A099920 -id:A099920 - cp's OEIS frontend

A099428 Duplicate of A099920.

0, 0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779

Offset: 0

dead

A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 6, 0, 5, 0, 1, 0, 4, 0, 10, 0, 6, 0, 1, 1, 0, 10, 0, 15, 0, 7, 0, 1, 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 1, 0, 15, 0, 35, 0, 28, 0, 9, 0, 1, 0, 6, 0, 35, 0, 56, 0, 36, 0, 10, 0, 1, 1, 0, 21, 0, 70, 0, 84, 0, 45, 0, 11, 0, 1

Offset: 0

Philippe Deléham, Nov 29 2009

Row sums: A000045(n+1), Fibonacci numbers.
A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - Philippe Deléham, Dec 02 2009
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014

			The triangle T(n,k) begins:
n\k 0  1   2   3   4    5    6    7    8    9  10  11  12  13 14 15 ...
0:  1
1:  0  1
2:  1  0   1
3:  0  2   0   1
4:  1  0   3   0   1
5:  0  3   0   4   0    1
6:  1  0   6   0   5    0    1
7:  0  4   0  10   0    6    0    1
8:  1  0  10   0  15    0    7    0    1
9:  0  5   0  20   0   21    0    8    0    1
10: 1  0  15   0  35    0   28    0    9    0   1
11: 0  6   0  35   0   56    0   36    0   10   0   1
12: 1  0  21   0  70    0   84    0   45    0  11   0   1
13: 0  7   0  56   0  126    0  120    0   55   0  12   0   1
14: 1  0  28   0 126    0  210    0  165    0  66   0  13   0  1
15: 0  8   0  84   0  252    0  330    0  220   0  78   0  14  0  1
... reformatted by _Wolfdieter Lang_, Jul 29 2014.
------------------------------------------------------------------------

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Tom Copeland, Addendum to Elliptic Lie Triad
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A162515 (rows reversed), A112552, A102426 (deflated).

A168561:=proc(n,k) if n-k mod 2 = 0 then binomial((n+k)/2,k) else 0 fi end proc:
seq(seq(A168561(n,k),k=0..n),n=0..12) ; # yields sequence in triangular form

Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)

T(n,k) = if ((n+k) % 2, 0, binomial((n+k)/2,k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Oct 09 2016

Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 09 2012
From R. J. Mathar, Feb 04 2022: (Start)
Sum_{k=0..n} T(n,k)*k = A001629(n+1).
Sum_{k=0..n} T(n,k)*k^2 = 0,1,4,11,... = 2*A055243(n)-A099920(n+1).
Sum_{k=0..n} T(n,k)*k^3 = 0,1,8,29,88,236,... = 12*A055243(n) -6*A001629(n+2) +A001629(n+1)-6*(A001872(n)-2*A001872(n-1)). (End)

Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010

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

Original entry on oeis.org

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

A208354 Number of compositions of n with at most one even part.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 72, 126, 219, 379, 653, 1121, 1918, 3272, 5567, 9449, 16003, 27049, 45636, 76866, 129267, 217079, 364057, 609793, 1020218, 1705036, 2846647, 4748101, 7912559, 13174889, 21919488, 36440646, 60538443, 100503667, 166744997, 276476129

Offset: 0

Alois P. Heinz, Feb 25 2012

Conjecture: a(n) is the number of compositions of n if all the 1's are constrained to be in a single run; for example, a(7) counts the compositions 4,1,1,1 and 1,1,1,4 but not the compositions 1,4,1,1 and 1,1,4,1. - Gregory L. Simay, Sep 29 2018

This also gives the number of ordered partitions of n into parts of sizes 1, 2, and 3 with at most one 3. - Jerrold Grossman, Apr 03 2024

			a(4) =  7: {4, 13, 31, 112, 121, 211, 1111}.
a(5) = 13: {5, 14, 41, 23, 32, 113, 131, 311, 1112, 1121, 1211, 2111, 11111}.
a(6) = 23: {6, 15, 51, 33, 114, 141, 411, 123, 132, 213, 231, 312, 321, 1113, 1131, 1311, 3111, 11112, 11121, 11211, 12111, 21111, 111111}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 11.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A010049, A211164.

T:=n->((2*n+3)*Fibonacci(n)-n*Fibonacci(n-1))/5; a:=List([0..40],n->T(n+1)-T(n-1)); # Muniru A Asiru, Oct 28 2018

I:=[1,1,2,4]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Oct 29 2018

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-2|1|2>>^n.
         <<1, 1, 2, 4>>)[1, 1]:
seq(a(n), n=0..40);

LinearRecurrence[{2, 1, -2, -1}, {1, 1, 2, 4}, 40] (* Jean-François Alcover, Feb 18 2017 *)
CoefficientList[Series[((-1 + x)^2 (1 + x))/(-1 + x + x^2)^2, {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

x='x+O('x^50); Vec((x+1)*(x-1)^2/(x^2+x-1)^2) \\ Altug Alkan, Oct 02 2018

G.f.: (x+1)*(x-1)^2/(x^2+x-1)^2.
a(n) = T(n+1) - T(n-1), where T(n) = ((2*n+3)*Fibonacci(n) - n*Fibonacci(n-1)) / 5 = A010049(n). - Gary Detlefs, Jan 19 2013
a(n) = (2*(A099920(n-2)+A000045(n+2)) + A099920(n-1)+A000045(n+1)) / 5. - Yuchun Ji, Mar 21 2019

A122491 a(n) = n * Fibonacci(n) - Sum_{i=0..n} Fibonacci(i).

Original entry on oeis.org

0, 0, 0, 2, 5, 13, 28, 58, 114, 218, 407, 747, 1352, 2420, 4292, 7554, 13209, 22969, 39748, 68494, 117590, 201210, 343275, 584087, 991440, 1679208, 2838408, 4789058, 8066669, 13566373, 22782892, 38209762, 64003002, 107083610, 178967807, 298803459, 498404504

Offset: 0

Ben Paul Thurston, Sep 16 2006

Similar to A190062.

Also the circuit rank and corank of the n-Lucas cube graph. - Eric W. Weisstein, Jul 28 2023

			a(5) = 13 because Fib(5) = 5, times 5 = 25 and subtract sum(Fib(5)) = 12 to get 13.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Carlos Alirio Rico Acevedo, Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Corank
Eric Weisstein's World of Mathematics, Lucas Cube Graph
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A000045.

with(combinat, fibonacci): for i from 1 to 30 do i*fibonacci(i) - sum(fibonacci(k), k=0..i); end do;

Table[n Fibonacci[n] - Fibonacci[n + 2] + 1, {n, 0, 40}] (* Stefan Steinerberger, Feb 22 2008 *)
LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 2, 5}, 40] (* Harvey P. Dale, May 17 2016 *)

a(n)=n*fibonacci(n) - fibonacci(n+2) + 1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n * Fibonacci(n) - Fibonacci(n+2) + 1. - Stefan Steinerberger, Feb 22 2008
G.f.: x^3*(2-x)/((1-x)*(1-x-x^2)^2). - Colin Barker, Feb 10 2012
a(n+2) = Sum_{k=0..n} A099920(k). - J. M. Bergot, Apr 13 2013
a(n) = 2*A006478(n)-A006478(n-1). - R. J. Mathar, May 04 2014

Edited by N. J. A. Sloane, Sep 17 2006

A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 8, 8, 8, 5, 15, 25, 20, 16, 8, 30, 55, 70, 48, 32, 13, 56, 125, 175, 184, 112, 64, 21, 104, 262, 440, 512, 464, 256, 128, 34, 189, 539, 1014, 1401, 1416, 1136, 576, 256, 55, 340, 1075, 2270, 3501, 4170, 3760, 2720, 1280, 512

Offset: 1

Mark Dols, Sep 03 2009

A164975 is jointly generated with A209125 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+ 2x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle T(n,k), 0 <= k < n, n >= 1, begins:
   1;
   1,   2;
   2,   3,   4;
   3,   8,   8,   8;
   5,  15,  25,  20,  16;
   8,  30,  55,  70,  48,  32;
  13,  56, 125, 175, 184, 112,  64;
  21, 104, 262, 440, 512, 464, 256, 128;
  ...
T(7,1) = 30 + 2*8 + 15 - 5 = 56.
T(6,1) = 15 + 2*5 +  8 - 3 = 30.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000045, A000079, A000244 (row sums).

A164975 := proc(n,k) option remember; if n <=0 or k > n or k< 1 then 0; elif k= 1 then combinat[fibonacci](n); else procname(n-1,k)+2*procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc: # R. J. Mathar, Jan 27 2011

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)
(* Clark Kimberling, Mar 05 2012 *)
With[{nmax = 10}, Rest[CoefficientList[CoefficientList[Series[ x/(1 - 2*y*x-x-x^2+y*x^2), {x,0,nmax}, {y,0,nmax}], x], y]]//Flatten] (* G. C. Greubel, Jan 14 2018 *)

T(n,n-1) = A000079(n-1).
T(n,n-2) = A001792(n-2). - R. J. Mathar, Jan 27 2011
T(n,1) = A099920(n-1). - R. J. Mathar, Jan 27 2011
G.f.: x/(1-2*y*x-x-x^2+y*x^2). - Philippe Deléham, Mar 21 2012
Sum_{k=0..n-1, n>0} T(n,k)*x^k = A000045(n), A000244(n-1), A004254(n), A186446(n-1), A190980(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 21 2012

Corrected by Philippe Deléham, Mar 21 2012

A108037 Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144

Offset: 0

N. J. A. Sloane, Jun 01 2005

			0; 1,1; 1,1,1; 2,2,2,2; 3,3,3,3,3; 5,5,5,5,5,5; ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A039913, A108036, A108035.

Cf. A099920 (row sums).

a108037 n k = a108037_tabl !! n !! k
a108037_row n = a108037_tabl !! n
a108037_tabl = zipWith replicate [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Table[Fibonacci[n],{n+1}],{n,0,12}]//Flatten (* Harvey P. Dale, May 07 2017 *)

from math import isqrt
from sympy import fibonacci
def A108037(n): return int(fibonacci((m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))) # Chai Wah Wu, Nov 07 2024

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

A146005 a(n) = n*Lucas(n).

Original entry on oeis.org

0, 1, 6, 12, 28, 55, 108, 203, 376, 684, 1230, 2189, 3864, 6773, 11802, 20460, 35312, 60707, 104004, 177631, 302540, 513996, 871266, 1473817, 2488368, 4194025, 7057518, 11858508, 19898116, 33345679, 55814940, 93320819, 155867104

Offset: 0

R. J. Mathar, Oct 26 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

I:=[0, 1, 6, 12]; [n le 4 select I[n] else 2*Self(n-1) + Self(n-2) - 2*Self(n-3) - Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 13 2012

Table[LucasL[n, 1]*n, {n, 0, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
CoefficientList[Series[x * (1 + 4*x - x^2)/(1 - x - x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
LinearRecurrence[{2,1,-2,-1},{0,1,6,12},40] (* Harvey P. Dale, Apr 03 2013 *)

a(n) = n*A000032(n).
G.f.: x(1+4x-x^2)/(1-x-x^2)^2.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4).
a(n) = A000045(n)-5*A000045(n+1)+5*A010049(n+1).
a(n) = A045925(n)+2*A099920(n-1).
E.g.f.: x*exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)). - G. C. Greubel, Jan 30 2016

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

A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x(x^2 - 2) + x(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2x and w = sqrt(5 + 4x).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 0, 5, 8, 3, 0, 16, 30, 16, 2, 0, 45, 104, 81, 24, 2, 0, 121, 340, 356, 170, 35, 2, 0, 320, 1068, 1411, 932, 315, 48, 2, 0, 841, 3262, 5209, 4396, 2079, 532, 63, 2, 0, 2205, 9760, 18281, 18784, 11440, 4144, 840, 80, 2, 0, 5776, 28746

Offset: 0

Franck Maminirina Ramaharo, Jul 09 2018

The triangle is related to the Kauffman bracket polynomial for the Turk's Head Knot ((3,n)-torus knot). Column 1 matches the determinant of the Turk's Head Knots THK(3,k) A004146.

			The triangle T(n,k) begins:
n\k: 0      1      2       3       4       5       6      7      8    9  10 11
0:   0      0      0       1
1:   0      1      2       1
2:   0      5      8       3
3:   0     16     30      16       2
4:   0     45    104      81      24       2
5:   0    121    340     356     170      35       2
6:   0    320   1068    1411     932     315      48      2
7:   0    841   3262    5209    4396    2079     532     63      2
8:   0   2205   9760   18281   18784   11440    4144    840     80    2
9:   0   5776  28746   61786   74838   55809   26226   7602   1260   99   2
10:  0  15125  83620  202841  282980  249815  144488  54690  13080 1815 120  2
...

Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., Vol. 318 (1990), 417-471.
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Alexander Stoimenow, Square numbers, spanning trees and invariants of achiral knots, Communications in Analysis and Geometry, Vol. 13 (2005), 591-631.

Row sums: A000302 (Powers of 4).
Row 1: row 1 of A300184, A300192 and row 0 of A300454.
Row 2: row 2 of A300454.
Cf. A137396, A299989, A300453.

v = 3 + 2*x; w = Sqrt[5 + 4*x];
row[n_] := CoefficientList[x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), x];
Array[row, 15, 0] // Flatten

v : 3 + 2*x$ w : sqrt(5 + 4*x)$
p(n, x) := expand(x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n))$
for n:0 thru 15 do print(makelist(ratcoef(p(n, x), x, k), k, 0, max(3, n + 1)));

T(n,1) = A004146(n).
T(n,2) = A122076(n,1) = A099920(2*n-1).
G.f.: (x^3 - 2*x)/(1 - y) + (2*x - 3*x*y - 2*x^2*y)/(1 - 3*y - 2*x*y + y^2 + 2*x*y^2 + x^2*y^2).

cp's OEIS Frontend

A099428 Duplicate of A099920.

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A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

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

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A208354 Number of compositions of n with at most one even part.

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A122491 a(n) = n * Fibonacci(n) - Sum_{i=0..n} Fibonacci(i).

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A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

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A108037 Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.

A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x(x^2 - 2) + x(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2x and w = sqrt(5 + 4x).