A023607 -id:A023607 - cp's OEIS frontend

A045925 a(n) = n*Fibonacci(n).

0, 1, 2, 6, 12, 25, 48, 91, 168, 306, 550, 979, 1728, 3029, 5278, 9150, 15792, 27149, 46512, 79439, 135300, 229866, 389642, 659111, 1112832, 1875625, 3156218, 5303286, 8898708, 14912641, 24961200, 41734339, 69705888, 116311074, 193898158, 322961275, 537492672

Offset: 0

Jeff Burch

Number of levels in all compositions of n+1 with only 1's and 2's.
Apart from first term: row sums of the triangle in A131410. - Reinhard Zumkeller, Oct 07 2012
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third one. - Sergey Kitaev, Dec 08 2020

Jean Paul Van Bendegem, The Heterogeneity of Mathematical Research, a chapter in Perspectives on Interrogative Models of Inquiry, Volume 8 of the series Logic, Argumentation & Reasoning pp 73-94, Springer 2015. See Section 2.1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Russell Euler, Problem B-670, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 28, No. 3 (1990), p. 277; Application of Generating Functions, Solution to Problem B-670 by Russell Jay Hendel, ibid., Vol. 29, No. 3 (1991), p. 278.
Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics, Vol. 26, No. 3 (2019), Article P3.26.
Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
A. G. Shannon, B. Kuloğlu, and E. Özkan, Rhaly terraced sequences their generalizations, properties and applications, Comp. Appl. Math. 44, 226 (2025). See p. 2.
Kai Ting Keshia Yap, David Wehlau and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Partial sums: A014286. Cf. A000045.

Cf. A099920, A023607.

a045925 n = a045925_list !! (n-1)
a045925_list = zipWith (*) [0..] a000045_list
-- Reinhard Zumkeller, Oct 01 2012

[n*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

a:= n-> n*(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..37);  # Alois P. Heinz, May 07 2021

Table[Fibonacci[n]*n, {n, 0, 33}] (* Zerinvary Lajos, Jul 09 2009 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 1, 2, 6}, 34] (* or *)
CoefficientList[ Series[(x + x^3)/(-1 + x + x^2)^2, {x, 0, 35}], x] (* Robert G. Wilson v, Nov 14 2015 *)

Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=polcoeff(sum(m=1,n,eulerphi(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n) \\ Paul D. Hanna, Jan 12 2012

a(n)=n*fibonacci(n) \\ Charles R Greathouse IV, Jan 12 2012

concat(0, Vec(x*(1+x^2)/(1-x-x^2)^2 + O(x^100))) \\ Altug Alkan, Oct 28 2015

G.f.: x*(1+x^2)/(1-x-x^2)^2.
G.f.: Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n, where phi(n) = A000010(n) and Lucas(n) = A000204(n). - Paul D. Hanna, Jan 12 2012
a(n) = a(n-1) + a(n-2) + L(n-1). - Gary Detlefs, Dec 29 2012
a(n) = F(n+1) + Sum_{k=1..n-2} F(k)*L(n-k), F = A000045 and L = A000032. - Gary Detlefs, Dec 29 2012
a(n) = F(2*n)/Sum_{k=0..floor(n/2)} binomial(n-k,k)/(n-k). - Gary Detlefs, Jan 19 2013
a(n) = A014965(n) * A104714(n). - Michel Marcus, Oct 24 2013
a(n) = 3*A001629(n+1) - A001629(n+2) + A000045(n-1). - Ralf Stephan, Apr 26 2014
a(n) = 2*n*(F(n-2) + floor(F(n-3)/2)) + (n^3 mod 3*n), F = A000045. - Gary Detlefs, Jun 06 2014
E.g.f.: x*(exp(-x/phi)/phi + exp(x*phi)*phi)/sqrt(5), where phi = (1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 28 2015
This is a divisibility sequence and is generated by  x^4 - 2*x^3 - x^2 + 2*x + 1. - R. K. Guy, Nov 13 2015
a(n) = L'(n, 1), the first derivative of the n-th Lucas polynomial evaluated at 1. - Andrés Ventas, Nov 12 2021
Sum_{n>=0} a(n)/2^n = 10 (Euler, 1990). - Amiram Eldar, Jan 22 2022

Incorrect formula removed by Gary Detlefs, Oct 27 2011

A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

Original entry on oeis.org

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652

Offset: 0

Paul Barry and Ralf Stephan, Oct 15 2004

A Fibonacci-Lucas convolution.
The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008
Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012
a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 2.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 35.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Lucas Cube Graph.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1)

Equals A010049(n) + A001629(n+1).

Cf. A000045, A000032, A045925, A023607.

a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

[(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)

a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(2-x)/(1-x-x^2)^2;
a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012
a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015
a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

A093967 a(n) = n * Pell(n).

Original entry on oeis.org

0, 1, 4, 15, 48, 145, 420, 1183, 3264, 8865, 23780, 63151, 166320, 434993, 1130948, 2925375, 7533312, 19323713, 49395780, 125877071, 319888560, 810893265, 2050891876, 5176349663, 13040153280, 32793453025, 82337215012, 206424991215

Offset: 0

Paul Barry, Apr 21 2004

Binomial transform of A093968.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A006645, A023607, A093835.

I:=[0,1,4,15]; [n le 4 select I[n] else 4*Self(n-1)-2*Self(n-2)-4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 20 2015

seq(fibonacci(n,2)*n, n=0..27); # Zerinvary Lajos, Apr 05 2008

LinearRecurrence[{4,-2,-4,-1}, {0,1,4,15}, 30] (* Vincenzo Librandi, Dec 20 2015 *)

{ default(realprecision, 100); s=sqrt(2); for (n=0, 100, a=n*round(((1+s)^n-(1-s)^n)/(2*s)); write("b093967.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 17 2009

[n*lucas_number1(n,2,-1) for n in (0..30)] # G. C. Greubel, Dec 28 2021

G.f.: x*(1+x^2)/(1 - 2*x - x^2)^2;
a(n) = n*((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2));
a(n) = n * A000129(n).

A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233

Offset: 1

Reinhard Zumkeller, May 20 2006

			Triangle begins as:
   1;
   2,  2;
   3,  4,  3;
   4,  6,  6,  5;
   5,  8,  9, 10,  8;
   6, 10, 12, 15, 16, 13;
   7, 12, 15, 20, 24, 26,  21;
   8, 14, 18, 25, 32, 39,  42,  34;
   9, 16, 21, 30, 40, 52,  63,  68,  55;
  10, 18, 24, 35, 48, 65,  84, 102, 110,  89;
  11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
  12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Fibonacci Number

Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Cf. A023652, A112469.

A119457:= func< n,k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025

(* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Second program *)
A119457[n_,k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n,k], {n,13}, {k,n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)

def A119457(n,k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Apr 16 2025

T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)

A016095 Triangular array T(n,k) read by rows, where T(n,k) = coefficient of x^n*y^k in 1/(1-x-y-(x+y)^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 3, 9, 9, 3, 5, 20, 30, 20, 5, 8, 40, 80, 80, 40, 8, 13, 78, 195, 260, 195, 78, 13, 21, 147, 441, 735, 735, 441, 147, 21, 34, 272, 952, 1904, 2380, 1904, 952, 272, 34, 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55

Offset: 0

N. J. A. Sloane, Jan 23 2001

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  3,  9,  9,  3;
  5, 20, 30, 20,  5;
  8, 40, 80, 80, 40, 8;
  ...

Columns include A000045, A023607. Central diagonal is A102307. Antidiagonal sums are in A063727.

read transforms; 1/(1-x-y-(x+y)^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);

T[n_, k_] := SeriesCoefficient[1/(1-x-y-(x+y)^2), {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2017 *)

G.f.: 1/(1-x-y-(x+y)^2).
T(n,k) = Fibonacci(n+1)*binomial(n,k) = A000045(n+1)*A007318(n,k). - Philippe Deléham, Oct 14 2006
Sum_{k=0..floor(n/2)} T(n-k,k) = A123392(n). - Philippe Deléham, Oct 14 2006
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x*(1+y)/((2*k+2+ x*(1+y))*x*(1+y)+ 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = T(n-1,k)+T(n-1,k-1)+T(n-2,k)+2*T(n-2,k-1)+T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = T(2,2) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013

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)

A108035 Triangle read by rows: n-th row consists of n copies of the n-th nonzero Fibonacci number.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233

Offset: 1

N. J. A. Sloane, Jun 01 2005

			1; 2,2; 3,3,3; 5,5,5,5; 8,8,8,8,8; ...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000045, A039913, A108036, A108037.

Cf. A023607 (row sums).

a108035 n k = a108035_tabl !! (n-1) !! (n-1)
a108035_row n = a108035_tabl !! (n-1)
a108035_tabl = zipWith replicate [1..] $ drop 2 a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Flatten[Table[Table[Fibonacci[n],{n-1}],{n,13}]] (* Harvey P. Dale, Jul 18 2015 *)

from math import isqrt
from sympy import fibonacci
def A108035(n): return int(fibonacci(1+(m:=isqrt(k:=n<<1))+(k>m*(m+1)))) # Chai Wah Wu, Nov 07 2024

G.f.: (1+x+y)/((1-x-x^2)*(1-y-y^2)). [U coordinates]

Definition clarified by N. J. A. Sloane, Nov 09 2024

A178521 The cost of all leaves in the Fibonacci tree of order n.

Original entry on oeis.org

0, 0, 3, 7, 17, 35, 70, 134, 251, 461, 835, 1495, 2652, 4668, 8163, 14195, 24565, 42331, 72674, 124354, 212155, 360985, 612743, 1037807, 1754232, 2959800, 4985475, 8384479, 14080601, 23614931, 39556030, 66181310, 110608187, 184670693, 308030923, 513334855

Offset: 0

Emeric Deutsch, Jun 14 2010

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 leaf 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 leaf.

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Colin Barker, Table of n, a(n) for n = 0..1000
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A136376, A354044.

# The function 'fibrec' is defined in A354044.
function A178521(n)
    n < 2 && return BigInt(0)
    a, b = fibrec(n - 1)
    a*n + (n - 1)*b
end
println([A178521(n) for n in 0:35]) # Peter Luschny, May 16 2022

with(combinat); seq(n*fibonacci(n+1)-fibonacci(n), n = 0 .. 35);

Table[n Fibonacci[n + 1] - Fibonacci[n], {n, 0, 40}]  (* Harvey P. Dale, Apr 21 2011 *)
Table[(n - 1) Fibonacci[n] + n Fibonacci[n - 1], {n, 0, 40}] (* Bruno Berselli, Dec 06 2013 *)

concat(vector(2), Vec(x^2*(x+3)/(x^2+x-1)^2 + O(x^50))) \\ Colin Barker, Jul 26 2017

a(n) = n*F(n+1) - F(n), where F(k) = A000045(k).
G.f.: x^2*(x+3)/(x^2+x-1)^2. - Colin Barker, Nov 11 2012
a(n) = Sum_{k=1..n-1} F(k) * L(n-k+1) where F(n) = A000045(n), L(n) = A000032(n). - Gary Detlefs, Dec 29 2012
a(n) = (n-1)*F(n) + n*F(n-1). - Bruno Berselli, Dec 06 2013
a(0) = 0, a(n) = A023607(n-1) + A099920(n-1). - Collin Berman, Dec 12 2016
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4). - Wesley Ivan Hurt, Dec 14 2016
a(n) = (n+1)*F(n+1) - F(n+2). - Bruno Berselli, Jul 26 2017
a(n) = (2^(-1-n)*(2*sqrt(5)*((1-sqrt(5))^n - (1+sqrt(5))^n) + (-(1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))*n))/5. - Colin Barker, Jul 26 2017
a(n) = (-n*sin(c*(-n - 1)) - sin(c*n)*i)/((-i)^(-n)*sqrt(5/4)) where c = arccos(i/2). - Peter Luschny, May 16 2022

A215082 Related to Fibonacci numbers, see the Formula section.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 8, 12, 17, 23, 35, 43, 66, 81, 124, 148, 229, 266, 414, 476, 742, 842, 1318, 1478, 2320, 2581, 4059, 4481, 7062, 7743, 12224, 13328, 21071, 22857, 36185, 39073, 61930, 66605, 105678, 113242, 179847, 192084, 305326, 325128, 517212, 549252

Offset: 0

Philippe Deléham, Aug 02 2012

			a(2) + a(3) = 2*2 = 4 -> a(3) = 3.
a(4) = a(3) + a(1) = 3 + 1 = 4.
a(4) + a(5) = 3*3 = 9 -> a(5) = 5.
a(6) = a(5) + a(3) = 5 + 3 = 8 , etc.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000045, A023607.

