A119473 -id:A119473 - cp's OEIS frontend

A159612 INVERT transform of (1, 3, 1, 3, 1, ...).

1, 4, 8, 24, 56, 152, 376, 984, 2488, 6424, 16376, 42072, 107576, 275864, 706168, 1809624, 4634296, 11872792, 30409976, 77901144, 199541048, 511145624, 1309309816, 3353892312, 8591131576, 22006700824, 56371227128, 144398030424, 369882938936, 947475060632, 2427006816376

Offset: 1

Gary W. Adamson, Apr 17 2009

The sequence 1,1,4,8,24,... is an eigensequence of the sequence triangle of 1,3,1,3,1,3,1,..., which is the Riordan array ((1+3x)/(1-x^2),x). - Paul Barry, Feb 10 2011
From Sean A. Irvine, Jun 07 2025: (Start)
Also, the number of walks of length n-1 starting at vertex 1 in the following graph:
  0   2
  |\ /|
  | 1 |
  |/ \|
  4   3. (End)

			a(4) = 24 = (1, 3, 1, 3) dot (8, 4, 1, 1) = (8 + 12, + 1 + 3).

Colin Barker, Table of n, a(n) for n = 1..1000
Sean A. Irvine, Walks on Graphs.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A006131, A119473.

Cf. A026597 (vertices 0, 2, 3, 4), A384604 (missing edge {0,4}).

LinearRecurrence[{1, 4}, {1, 4}, 50] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

Vec(x*(1+3*x)/(1-x-4*x^2) + O(x^40)) \\ Colin Barker, Dec 22 2016

G.f.: x*(1+3*x)/(1-x-4*x^2). - Philippe Deléham, Mar 01 2012
a(n) = a(n-1) + 4*a(n-2), a(1)=1, a(2)=4. - Vincenzo Librandi, Mar 11 2011
a(n+1) = Sum_{k=0..n} A119473(n,k)*3^k. - Philippe Deléham, Oct 05 2012
a(n) = 2^(-3-n)*((1-sqrt(17))^n*(-5+3*sqrt(17)) + (1+sqrt(17))^n*(5+3*sqrt(17))) / sqrt(17) for n > 0. - Colin Barker, Dec 22 2016
a(n) = A006131(n)+3*A006131(n-1). - R. J. Mathar, Jun 07 2025
E.g.f.: (exp(x/2)*(51*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2)) - 51)/68. - Stefano Spezia, Jun 07 2025

A189732 a(1)=1, a(2)=5, a(n) = a(n-1) + 5*a(n-2).

Original entry on oeis.org

1, 5, 10, 35, 85, 260, 685, 1985, 5410, 15335, 42385, 119060, 330985, 926285, 2581210, 7212635, 20118685, 56181860, 156775285, 437684585, 1221561010, 3409983935, 9517788985, 26567708660, 74156653585, 206995196885, 577778464810, 1612754449235, 4501646773285

Offset: 1

Harvey P. Dale, Apr 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
Index entries for linear recurrences with constant coefficients, signature (1, 5).

Cf. A000045, A000079, A105476, A159612, A080040, A135522, A103435, A189734, A189735, A189736, A189737, A189738, A189739, A189741, A189742, A189743, A189744, A189745, A189746, A189747, A189748, A189749.

Mathematica
```
LinearRecurrence[{1,5},{1,5},40]
```

a[1]:1$ a[2]:5$ a[n]:=a[n-1]+5*a[n-2]$ makelist(a[n], n, 1, 29);  /* Bruno Berselli, May 24 2011 */

a(n)=([0,1; 5,1]^(n-1)*[1;5])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

G.f.: x*(1+4*x)/(1-x-5*x^2). - Bruno Berselli, May 24 2011

a(n+1) = Sum_{k=0..n} A119473(n,k)*4^k. - Philippe Deléham, Oct 05 2012

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

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A159612 INVERT transform of (1, 3, 1, 3, 1, ...).

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A189732 a(1)=1, a(2)=5, a(n) = a(n-1) + 5*a(n-2).

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A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

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