A139818 -id:A139818 - cp's OEIS frontend

A140966 a(n) = (5 + (-2)^n)/3.

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

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

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).

Original entry on oeis.org

1, 9, 9, 25, 9, 81, 25, 121, 9, 81, 81, 225, 25, 225, 121, 441, 9, 81, 81, 225, 81, 729, 225, 1089, 25, 225, 225, 625, 121, 1089, 441, 1849, 9, 81, 81, 225, 81, 729, 225, 1089, 81, 729, 729, 2025, 225, 2025, 1089, 3969, 25, 225, 225, 625, 225, 2025, 625, 3025, 121, 1089, 1089, 3025, 441, 3969, 1849, 7225, 9, 81, 81, 225, 81, 729, 225

Offset: 0

N. J. A. Sloane, Aug 20 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
This is the odd-rule cellular automaton defined by OddRule 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of {A001045(k+2)^2} (or of A139818).
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

			Here is the neighborhood:
[X, X, X]
[X, X, X]
[X, X, X]
which contains a(1) = 9 ON cells.
.
From Omar E. Pol, Mar 17 2015: (Start)
Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A139818(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below:
..
9;
...
9;
25;
..........
9,     81;
25;
121;
....................
9,     81,  81, 225;
25,   225;
121;
441;
........................................
9,     81,  81, 225, 81, 729, 225, 1089;
25,   225, 225, 625;
121, 1089;
441;
1849;
...
Note that every row r is equal to A139818(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number: T(s,r,k) = T(s+1,r,k).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8192
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034.

Cf. A001045, A139818.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=(1/x+1+x)*(1/y+1+y);
OddCA(f, 70);

b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)];
a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 30 2017 *)

a(n) = A071053(n)^2.

A206472 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having zero permanent.

Original entry on oeis.org

25, 121, 121, 441, 1411, 441, 1849, 11025, 11025, 1849, 7225, 106891, 110889, 106891, 7225, 29241, 958441, 1896129, 1896129, 958441, 29241, 116281, 8963667, 23707161, 68352739, 23707161, 8963667, 116281, 466489, 82609921, 356190129, 1693734025

Offset: 1

R. H. Hardin Feb 08 2012

Table starts
.....25.......121.........441...........1849.............7225
....121......1411.......11025.........106891...........958441
....441.....11025......110889........1896129.........23707161
...1849....106891.....1896129.......68352739.......1693734025
...7225....958441....23707161.....1693734025......61244865529
..29241...8963667...356190129....52552339515....3420616959081
.116281..82609921..4803737481..1457044540561..144660587865049
.466489.767387611.69030731169.43214597865811.7310031634630729

			Some solutions for n=4 k=3
..0..2..0..2....0..0..0..1....2..0..1..2....2..2..0..0....1..2..0..0
..0..1..0..0....1..0..0..1....1..0..0..0....0..0..0..1....0..0..0..2
..0..2..0..1....0..0..0..0....1..0..2..0....2..0..0..0....1..0..0..0
..0..0..0..0....1..2..0..1....1..0..0..0....2..0..0..1....0..0..1..0
..0..2..2..2....0..0..0..1....0..0..2..1....2..0..0..2....1..0..2..0

R. H. Hardin, Table of n, a(n) for n = 1..684

Column 1 is A139818(n+3)

A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485

Offset: 0

Paul Curtz, Oct 11 2019

nonn

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000129, A001045, A054977, A139756, A141413, A166444, A183190.

