10}, {k,1,n}] //TableForm (* Rigoberto Florez, Oct 03 2019 *)

Write w(n,k) for the triangle at A094585.  The triangle at A094585 is then given by w(n,n-k).
T(n,k) = Fibonacci(n+3) - Fibonacci(k+2) for n > 0 and 1 <= k <= n. - Rigoberto Florez, Oct 03 2019
G.f.: x*y*(x*y+x+1)/((1-x)*(x^2+x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 20 2025

Offset 1 from Muniru A Asiru, Apr 29 2019

A094584 Dot product of (1,2,...,n) and first n distinct Fibonacci numbers.

Original entry on oeis.org

1, 5, 14, 34, 74, 152, 299, 571, 1066, 1956, 3540, 6336, 11237, 19777, 34582, 60134, 104062, 179320, 307855, 526775, 898706, 1529160, 2595624, 4396224, 7431049, 12537917, 21118814, 35517226, 59646386, 100034456, 167562035, 280348531, 468543802, 782277612

Offset: 1

Clark Kimberling, May 13 2004

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

			a(4) = (1,2,3,4)*(1,2,3,5) = 1+4+9+20 = 34.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.
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]

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. [From _Emeric Deutsch_, Jun 14 2010]
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A000045, A094585.

Partial sums of A023607.

List([1..40],n->(n+1)*Fibonacci(n+3)-Fibonacci(n+5)+3); # Muniru A Asiru, Apr 27 2019

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

[n*Fibonacci(n+3)-Fibonacci(n+4)+3: n in [1..40]]; // G. C. Greubel, Apr 28 2019

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

Table[Range[n].Fibonacci[Range[2,n+1]],{n,40}] (* Harvey P. Dale, Aug 21 2011 *)

{a(n) = n*fibonacci(n+3) - fibonacci(n+4) +3}; \\ G. C. Greubel, Apr 28 2019

[n*fibonacci(n+3) - fibonacci(n+4) +3 for n in (1..40)] # G. C. Greubel, Apr 28 2019

a(n) = F(2) + 2*F(3) + 3*F(4) + ... + n*F(n+1) = (n+1)*F(n+3) - F(n+5) + 3.
G.f.: x*(1+2*x)/((1-x)*(1-x-x^2)^2). - Colin Barker, Nov 11 2012
From Wesley Ivan Hurt, Mar 10 2015: (Start)
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).
a(n) = Sum_{i=1..n+2} (n-i+1) * F(n-i+2).
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)

A094586 Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.

Original entry on oeis.org

1, 5, 16, 47, 131, 356, 953, 2529, 6676, 17567, 46135, 121016, 317201, 831053, 2176712, 5700303, 14926171, 39081404, 102323209, 267896585, 701380076, 1836265535, 4807451951, 12586147632, 32951083681, 86267253461, 225850919488

Offset: 1

Clark Kimberling, May 13 2004

As T is also the triangle of sums of consecutive distinct Fibonacci numbers, a(n) is such a sum, namely Sum_{j=n+1..2n} Fibonacci(j).

			a(4) = F(10)-F(6) = 55-8 = 47.

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A094585.

List([1..30],n->Fibonacci(2*n+2)-Fibonacci(n+2)); # Muniru A Asiru, Apr 28 2019

F:=Fibonacci; [F(2*n+2)-F(n+2): n in [1..30]]; // G. C. Greubel, Jul 14 2019

Table[Sum[Fibonacci[n+i], {i,n}], {n,30}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{F=Fibonacci}, Table[F[2n+2]-F[n+2], {n,30}]] (* G. C. Greubel, Jul 14 2019 *)
LinearRecurrence[{4,-3,-2,1},{1,5,16,47},30] (* Harvey P. Dale, Dec 31 2024 *)

vector(30, n, f=fibonacci; f(2*n+2)-f(n+2)) \\ G. C. Greubel, Jul 14 2019

f=fibonacci; [f(2*n+2)-f(n+2) for n in (1..30)] # G. C. Greubel, Jul 14 2019

a(n) = Fibonacci(2n+2) - Fibonacci(n+2) = A094585(2n-1, n).

G.f.: x*(1+x-x^2)/((1-x-x^2)*(1-3*x+x^2)). - Colin Barker, Sep 16 2012

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A193999 Mirror of the triangle A094585.

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A094584 Dot product of (1,2,...,n) and first n distinct Fibonacci numbers.

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A094586 Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula