A084179 -id:A084179 - cp's OEIS frontend

A052964 Expansion of (1-x)/((1-2x)(1+x-x^2)).

1, 0, 3, 1, 10, 7, 35, 36, 127, 165, 474, 715, 1807, 3004, 6995, 12393, 27370, 50559, 107883, 204820, 427351, 826045, 1698458, 3321891, 6765175, 13333932, 26985675, 53457121, 107746282, 214146295, 430470899, 857417220, 1720537327

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Number of walks of length n+1 between two adjacent vertices in the cycle graph C_5. Example: a(2)=3 because in the cycle ABCDE we have three walks of length 3 between A and B: ABAB, ABCB and AEAB. - Emeric Deutsch, Apr 01 2004
In general a(n,m)=2^n/m*Sum(k,0,m-1,Cos(2Pi*k/m)^(n+1)) gives number of walks of length n between two adjacent vertices in the cycle graph C_m. Here we have m=5. - Herbert Kociemba, May 31 2004
Counts walks of length n at the vertex of degree 3 of the graph with adjacency matrix A=[0,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Binomial transform is (L(n-2)+2*3^n)/5, or A099159. - Paul Barry, Oct 01 2004
Also, the cogrowth sequence for the 10-element dihedral group D5; that is a(n) is the number of words of length n+2 that reduce to the identity using the presentation . - Sean A. Irvine, Nov 04 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1035
Index entries for linear recurrences with constant coefficients, signature (1,3,-2).

spec := [S,{S=Sequence(Prod(Union(Prod(Sequence(Z),Z),Z,Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

CoefficientList[Series[(1-x)/((1-2x)(1+x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,-2},{1,0,3},40] (* Harvey P. Dale, Jun 03 2019 *)

G.f.: -(-1+x)/(1-x-3*x^2+2*x^3)
Recurrence: {a(1)=0, a(0)=1, a(2)=3, 2*a(n)-3*a(n+1)-a(n+2)+a(n+3)=0}
Sum(-1/25*(-1-11*_alpha+6*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-_Z-3*_Z^2+2*_Z^3))
a(n-1)=2^n/5*Sum(k, 0, 4, Cos(2Pi*k/5)^(n+1)), n>=1 - Herbert Kociemba, May 31 2004
a(n)=((sqrt(5)-1)/2)^n(3/10-sqrt(5)/10)+((-sqrt(5)-1)/2)^n(3/10+sqrt(5)/10)+2^(n+1)/5 - Paul Barry, Oct 01 2004
a(n) = (2^(n+1) + A000032(n+2)*(-1)^n)/5 - Ross La Haye, May 31 2006
a(n) = |A084179(n+1)|-|A084179(n)|. - R. J. Mathar, Feb 27 2019

A226205 a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

Original entry on oeis.org

1, 0, 3, 5, 16, 39, 105, 272, 715, 1869, 4896, 12815, 33553, 87840, 229971, 602069, 1576240, 4126647, 10803705, 28284464, 74049691, 193864605, 507544128, 1328767775, 3478759201, 9107509824, 23843770275, 62423800997, 163427632720, 427859097159, 1120149658761

Offset: 1

Michael Somos, Jun 06 2013

A001519(n)^2 = A079472(n)^2 + a(n)^2 and (A001519(n), A079472(n), a(n)) is a Pythagorean triple.
INVERT transform is A052156. PSUM transform is A007598. SUMADJ transform is A088305. BINOMIAL transform is A039717. BINOMIAL transform with 0 prepended is A112091 with 0 prepended. BINOMIAL transform inverse is A084179(n+1).
In general, the difference between squares of two consecutive terms of a second order linear recurrence having a signature of (c,d) will be a third order recurrence with signature (c^2+d,(c^2+d)*d,-d^3). - Gary Detlefs, Mar 13 2025

			G.f. = x + 3*x^3 + 5*x^4 + 16*x^5 + 39*x^6 + 105*x^7 + 272*x^8 + 715*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John P. Bonomo and Montana Ferita, A Small Fib, College Math. J., 2023.
Nurettin Irmak, Product of arbitrary Fibonacci numbers with distance 1 to Fibonomial coefficient, Turk J Math, (2017) 41: 825-828. See p. 828.
C.-A. Laisant, Observations sur les triangles rectangles en nombres entiers et les suites de Fibonacci, Nouvelles Annales de Math. (1919, in French) Série 4, Vol. 19, 391-397.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001519, A001622, A001654, A007598, A033999, A039717, A052156, A079472, A084179, A088305, A112091, A121646, A290565.
Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.
Cf. A260259: numbers of the form F(n)*F(n+1)-(-1)^n. - Bruno Berselli, Nov 02 2015

[Fibonacci(n)^2-Fibonacci(n-1)^2: n in [1..40]]; // Vincenzo Librandi, Jun 18 2014

