A081567 -id:A081567 - cp's OEIS frontend

A014445 Even Fibonacci numbers; or, Fibonacci(3*n).

0, 2, 8, 34, 144, 610, 2584, 10946, 46368, 196418, 832040, 3524578, 14930352, 63245986, 267914296, 1134903170, 4807526976, 20365011074, 86267571272, 365435296162, 1548008755920, 6557470319842, 27777890035288, 117669030460994, 498454011879264, 2111485077978050

Offset: 0

Mohammad K. Azarian

a(n) = 3^n*b(n;2/3) = -b(n;-2), but we have 3^n*a(n;2/3) = F(3n+1) = A033887 and a(n;-2) = F(3n-1) = A015448, where a(n;d) and b(n;d), n=0,1,...,d, denote the so-called delta-Fibonacci numbers (the argument "d" of a(n;d) and b(n;d) is abbreviation of the symbol "delta") defined by the following equivalent relations: (1 + d*((sqrt(5) - 1)/2))^n = a(n;d) + b(n;d)*((sqrt(5) - 1)/2) equiv. a(0;d)=1, b(0;d)=0, a(n+1;d) = a(n;d) + d*b(n;d), b(n+1;d) = d*a(n;d) + (1-d)b(n;d) equiv. a(0;d)=a(1;d)=1, b(0;1)=0, b(1;d)=d, and x(n+2;d) + (d-2)*x(n+1;d) + (1-d-d^2)*x(n;d) = 0 for every n=0,1,...,d, and x=a,b equiv. a(n;d) = Sum_{k=0..n} C(n,k)*F(k-1)*(-d)^k, and b(n;d) = Sum_{k=0..n} C(n,k)*(-1)^(k-1)*F(k)*d^k equiv. a(n;d) = Sum_{k=0..n} C(n,k)*F(k+1)*(1-d)^(n-k)*d^k, and b(n;d) = Sum_{k=1..n} C(n;k)*F(k)*(1-d)^(n-k)*d^k. The sequences a(n;d) and b(n;d) for special values d are connected with many known sequences: A000045, A001519, A001906, A015448, A020699, A033887, A033889, A074872, A081567, A081568, A081569, A081574, A081575, A163073 (see also the papers of Witula et al.). - Roman Witula, Jul 12 2012
For any odd k, Fibonacci(k*n) = sqrt(Fibonacci((k-1)*n) * Fibonacci((k+1)*n) + Fibonacci(n)^2). - Gary Detlefs, Dec 28 2012
The ratio of consecutive terms approaches the continued fraction 4 + 1/(4 + 1/(4 +...)) = A098317. - Hal M. Switkay, Jul 05 2020

			G.f. = 2*x + 8*x^2 + 34*x^3 + 144*x^4 + 610*x^5 + 2584*x^6 + 10946*x^7 + ...

Arthur T. Benjamin and Jennifer J. Quinn,, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, id. 232.

T. D. Noe, Table of n, a(n) for n = 0..200
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38 (2012), pp. 1871-1876 (See Corollary 1 (v)).
H. H. Ferns, Problem B-115, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 2 (1967), p. 202; Identities for F_{kn} and L{kn}, Solution to Problem B-115 by Stanley Rabinowitz, ibid., Vol. 6, No. 1 (1968), pp. 92-93.
Ira M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq., Vol. 16 (2013), Article 13.4.5.
Edyta Hetmaniok, Bozena Piatek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Math., Vol. 15 (2017), pp. 477-485.
Tanya Khovanova, Recursive Sequences.
Ron Knott, Mathematics of the Fibonacci Series.
Bahar Kuloğlu, Engin Özkan, and Marin Marin, Fibonacci and Lucas Polynomials in n-gon, An. Şt. Univ. Ovidius Constanţa (Romania 2023) Vol. 31, No 2, 127-140.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Roberto Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, JIS 19 (2016) # 16.5.4
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math., Vol. 3, No. 2 (2009), pp. 310-329, MR2555042.
Roman Witula, Binomials transformation formulae of scaled Lucas numbers, Demonstratio Math., Vol. 46 (2013), pp. 15-27.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A001076, A049310, A111125.
Cf. A001519, A001906, A015448, A020699, A033887, A033889, A074872, A081567, A081568, A081569, A081574, A081575, A098317, A163073.
First differences of A099919.
Third column of array A102310.

