A131322 -id:A131322 - cp's OEIS frontend

A124400 a(n) = a(n-1) + 3*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=4, a(3)=7.

1, 1, 4, 7, 18, 38, 88, 195, 441, 988, 2223, 4992, 11220, 25208, 56645, 127277, 285992, 642615, 1443946, 3244514, 7290360, 16381287, 36808421, 82707768, 185842671, 417584688, 938304280, 2108350576, 4737420745, 10644887785, 23918845740

Offset: 0

Philippe Deléham, Dec 14 2006

Unsigned version of A077920.

The sequence is the INVERT transform of the aerated even-indexed Fibonacci numbers (i.e., of (1, 0, 3, 0, 8, 0, ...)). Sequence A131322 is the INVERT transform of the aerated odd-indexed Fibonacci numbers. - Gary W. Adamson, Feb 07 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,0,-1).

Cf. A131322.

a:=[1,1,4,7];; for n in [5..35] do a[n]:=a[n-1]+3*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Dec 25 2019

I:=[1,1,4,7]; [n le 2 select I[n] else Self(n-1) +3*Self(n-2) -Self(n-4): n in [1..35]]; // G. C. Greubel, Dec 25 2019

seq(coeff(series(1/(1-x-3*x^2+x^4), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 25 2019

LinearRecurrence[{1,3,0,-1}, {1,1,4,7}, 35] (* G. C. Greubel, Dec 25 2019 *)
CoefficientList[Series[1/(1-x-3x^2+x^4),{x,0,30}],x] (* Harvey P. Dale, Feb 01 2022 *)

my(x='x+O('x^35)); Vec(1/(1-x-3*x^2+x^4)) \\ G. C. Greubel, Dec 25 2019

def A124400_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-3*x^2+x^4) ).list()
A124400_list(35) # G. C. Greubel, Dec 25 2019

G.f.: 1/(1-x-3*x^2+x^4).

A131321 Triangle read by rows: A168561^2.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 5, 0, 6, 0, 1, 0, 14, 0, 8, 0, 1, 13, 0, 27, 0, 10, 0, 1, 0, 46, 0, 44, 0, 12, 0, 1, 34, 0, 107, 0, 65, 0, 14, 0, 1, 0, 145, 0, 204, 0, 90, 0, 16, 0, 1, 89, 0, 393, 0, 345, 0, 119, 0, 18, 0, 1, 0, 444, 0, 854, 0, 538, 0, 152, 0, 20, 0, 1

Offset: 0

Gary W. Adamson, Jun 28 2007

Left border, nonzero terms = odd indexed Fibonacci numbers: (1, 2, 5, 13, ...). Next column, nonzero terms = A030267: (1, 4, 14, 46, 145, ...). Row sums = A131322: (1, 1, 3, 5, 12, 23, 51, ...).

Riordan array (f(x),x*f(x)) where f(x) = (1-x^2)/(1-3*x^2+x^4). Aerated version of triangle in A188137. - Philippe Deléham, Jan 26 2012

			First few rows of the triangle are:
   1;
   0,  1;
   2,  0,  1;
   0,  4,  0,  1;
   5,  0,  6,  0,  1;
   0, 14,  0,  8,  0,  1;
  13,  0, 27,  0, 10,  0,  1;
  ...

Cf. A049310, A030267, A168561, A188137.

F:= (n, k)-> coeff(combinat[fibonacci](n+1, x), x, k):
T:= (n, k)-> add(F(n, j)*F(j, k), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Dec 12 2019

A168561 squared, as an infinite lower triangular matrix.

A166482 a(n) = Sum_{k=0..n} binomial(n+k,2k)*Fibonacci(2k+1).

Original entry on oeis.org

1, 3, 12, 51, 221, 965, 4227, 18540, 81363, 357145, 1567849, 6883059, 30218028, 132664227, 582428789, 2557009709, 11225925267, 49284687948, 216372426339, 949930508209, 4170438905425, 18309298027683, 80382521554380

Offset: 0

Paul Barry, Oct 14 2009

Conjecture: a(n) is the number of tilings of a 4 X 4n rectangle into L tetrominoes (no reflections, only rotations). - Nicolas Bělohoubek, Feb 12 2022

The conjecture above was confirmed by Nicolas Bělohoubek and Antonín Slavík. (See links.) - Nicolas Bělohoubek, Jan 21 2025

G. C. Greubel, Table of n, a(n) for n = 0..500
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 5.
Index entries for linear recurrences with constant coefficients, signature (7,-13,7,-1).

Cf. A131322.

CoefficientList[Series[(1-4x+4x^2-x^3)/(1-7x+13x^2-7x^3+x^4), {x,0,30}],x]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{7, -13, 7, -1}, {1, 3, 12, 51}, 50] (* G. C. Greubel, May 15 2016 *)

G.f.: (1 - 4x + 4x^2 - x^3)/(1 - 7x + 13x^2 - 7x^3 + x^4).
a(n) = Sum_{k=0..n} binomial(n+k,2k) * Sum_{j=0..k} binomial(k+j,2j).
a(n) ~ (1 + 1/sqrt(5) + 2*sqrt(31/290 + 13/(58*sqrt(5)))) * ((7 + sqrt(5) + sqrt(38 + 14*sqrt(5)))^n / 2^(2*n+2)). - Vaclav Kotesovec, Feb 22 2022
a(n) = A131322(2*n). - Nicolas Bělohoubek, Jan 21 2025

cp's OEIS Frontend

A124400 a(n) = a(n-1) + 3*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=4, a(3)=7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A131321 Triangle read by rows: A168561^2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A166482 a(n) = Sum_{k=0..n} binomial(n+k,2k)*Fibonacci(2k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula