A007070 -id:A007070 - cp's OEIS frontend

A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

1, 1, 4, 11, 32, 84, 230, 597, 1567, 4020, 10286, 25994, 65387, 163065, 404617, 997687, 2448220, 5977334, 14530835, 35173496, 84814982, 203760809, 487845377, 1164191563, 2769721073, 6570218773, 15542642042, 36671354125, 86306246887, 202637312099, 474684979292, 1109539437382

Offset: 0

Ilya Gutkovskiy, Oct 01 2018

nonn

Convolution of A010815 and A034899.

Euler transform of A000225.

Alois P. Heinz, Table of n, a(n) for n = 0..3171
N. J. A. Sloane, Transforms

Cf. A000225, A007070, A010815, A034691, A034899, A319919.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
       d*(2^d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 13 2021

nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Exp[Sum[x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ A247003^2 * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A065446 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 15 2021

A059474 Triangle read by rows: T(n,k) is coefficient of z^nw^k in 1/(1 - 2z - 2w + 2z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 16, 16, 8, 16, 40, 52, 40, 16, 32, 96, 152, 152, 96, 32, 64, 224, 416, 504, 416, 224, 64, 128, 512, 1088, 1536, 1536, 1088, 512, 128, 256, 1152, 2752, 4416, 5136, 4416, 2752, 1152, 256, 512, 2560, 6784, 12160, 16032, 16032, 12160, 6784, 2560, 512

Offset: 0

N. J. A. Sloane, Feb 03 2001; revised Jun 12 2005

Pascal-like triangle: start with 1 at top; every subsequent entry is the sum of everything above you, plus 1.

			Triangle begins as:
   n\k [0]  [1]  [2]  [3]  [4]  [5]  [6] ...
  [0]   1;
  [1]   2,   2;
  [2]   4,   6,   4;
  [3]   8,  16,  16,   8;
  [4]  16,  40,  52,  40,  16;
  [5]  32,  96, 152, 152,  96,  32;
  [6]  64, 224, 416, 504, 416, 224,  64;
       ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A001792, A007070, A057711, A077957, A084773.

See A059576 for a similar triangle.

A059474:= func< n,k | (&+[(-1)^j*2^(n-j)*Binomial(n-k,j)*Binomial(n-j,n-k): j in [0..n-k]]) >;
[A059474(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 21 2023

read transforms; SERIES2(1/(1-2*z-2*w+2*z*w),x,y,12): SERIES2TOLIST(%,x,y,12);
# Alternative
T := (n, k) -> 2^n*binomial(n, k)*hypergeom([-k, -n + k], [-n], 1/2):
for n from 0 to 10 do seq(simplify(T(n, k)), k = 0 .. n) end do; # Peter Luschny, Nov 26 2021

Table[(-1)^k*2^n*JacobiP[k, -n-1,0,0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 04 2017; May 21 2023 *)

def A059474(n,k): return 2^n*binomial(n, k)*simplify(hypergeometric([-k, k-n], [-n], 1/2))
flatten([[A059474(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 21 2023

G.f.: 1/(1 - 2*z - 2*w + 2*z*w).
T(n, k) = Sum_{j=0..n} (-1)^j*2^(n + k - j)*C(n, j)*C(n + k - j, n).
T(n, 0) = T(n, n) = A000079(n).
T(2*n, n) = A084773(n).
T(n, k) = 2^n*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2). - Peter Luschny, Nov 26 2021
From G. C. Greubel, May 21 2023: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = A007070(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
T(n, 1) = A057711(n+1) = 2*A001792(n) - [n=0].
T(n, 2) = 4*A049611(n-1). (End)

A116414 Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

Original entry on oeis.org

1, 4, 1, 13, 8, 1, 40, 42, 12, 1, 121, 184, 87, 16, 1, 364, 731, 496, 148, 20, 1, 1093, 2736, 2454, 1040, 225, 24, 1, 3280, 9844, 11064, 6170, 1880, 318, 28, 1, 9841, 34448, 46738, 32624, 13015, 3080, 427, 32, 1, 29524, 118101, 188208, 158724, 79044, 24381, 4704

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A116415. Diagonal sums are A007070. First column is A003462(n+1). Product of A007318 and A116412.

Subtriangle of triangle given by (0, 4, -3/4, 3/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
    1;
    4,   1;
   13,   8,   1;
   40,  42,  12,   1;
  121, 184,  87,  16,  1;
  364, 731, 496, 148, 20, 1;
Triangle T(n,k), 0 <= k <= n, given by (0, 4, -3/4, 3/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,   1;
  0,   4,   1;
  0,  13,   8,   1;
  0,  40,  42,  12,   1;
  0, 121, 184,  87,  16,  1;
  0, 364, 731, 496, 148, 20, 1;
  ... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A003462, A116412, A110441.