Cf. A000079, A005578, A063524, A086445, A139818, A257113.

a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* Jean-François Alcover, Oct 16 2019 *)

a(n) is the fourth row of the following array:
  0,  0,  0,  0, 0, 1, 3, 7, 14, 27, 51, 97, ...
  0,  0,  0,  0, 1, 2, 4, 7, 13, 24, 46, 89, ... = A086445
  0,  0,  0,  1, 1, 2, 3, 6, 11, 22, 43, 86, ... = 0, 0, 0, A005578(n)
  0,  0,  1,  0, 1, 1, 3, 5, 11, 21, 43, 85, ... = a(n)
  0,  1, -1,  1, 0, 2, 2, 6, 10, 22, 42, 86, ...
  1, -2,  2, -1, 2, 0, 4, 4, 12, 20, 44, 84, ...
From the main diagonal onward, every row is an autosequence of the first kind.
From Stefano Spezia, Oct 16 2019: (Start)
O.g.f.: x^2*(-1 + x + x^2)/(-1 + x + 2*x^2).
E.g.f.: (1/24)*exp(-x)*(8 - 9*exp(x) + exp(3*x) + 6*exp(x)*x + 6*exp(x)*x^2).
a(n) = a(n-1) + 2*a(n-2) for n > 4. (End)
a(n) = Sum_{k=0..n-1} A183190(n-k-2, n-2*k-2). - Jean-François Alcover, Nov 10 2019

Partially edited by Peter Luschny, Nov 12 2019

A328824 Numerators of A113405(-n) (see the comment for details).

Original entry on oeis.org

0, 1, 1, 1, -7, -7, -7, 57, 57, 57, -455, -455, -455, 3641, 3641, 3641, -29127, -29127, -29127, 233017, 233017, 233017, -1864135, -1864135, -1864135, 14913081, 14913081, 14913081, -119304647, -119304647, -119304647

Offset: 0

Paul Curtz, Oct 28 2019

Let A(n) = (2^n + (-1)^(n+1) - 2*sqrt(3)*sin((Pi*n)/3))/9. Then A(n) = A113405(n) and a(n) = numerator(A(-n)).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-8,8).

Cf. A000079, A014990, A015565, A113405, A139818.

gf := x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)): ser := series(gf, x, 36):
seq(coeff(ser,x,n),n=0..30); # Peter Luschny, Nov 11 2019

LinearRecurrence[{1,0,-8,8},{0,1,1,1},50] (* Paolo Xausa, Nov 13 2023 *)

concat(0, Vec(x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, Nov 11 2019

From Colin Barker, Nov 11 2019: (Start)
G.f.: x / ((1 - x)*(1 + 2*x)*(1 - 2*x + 4*x^2)).
a(n) = a(n-1) - 8*a(n-3) + 8*a(n-4) for n>3. (End)

A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

Original entry on oeis.org

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Ralf Steiner, May 16 2020

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
x(n) = A084175(n+2).
y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
     = 2*A192382(n+1) = 4*A003683(n+1).
z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
     = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
     = A000302(n+1) + A139818(n+1).
u(n) = A000079(n+1) = 2^(n+1).
v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).

G. C. Greubel, Table of n, a(n) for n = 0..825
V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020
E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023

A374098 a(n) = A112387(n)^2.

Original entry on oeis.org

1, 1, 4, 1, 16, 9, 64, 25, 256, 121, 1024, 441, 4096, 1849, 16384, 7225, 65536, 29241, 262144, 116281, 1048576, 466489, 4194304, 1863225, 16777216, 7458361, 67108864, 29822521, 268435456, 119311929, 1073741824, 477204025, 4294967296, 1908903481, 17179869184

Offset: 0

Paul Curtz, Jun 28 2024

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A003683, A084175, A112387, A139818.

LinearRecurrence[{0, 3, 0, 6, 0, -8}, {1, 1, 4, 1, 16, 9}, 35] (* Amiram Eldar, Jul 01 2024 *)

a(2*n) = A000302(n); a(2*n+1) = A139818(n+1).
(a(2*n) + a(2*n-1))^2 = A084175(n+1)^2 + 16*A003683(n)^2, for n >= 1. - Thomas Scheuerle, Jun 28 2024
G.f. ( 1+x+x^2-2*x^3-2*x^4 ) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x)*(2*x^2+1) ). - R. J. Mathar, Aug 02 2024

A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.