[Fibonacci(3*n): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a:= n-> (<<0|1>, <1|1>>^(3*n))[1,2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 03 2023

Table[Fibonacci[3n], {n,0,30}] (* Stefan Steinerberger, Apr 07 2006 *)
LinearRecurrence[{4,1},{0,2},30] (* Harvey P. Dale, Nov 14 2021 *)
Table[ SeriesCoefficient[2*x/(1 - 4*x - x^2), {x, 0, n}], {n, 0, 20}] (* Nikolaos Pantelidis, Feb 02 2023 *)

numlib::fibonacci(3*n) $ n = 0..30; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(3*n) \\ Charles R Greathouse IV, Oct 25 2012

[fibonacci(3*n) for n in range(0, 30)] # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} binomial(n, k)*F(k)*2^k. - Benoit Cloitre, Oct 25 2003
From Lekraj Beedassy, Jun 11 2004: (Start)
a(n) = 4*a(n-1) + a(n-2), with a(-1) = 2, a(0) = 0.
a(n) = 2*A001076(n).
a(n) = (F(n+1))^3 + (F(n))^3 - (F(n-1))^3. (End)
a(n) = Sum_{k=0..floor((n-1)/2)} C(n, 2*k+1)*5^k*2^(n-2*k). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) = Sum_{k=0..n} F(n+k)*binomial(n, k). - Benoit Cloitre, May 15 2005
O.g.f.: 2*x/(1 - 4*x - x^2). - R. J. Mathar, Mar 06 2008
a(n) = second binomial transform of (2,4,10,20,50,100,250). This is 2* (1,2,5,10,25,50,125) or 5^n (offset 0): *2 for the odd numbers or *4 for the even. The sequences are interpolated. Also a(n) = 2*((2+sqrt(5))^n - (2-sqrt(5))^n)/sqrt(20). - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009
a(n) = 3*F(n-1)*F(n)*F(n+1) + 2*F(n)^3, F(n)=A000045(n). - Gary Detlefs, Dec 23 2010
a(n) = (-1)^n*3*F(n) + 5*F(n)^3, n >= 0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - Wolfdieter Lang, Aug 31 2012
With L(n) a Lucas number, F(3*n) = F(n)*(L(2*n) + (-1)^n) = (L(3*n+1) + L(3*n-1))/5 starting at n=1. - J. M. Bergot, Oct 25 2012
a(n) = sqrt(Fibonacci(2*n)*Fibonacci(4*n) + Fibonacci(n)^2). - Gary Detlefs, Dec 28 2012
For n > 0, a(n) = 5*F(n-1)*F(n)*F(n+1) - 2*F(n)*(-1)^n. - J. M. Bergot, Dec 10 2015
a(n) = -(-1)^n * a(-n) for all n in Z. - Michael Somos, Nov 15 2018
a(n) = (5*Fibonacci(n)^3 + Fibonacci(n)*Lucas(n)^2)/4 (Ferns, 1967). - Amiram Eldar, Feb 06 2022
a(n) = 2*i^(n-1)*S(n-1,-4*i), with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(-1, x) = 0. From the simplified trisection formula. - Gary Detlefs and Wolfdieter Lang, Mar 04 2023
E.g.f.: 2*exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - Stefano Spezia, Jun 03 2024
a(n) = 2*F(n) + 3*Sum_{k=0..n-1} F(3*k)*F(n-k). - Yomna Bakr and Greg Dresden, Jun 10 2024

A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

Original entry on oeis.org

1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset: 0

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004
Binomial transform of A002878. - Philippe Deléham, Oct 04 2005
Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013
Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - Wolfdieter Lang, Nov 16 2023~

			G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0 ,b=1; p=5, q=-5.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=5.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000032, A000045, A001622.

Cf. A002878, A039717, A081567, A109466, A216219.

a:=[1,5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

I:=[1,5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

Table[MatrixPower[{{2,1},{1,3}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)
Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n,0,30}] (* G. C. Greubel, Dec 28 2019 *)

{a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 */

[lucas_number1(n,5,5) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).
G.f.: 1/(1 - 5*x + 5*x^2).
a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).
a(2*k) = 5^k*Lucas(2*k+1).
a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - Benoit Cloitre, Jun 21 2003
a(n) = 5*a(n-1) - 5*a(n-2). - Benoit Cloitre, Oct 23 2003
a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004
a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - Philippe Deléham, Nov 28 2006
a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013
a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013
G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013
a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015
E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019
a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - Philippe Deléham, Mar 10 2023

A178381 Number of paths of length n starting at initial node of the path graph P_9.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 125, 250, 450, 900, 1625, 3250, 5875, 11750, 21250, 42500, 76875, 153750, 278125, 556250, 1006250, 2012500, 3640625, 7281250, 13171875, 26343750, 47656250, 95312500, 172421875, 344843750

Offset: 0

Johannes W. Meijer, May 27 2010, May 29 2010

Counts all paths of length n, n>=0, starting at initial node on the path graph P_9, see the Maple program.
The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and g4; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3 and g5.
The path graphs P_(2*p) have as limit(a(n+1)/a(n), n =infinity) = 2 resp. hypergeom([(p-1)/(2*p+1),(p+2)/(2*p+1)],[1/2],3/4) and the path graphs P_(2*p+1) have as limit(a(n+1)/a(n), n =infinity) = 1+cos(Pi/(p+1)), p>=1; see the crossrefs. - Johannes W. Meijer, Jul 01 2010

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 70*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Trigonometric Identities.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5).

This is row 9 of A094718.
a(2*n) = A147748(n) and a(2*n+1) = A081567(n).
a(4*n+2) = A109106(n) and a(4*n+3) = A179135(n).
Cf. A000007 (P_1), A000012 (P_2), A016116 (P_3), A000045 (P_4), A038754 (P_5), A028495 (P_6), A030436 (P_7), A061551 (P_8), this sequence (P_9), A336675 (P_10), A336678 (P_11), and A001405 (P_infinity).
Cf. A216212 (P_9 starting in the middle).
Cf. A033191, A179131, A179132, A128052, A179133.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4))); // G. C. Greubel, Sep 18 2018

with(GraphTheory): P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=36; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P); od: seq(a(n),n=0..nmax);
r := j -> (-1)^(j/10) - (-1)^(1-j/10):
a := k -> add((2 + r(j))*r(j)^k, j in [1, 3, 5, 7, 9])/10:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 18 2020

CoefficientList[Series[(1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4), {x,0,50}], x] (* G. C. Greubel, Sep 18 2018 *)

x='x+O('x^50); Vec((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4)) \\ G. C. Greubel, Sep 18 2018

G.f.: (1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=5 with a(0)=1, a(1)=1, a(2)=2, a(3)=3 and a(4)=6.
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x / (1 - x / (1 + x / (1 + x)))))))). - Michael Somos, Feb 08 2015

A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 5, 12, 12, 4, 1, 8, 25, 30, 20, 5, 1, 13, 48, 75, 60, 30, 6, 1, 21, 91, 168, 175, 105, 42, 7, 1, 34, 168, 364, 448, 350, 168, 56, 8, 1, 55, 306, 756, 1092, 1008, 630, 252, 72, 9, 1, 89, 550, 1530, 2520, 2730, 2016, 1050, 360, 90, 10, 1

Offset: 0

Clark Kimberling, May 03 2004

Triangle of coefficients of polynomials u(n,x) jointly generated with A209415; see the Formula section.
Column 1:  Fibonacci numbers:  F(n)=A000045(n)
Column 2:  n*F(n)
Row sums:  odd-indexed Fibonacci numbers
Alternating row sums: signed Fibonacci numbers
Coefficient of x^n in u(n,x):  1
Coefficient of x^(n-1) in u(n,x):  n
Coefficient of x^(n-2) in u(n,x):  n(n+1)
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012
Row n shows the coefficients of the numerator of the n-th derivative of (1/n!)*(x+1)/(1-x-x^2); see the Mathematica program. - Clark Kimberling, Oct 22 2019

			First five rows:
  1;
  1,  1;
  2,  2,  1;
  3,  6,  3,  1;
  5, 12, 12,  4,  1;
First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
From _Philippe Deléham_, Mar 27 2012: (Start)
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  3,  6,  3,  1;
  0,  5, 12, 12,  4,  1. (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
E. Kiliç, H. Belbachir, Generalized double binomial sums families by generating functions, 2014.

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094442, A094443, A094444.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+1) ))); # G. C. Greubel, Oct 30 2019

[Binomial(n,k)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019

with(combinat); seq(seq(fibonacci(n-k+1)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A094441 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A094442 *)
(* Next program outputs polynomials having coefficients T(n,k) *)
g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(x + 1)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* Clark Kimberling, Oct 22 2019 *)
(* Second program *)
Table[Fibonacci[n-k+1]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = binomial(n,k)*fibonacci(n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[binomial(n,k)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Sum_{k=0..n} T(n,k)*x^k = A039834(n-1), A000045(n+1), A001519(n+1), A081567(n), A081568(n), A081569(n), A081570(n), A081571(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Dec 14 2009
From Clark Kimberling, Mar 09 2012: (Start)
A094441 shows the coefficient of the polynomials u(n,x) which are jointly generated with polynomials v(n,x) by these rules:
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
(End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 27 2012
G.f. (1-x*y)/(1 - 2*x*y - x - x^2 + x^2*y + x^2*y^2). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+1).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-1). (End)

A039717 Row sums of convolution triangle A030523.