With[{n = 10}, DeleteCases[#, 0] & /@ Rest@ CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 4 x + 3 x^2 - x y), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Riordan array (1/(1-4x+3x^2), x/(1-4x+3x^2)); number triangle T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k+j,j)*3^j.
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(2,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Oct 31 2013
G.f.: (1-4*x+3*x^2)/(1-4*x+3*x^2-x*y). - Philippe Deléham, Oct 31 2013
From Peter Bala, Oct 07 2019: (Start)
O.g.f.: 1/(1 - 4*x + 3*x^2 - x*y) = 1 + (4 + y)*x + (13 + 8*y + y^2)*x^2 + ....
Recurrence for row polynomials: R(n,y) = (4 + y)*R(n-1,y) - 3*R(n-2,y) with R(0,y) = 1 and R(1,y) = 4 + y.
The row reverse polynomial y^n*R(n,1/y) is equal to the numerator polynomial of the finite continued fraction 1 + y/(1 + 3*y/(1 + ... + y/(1 + 3*y/(1)))) (with 2*n partial numerators). Cf. A110441. (End)

A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

Original entry on oeis.org

1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 01 2006

The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.

Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - Philippe Deléham, Mar 04 2016

			Triangle begins as:
      1;
      4,     -1;
     15,     -8,     1;
     56,    -46,    12,    -1;
    209,   -232,    93,   -16,    1;
    780,  -1091,   592,  -156,   20,   -1;
   2911,  -4912,  3366, -1200,  235,  -24,  1;
  10864, -21468, 17784, -8010, 2120, -330, 28, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Joanne Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

Cf. A000302, A001353, A001906, A004254, A007070, A123966.

Cf. A139278, A159764, A181983, A207823 (absolute values).

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n,k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
[A124029(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

A123966x := proc(n,x)
    local A,r,c ;
    A := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
            A[r,c] :=0 ;
        if r = c then
            A[r,c] := A[r,c]+4 ;
        elif abs(r-c)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A123966 := proc(n,k)
    coeftayl( A123966x(n,x),x=0,k) ;
end proc:
seq(seq(A123966(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Table[M[d], {d,10}]
Table[Det[M[d]], {d,10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
(* Recursive Polynomial form*)
p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

def A124029(n,k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
flatten([[A124029(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
Sum_{k=0..n} T(n, k) = A001906(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).
T(n, n) = (-1)^n.
T(n, n-1) = 4*A181983(n), n >= 1.
T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.
T(n, 0) = A001353(n+1). (End)

A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

Original entry on oeis.org

1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

nonn

Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - Sergio Falcon, Sep 21 2009; edited by Klaus Brockhaus, Oct 11 2009
Eighth binomial transform of A048697.
First differences are in A164600.
lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (18,-79).

Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - Sergio Falcon, Sep 21 2009

Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-79},{1,18},25] (* G. C. Greubel, Aug 22 2016 *)

a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009
E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A163608 a(n) = ((5 + 2sqrt(2))(2 + sqrt(2))^n + (5 - 2sqrt(2))(2 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 14, 46, 156, 532, 1816, 6200, 21168, 72272, 246752, 842464, 2876352, 9820480, 33529216, 114475904, 390845184, 1334428928, 4556025344, 15555243520, 53108923392, 181325206528, 619082979328, 2113681504256, 7216560058368

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163607. Inverse binomial transform of A163609.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A163607, A163609.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(2+r)^n+(5-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{4,-2},{5,14},30] (* Harvey P. Dale, Jan 31 2017 *)

x='x+O('x^50); Vec((5-6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 5, a(1) = 14.
G.f.: (5-6*x)/(1-4*x+2*x^2).
a(n) = 5*A007070(n) - 6*A007070(n-1). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(2*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920

Offset: 1

Clark Kimberling, Mar 25 2012

Column 1: triangular numbers, A000217
Coefficient of v(n,x):  2^(n-1)
Row sums:  A035344
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017

			First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2

Cf. A210753, A208510.

Cf. A056242, A165241.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A214996 Power floor-ceiling sequence of 2+sqrt(2).