a:= n-> (Matrix(6, (i, j)-> `if`(i=j-1, 1, `if`(i=6, [-1, -3, -2, 1, 2, 1][j], 0)))^iquo(n, 2, 'r'). `if`(r=0, <<0, 1, 4, 8, 17, 35>>, <<1, 3, 5, 12, 23, 43>>))[1, 1]: seq (a(n), n=0..50);  # Alois P. Heinz, Aug 02 2012

a(0) = 0, a(1) = 1, a(2) = 1, a(2n) + a(2n+1) = (n+1)*Fibonacci(n+2), a(2n) = a(2n-1) + a(2n-3).

G.f.: x*(2*x^2+1)*(x^3+x+1) / ((x^2-x+1)*(x^2+x+1)*(x^4+x^2-1)^2). - Alois P. Heinz, Aug 02 2012

A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.

Original entry on oeis.org

0, -1, 1, 1, 5, 10, 22, 43, 83, 155, 285, 516, 924, 1639, 2885, 5045, 8773, 15182, 26162, 44915, 76855, 131119, 223101, 378696, 641400, 1084175, 1829257, 3081193, 5181893, 8702290, 14594830, 24446971, 40902299, 68359619, 114132765, 190373580, 317258388, 528265207

Offset: 0

Bruno Berselli, Nov 04 2015

a(n) is prime for n = 4, 7, 8, 26, 28, 52, 86, 87, 93, 97, 158, 196, 303, 2908, 3412, 4111, 4208, 6183, 6337, 9878, ...

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A001629, A007502, A354044.
Cf. A178521: n*F(n+1) + (n+1)*F(n).
Cf. A094588: n*F(n-1) + F(n).
Cf. A099920: Sum_{i=0..n} F(i)*L(n-i).
Cf. A023607: Sum_{i=0..n} F(i)*L(n+1-i).

# The function 'fibrec' is defined in A354044.
function A264147(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n)
    n*b - a*(n + 1)
end # Peter Luschny, May 16 2022

[n*Fibonacci(n+1)-(n+1)*Fibonacci(n): n in [0..40]];

A264147 := proc(n)
    n*combinat[fibonacci](n+1)-(n+1)*combinat[fibonacci](n) ;
end proc:
seq(A264147(n),n=0..10) ; # R. J. Mathar, Jun 02 2022

Table[n Fibonacci[n + 1] - (n + 1) Fibonacci[n], {n, 0, 40}]

makelist(n*fib(n+1)-(n+1)*fib(n), n, 0, 40);

for(n=0, 40, print1(n*fibonacci(n+1)-(n+1)*fibonacci(n)", "));

concat(0, Vec(-x*(1 - 3*x) / (1 - x - x^2)^2 + O(x^50))) \\ Colin Barker, Jul 27 2017

[n*fibonacci(n+1)-(n+1)*fibonacci(n) for n in (0..40)]

G.f.:  x*(-1 + 3*x)/(1 - x - x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
a(n) = n*F(n-1) - F(n).
a(n) = Sum_{i=0..n} F(i)*L(n-1-i), where L() is a Lucas number (A000032).
a(n) = 3*A001629(n) - A001629(n+1).
a(n) = -(-1)^n*A178521(-n).
a(n+2) - a(n) = A007502(n+1).
Sum_{i>0} 1/a(i) = 1.39516607051636028893879220294180374...
a(n) = (-((1+sqrt(5))/2)^n*(2*sqrt(5) + (-5+sqrt(5))*n) + ((1-sqrt(5))/2)^n*(2*sqrt(5) + (5+sqrt(5))*n)) / 10. - Colin Barker, Jul 27 2017
a(n) = (-i)^n*(n*sin(c*(n+1)) - (n+1)*sin(c*n)*i)/sqrt(5/4) where c = arccos(i/2). - Peter Luschny, May 16 2022

cp's OEIS Frontend

A045925 a(n) = n*Fibonacci(n).

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A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

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A093967 a(n) = n * Pell(n).

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A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

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A016095 Triangular array T(n,k) read by rows, where T(n,k) = coefficient of x^n*y^k in 1/(1-x-y-(x+y)^2).

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

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A264147 a(n) = nF(n+1) - (n+1)F(n), where F = A000045.