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,0,3>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

a[ n_] := Fibonacci[n + 1] Fibonacci[n - 2]; (* Michael Somos, Jun 17 2014 *)
CoefficientList[Series[(1 - x)^2/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)

{a(n) = fibonacci( n + 1) * fibonacci( n - 2)};

a(n) = round(2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5) \\ Colin Barker, Sep 28 2016

lista(nn) = {my(p = (3*x-1)/(x^3-2*x^2-2*x+1)); for (n=1, nn, p = deriv(p, x); print1(subst(p, x, 0)/n!, ", "); ); } \\ Michel Marcus, May 22 2018

G.f.: x * (1 - x)^2 / ((1 + x) * (1 -3*x + x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = - A121646(n).
a(n) = -a(1-n) for all n in Z.
a(n) = A121801(n+1) / 2. - Michael Somos, Jun 17 2014
a(n) = a(n-1) + A000045(n-1)^2 - 2*(-1)^n, for n>1. - Alexander Samokrutov, Sep 07 2015
a(n) = F(n-1)*F(n) - (-1)^n. - Bruno Berselli, Oct 30 2015
a(n) = 2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5. - Colin Barker, Sep 28 2016
From Amiram Eldar, Oct 06 2020: (Start)
Sum_{n>=3} 1/a(n) = (1/2) * A290565 - 1/4.
Sum_{n>=3} (-1)^(n+1)/a(n) = (3/2) * (1/phi - 1/2), where phi is the golden ratio (A001622). (End)

A084178 Inverse binomial transform of Fibonacci oblongs.

Original entry on oeis.org

0, 1, 0, 3, -1, 10, -7, 35, -36, 127, -165, 474, -715, 1807, -3004, 6995, -12393, 27370, -50559, 107883, -204820, 427351, -826045, 1698458, -3321891, 6765175, -13333932, 26985675, -53457121, 107746282, -214146295, 430470899, -857417220, 1720537327, -3431847189

Offset: 0

Paul Barry, May 18 2003

Inverse binomial transform of A001654.

Index entries for linear recurrences with constant coefficients, signature (-1,3,2).

Cf. A000045, A052964, A084179.

LinearRecurrence[{-1,3,2},{0,1,0},40] (* Harvey P. Dale, Nov 24 2020 *)

a(n)=((1/2+sqrt(5)/2)^(n+1)+(1/2-sqrt(5)/2)^(n+1)-(-2)^n)/5;
G.f.: x(1+x)/(1+x-3x^2-2x^3)=x(1-x)/((1+2x)(1-x-x^2)).
a(n) = A084179(n)+A084179(n-1). - R. J. Mathar, Dec 10 2014

A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).

Original entry on oeis.org

1, 1, 1, 2, -1, 1, 3, 4, -3, 1, 5, -5, 10, -5, 1, 8, 15, -25, 20, -7, 1, 13, -22, 65, -65, 34, -9, 1, 21, 57, -152, 195, -133, 52, -11, 1, 34, -93, 361, -542, 461, -237, 74, -13, 1, 55, 220, -815, 1445, -1464, 935, -385, 100, -15, 1, 89, -385, 1850, -3705

Offset: 0

Philippe Deléham, Feb 08 2014

First column: Fibonacci numbers A000045(n+1).

			Triangle begins:
   1;
   1,    1;
   2,   -1,    1;
   3,    4,   -3,    1;
   5,   -5,   10,   -5,   1;
   8,   15,  -25,   20,  -7,   1;
  13,  -22,   65,  -65,  34,  -9,  1;
  ...
Production matrix is:
   1,  1;
   1, -2,  1;
   2,  0, -2,  1;
   4,  0,  0, -2,  1;
   8,  0,  0,  0, -2,  1;
  16,  0,  0,  0,  0, -2,  1;
  32,  0,  0,  0,  0,  0, -2,  1;
  64,  0,  0,  0,  0,  0,  0, -2,  1;
  ...

Indranil Ghosh, Rows 0..100, flattened

Columns: A000045, A084179.

nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 15 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A098600(n), A000032(n+1), A027961(n+1), A027974(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,k) = T(n-1,k-1) - T(n-1,k) + 3*T(n-2,k) - T(n-2,k-1) + 2*T(n-3,k) - T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = -1, T(n,k) = 0 if k<0 or if k>n.
T(n,0) = T(n-1,0) + T(n-2,0) with T(0,0) = T(1,0) = 1, T(n,k) = T(n-1,k-1) - 2*T(n-1,k) for k>=1.
G.f.: (1+2*x)/((1+2*x-y*x)*(1-x-x^2)).

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A052964 Expansion of (1-x)/((1-2x)(1+x-x^2)).

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A226205 a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

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A084178 Inverse binomial transform of Fibonacci oblongs.

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A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).

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