Original entry on oeis.org

1, 4, 15, 55, 200, 725, 2625, 9500, 34375, 124375, 450000, 1628125, 5890625, 21312500, 77109375, 278984375, 1009375000, 3651953125, 13212890625, 47804687500, 172958984375, 625771484375, 2264062500000, 8191455078125

Offset: 1

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.
With offset 0 = INVERT transform of A001792: (1, 3, 8, 20, 48, 112, ...). - Gary W. Adamson, Oct 26 2010
From Tom Copeland, Nov 09 2014: (Start)
The array belongs to a family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the o.g.f. (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse x*(1-x)/(1 + (t-1)*x*(1-x)). See A091867 for more info on this family. Here t = -4 (mod signs in the results).
Let C(x) = (1 - sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f.: G(x) = x*(1-x)/(1 - 5x*(1-x)) = P(Cinv(x),-5).
Inverse O.g.f.: Ginv(x) = (1 - sqrt(1 - 4*x/(1+5x)))/2 = C(P(x,5)) (signed A026378). Cf. A030528. (End)
p-INVERT of (2^n), where p(s) = 1 - s - s^2; see A289780. - Clark Kimberling, Aug 10 2017

Michel Marcus, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Appears in A109106.

Cf. A000045, A000108, A001792, A005043, A091867, A026378, A030528.

CoefficientList[Series[(1 - x) / (1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

Vec(x*(1-x)/(1-5*x+5*x^2) + O(x^40)) \\ Altug Alkan, Nov 20 2015

G.f.: x*(1-x)/(1-5*x+5*x^2) = g1(3, x)/(1-g1(3, x)), g1(3, x) := x*(1-x)/(1-2*x)^2 (g.f. first column of A030523).
From Paul Barry, Apr 16 2004: (Start)
Binomial transform of Fibonacci(2n+2).
a(n) = (sqrt(5)/2 + 5/2)^n*(3*sqrt(5)/10 + 1/2) - (5/2 - sqrt(5)/2)^n*(3*sqrt(5)/10 - 1/2). (End)
a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)*sin(r*Pi/2)*(2*cos(r*Pi/10))^(2*n).
a(n) = 5*a(n-1) - 5*a(n-2).
a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(n, i)*binomial(k+i+1, 2k+1). - Paul Barry, Jun 22 2004
From Johannes W. Meijer, Jul 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A020876(n) + A093131(n)*sqrt(5))/2.
Limit_{n->oo} A020876(n)/A093131(n) = sqrt(5).
(End)
From Benito van der Zander, Nov 19 2015: (Start)
Limit_{k->oo} a(k+1)/a(k) = 1 + phi^2 = (5 + sqrt(5)) / 2.
a(n) = a(n-1) * 3 + A081567(n-2) (not proved).
(End)
E.g.f.: exp(x*5/2) * (cosh(x*sqrt(5)/2) + (3/sqrt(5))*sinh(x*sqrt(5)/2)). - Fabian Pereyra, Oct 29 2024

A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 0, 5, 10, 10, 5, 0, 0, 5, 15, 20, 15, 5, 0, 0, 0, 20, 35, 35, 20, 0, 0, 0, 0, 20, 55, 70, 55, 20, 0, 0, 0, 0, 0, 75, 125, 125, 75, 0, 0, 0, 0, 0, 0, 75, 200, 250, 200

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1,  1,  1,   1,   0,   0,   0,    0,    0, 0, ...
1, 2,  3,  4,   5,   5,   0,   0,    0,    0, 0, ...
1, 3,  6, 10,  15,  20,  20,   0,    0,    0, 0, ...
1, 4, 10, 20,  35,  55,  75,  75,    0,    0, 0, ...
1, 5, 15, 35,  70, 125, 200, 275,  275,    0, 0, ...
0, 5, 20, 55, 125, 250, 450, 725, 1000, 1000, 0, ...
0, 0, 20, 75, 200, 450, 900, ...

Cf. A030191, A039717, A068913, A081567, A147748, A216210, A216212, A216218

T(n,n) = A147748(n).
T(n+1,n) = T(n,n+1) = A081567(n).
T(n+2,n) = T(n,n+2) = A039717(n+1).
T(n+3,n) = T(n+4,n) = T(n,n+3) = T(n,n+4) = A030191(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A068913(4,n) = A216212(n).

A033191 Binomial transform of [ 1, 0, 1, 1, 3, 6, 15, 36, 91, 231, 595, ... ], which is essentially binomial(Fibonacci(k) + 1, 2).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, 58598, 206516, 732825, 2613834, 9358677, 33602822, 120902914, 435668420, 1571649221, 5674201118, 20497829133, 74079051906, 267803779710, 968355724724, 3502058316337, 12666676646162, 45818284122149

Offset: 0

Simon P. Norton

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 1. - Herbert Kociemba, Jun 14 2004
The sequence 1,2,5,14,... has g.f. 1/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x)))) = (1-6x+10x^2-4x^3)/(1-8x+21x^2-20x^3+5x^4), and is the second binomial transform A001519 aerated. - Paul Barry, Dec 17 2009
Counts all paths of length (2*n), n>=0, starting and ending at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010

			1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + ...

Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=8, pages 10-11). [_Bruno Berselli_, May 12 2012]
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 9.
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A033192.
Cf. A081567, A147748 and A178381.
Cf. A211216.

with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):=B(n)[1,1]; od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

CoefficientList[Series[(1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4), {x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{8,-21,20,-5},{1,2,5,14}, 30]]  (* Harvey P. Dale, Apr 26 2011 *)

{a(n) = local(A); A = 1; for( i=1, 8, A = 1 / (1 - x*A)); polcoeff( A + x * O(x^n), n)} /* Michael Somos, May 12 2012 */

G.f.: (1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4). - Ralf Stephan, May 13 2003
From Herbert Kociemba, Jun 14 2004: (Start)
a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)^2*(2*cos(r*Pi/10))^(2n), n >= 1;
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), n >= 5. (End)
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x )))))))). - Michael Somos, May 12 2012

A081568 Third binomial transform of Fibonacci(n+1).

Original entry on oeis.org

1, 4, 17, 75, 338, 1541, 7069, 32532, 149965, 691903, 3193706, 14745009, 68084297, 314394980, 1451837593, 6704518371, 30961415074, 142980203437, 660285858245, 3049218769908, 14081386948661, 65028302171639, 300302858766202, 1386808687475385, 6404329365899473

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081567.
Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2) for n >= 2, with a(0) = 1 and a(1) = k + 1.
a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n = 0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (7,-11).

Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A028387, A081567, A081569 (binomial transform), A094441, A099453.

a:=[1,4];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-3*x)/(1-7*x+11*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-3x)/(1 -7x +11x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{7,-11},{1,4},30] (* Harvey P. Dale, Feb 01 2015 *)

Vec((1-3*x)/(1-7*x+11*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015

def A081568_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-7*x+11*x^2)).list()
A081568_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 7*a(n-1) - 11*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 4.
a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n) - 3*A099453(n-1).
G.f.: (1 - 3*x)/(1 - 7*x + 11*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - Philippe Deléham, Dec 14 2009
G.f.: Q(0,u)/x - 1/x, where u = x/(1 - 3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k + 1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).

Original entry on oeis.org

5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500, 183166503906250, 662702636718750

Offset: 0

L. Edson Jeffery, Apr 20 2011

Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 2 0).
Then a(n) = Trace(A^(2*n)).
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0From Tom Copeland, Dec 08 2015: (Start)
These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.
a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).
-log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning  with the first power. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A147748, A189316, A189317, A189318.

Cf. A025192, A081567, A127672, A265185.

I:=[5,10,30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015

CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2),{x,0,40}],x] (* or *)
Join[{5},LinearRecurrence[{5,-5},{10,30},40]]  (* Harvey P. Dale, Apr 25 2011 *)

Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.
a(n) = Sum_{k=1..5} (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.
a(n) = 5*A147748(n).
E.g.f.: 1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Jul 09 2024

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cp's OEIS Frontend

A081666 n*3^(n-1)+A081567(n).

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A014445 Even Fibonacci numbers; or, Fibonacci(3*n).

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A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

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A178381 Number of paths of length n starting at initial node of the path graph P_9.

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A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.

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A039717 Row sums of convolution triangle A030523.

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A189315 Expansion of g.f. 5(1-3x+x^2)/(1-5x+5x^2).