Original entry on oeis.org

3, 11, 37, 127, 433, 1479, 5049, 17239, 58857, 200951, 686089, 2342455, 7997641, 27305655, 93227337, 318298039, 1086737481, 3710353847, 12667940425, 43251054007, 147668335177, 504171232695, 1721348260425, 5877050576311, 20065505784393, 68507921984951

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power floor-ceiling sequence and power floor-ceiling function, p2(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p2(r) = (11 + 8*sqrt(2))/7.
From Greg Dresden, Jun 02 2020: (Start)
a(n) is the number of ways to tile a 2 X (n+1) strip, with one extra square at the top left corner, using 1 X 1 squares, 2 X 2 squares, and 1 X 2 dominoes (either horizontal or vertical). This picture shows a(1) = 11.
                                                     _
  ||  ||  | |  ||_  ||  ||  ||  | |  | |  ||  ||
  ||| |   | ||| | || || | |_| ||| ||| || | |__| | | |
  ||| |_| ||| ||| ||| ||| |_| |_| ||| |_| |||
(End)

			a(0) = floor(r) = 3, where r = 2+sqrt(2).
a(1) = ceiling(3*r) = 11; a(2) = floor(11*r) = 37.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Cf. A214992, A007052, A214997, A007070, A052543.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((3+2*x-2*x^2)/(1-3*x-2*x^2+2*x^3))) // G. C. Greubel, Feb 02 2018

x = 2 + Sqrt[2]; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A007052 *)
Table[p2[n], {n, 0, z}]  (* A214996 *)
Table[p3[n], {n, 0, z}]  (* A214997 *)
Table[p4[n], {n, 0, z}]  (* A007070 *)

Vec((3 + 2*x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = ceiling(x*a(n-1)) if n is odd, a(n) = floor(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = floor(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (3 + 2*x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/7)*((-1)^(1+n) + (11-8*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(11+8*sqrt(2))). - Colin Barker, Nov 13 2017

A214997 Power ceiling-floor sequence of 2+sqrt(2).

Original entry on oeis.org

4, 13, 45, 153, 523, 1785, 6095, 20809, 71047, 242569, 828183, 2827593, 9654007, 32960841, 112535351, 384219721, 1311808183, 4478793289, 15291556791, 52208640585, 178251448759, 608588513865, 2077851157943, 7094227604041, 24221208100279, 82696377193033

Offset: 0

Clark Kimberling, Nov 10 2012

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p3(r) = 3.8478612632206289...
a(n) is the number of words over {0,1,2,3} of length n+1 that avoid 23, 32, and 33. As an example, a(2)=45 corresponds to the 45 such words of length 3; these are all 64 words except for the 19 prohibited cases which are 320, 321, 322, 323, 230, 231, 232, 233, 330, 331, 332, 333, 023, 123, 223, 032, 132, 033, 133. - Greg Dresden and Mina BH Arsanious, Aug 09 2023
Let M denote the 4 X 4 matrix = [[1,1,1,1], [1,1,1,1], [1,1,1,0], [1,1,0,0]] and A(n) = the column vector (p(n),q(n),r(n),s(n)) = M^n * A(0), where A(0) = (1,1,1,1), then a(n) = p(n)+q(n)+r(n)+s(n) = p(n+1). - Mina BH Arsanious, Jan 18 2025
Sum_{k=0..n} a(k) = (r(n-2)-3)/2 where r(n) is defined in previous comment. - Mina BH Arsanious, May 21 2025

			a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Cf. A214992, A007052, A214996, A007070.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((4 +x-2*x^2)/(1-3*x-2*x^2+2*x^3))); // G. C. Greubel, Feb 01 2018

(See A214996.)
CoefficientList[Series[(4+x-2*x^2)/(1-3*x-2*x^2+2*x^3), {x,0,50}], x] (* G. C. Greubel, Feb 01 2018 *)

Vec((4 + x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = ceiling(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (4 + x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/14)*(2*(-1)^n + (27-19*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(27+19*sqrt(2))). - Colin Barker, Nov 13 2017

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0

Offset: 0

Seiichi Manyama, Mar 01 2021

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  0, 2,   4,    6,     8,    10, ...
  0, 3,  14,   33,    60,    95, ...
  0, 4,  48,  180,   448,   900, ...
  0, 5, 164,  981,  3344,  8525, ...
  0, 6, 560, 5346, 24960, 80750, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
Rows 0..2 give A000012, A005843, A033991.
Main diagonal gives (-1) * A109520(n+1).
Cf. A342120, A342129, A342134.

T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021

T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)

T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));

T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.

cp's OEIS Frontend

A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

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A059474 Triangle read by rows: T(n,k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w + 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

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A116414 Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

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A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

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A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

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A163608 a(n) = ((5 + 2*sqrt(2))*(2 + sqrt(2))^n + (5 - 2*sqrt(2))*(2 - sqrt(2))^n)/2.

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A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

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A059474 Triangle read by rows: T(n,k) is coefficient of z^nw^k in 1/(1 - 2z - 2w + 2z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

A163608 a(n) = ((5 + 2sqrt(2))(2 + sqrt(2))^n + (5 - 2sqrt(2))(2 - sqrt(2))^n)/2.

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).