Original entry on oeis.org

1, 9, 9, 81, 225, 1089, 3969, 16641, 65025, 263169, 1046529, 4198401, 16769025, 67125249, 268402689, 1073807361, 4294836225, 17180131329, 68718952449, 274878955521, 1099509530625, 4398050705409, 17592177655809, 70368760954881, 281474943156225, 1125899973951489

Offset: 0

Peter Luschny, Jan 09 2019

Colin Barker conjectures that A208556 is a shifted version of this sequence.

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Cf. A001045, A062510, A139818, A208556, A322942, A323232.

gf := (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)):
ser := series(gf,x,32): seq(coeff(ser,x,n), n=0..25);

LinearRecurrence[{3, 6, -8}, {1, 9, 9, 81}, 25]

# Demonstrates the product formula.
CC = ComplexField(200)
def t(n,k): return CC(3)*cos(CC(pi*k/n)) - CC(i)*sin(CC(pi*k/n))
def T(n,k): return t(n,k)*(t(n,k).conjugate())
def a(n): return prod(T(n,k) for k in (1..n))
print([a(n).real().round() for n in (0..29)])

a(n) = Product_{k=1..n} T(n, k) where T(n, k) = t(n,k)*conjugate(t(n,k)) and t(n,k) = 3*cos(Pi*k/n) - i*sin(Pi*k/n), i is the imaginary unit.
a(n) = [x^n] (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)).
a(n) = n! [x^n] (1 + exp(x) - 2*exp(-2*x) + exp(4*x)).
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) for n >= 4.
A062510(n) = sqrt(a(n)) for n > 0.
a(n) = 4^n-2*(-2)^n+1, n>0. - R. J. Mathar, Mar 06 2022

A362379 Convolution triangle of A052547(n).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 4, 0, 1, 5, 2, 6, 0, 1, 5, 14, 3, 8, 0, 1, 14, 14, 27, 4, 10, 0, 1, 19, 49, 27, 44, 5, 12, 0, 1, 42, 68, 113, 44, 65, 6, 14, 0, 1, 66, 175, 159, 214, 65, 90, 7, 16, 0, 1, 131, 286, 465, 304, 360, 90, 119, 8, 18, 0, 1

Offset: 0

Philippe Deléham, Apr 20 2023

			Triangle begins, for n>=0, 0<=k<=n :
   1 ;
   0,  1 ;
   2,  0,   1 ;
   1,  4,   0,  1 ;
   5,  2,   6,  0,  1 ;
   5, 14,   3,  8,  0,  1 ;
  14, 14,  27,  4, 10,  0,  1 ;
  19, 49,  27, 44,  5, 12,  0, 1 ;
  42, 68, 113, 44, 65,  6, 14, 0, 1 ;
  ...

Cf. A052547, A077998 (row sums), A052964 (diagonal sums).

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) - T(n-2,k-1) - T(n-3,k) ; T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = 0, T(2,0) = 2, T(n,k) = 0 if k<0 or if k>n .
Sum_{k = 0..n} T(n,k)*x^k = A052547(n), A077998(n), A052536(n), A052941(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..n} T(n,k)*2^(n-k) = A139818(n+1) = A001045(n+1)^2.

A374927 a(n) = A135318(n)^2.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 25, 64, 121, 256, 441, 1024, 1849, 4096, 7225, 16384, 29241, 65536, 116281, 262144, 466489, 1048576, 1863225, 4194304, 7458361, 16777216, 29822521, 67108864, 119311929, 268435456, 477204025, 1073741824, 1908903481, 4294967296, 7635439161

Offset: 0

Paul Curtz, Jul 24 2024

A374098 terms swapped by pairs.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A112387, A135318, A139818, A374098.

a(2*n) = A139818(n+1).
a(2*n+1) = 4^n = A000302(n).
a(n) = 3*a(n-2) +6*a(n-4) -8*a(n-6).

cp's OEIS Frontend

A140966 a(n) = (5 + (-2)^n)/3.

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A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).

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A206472 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having zero permanent.

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A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

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A328824 Numerators of A113405(-n) (see the comment for details).

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A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

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A374098 a(n) = A112387(n)^2.

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A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.

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