A199923 -id:A199923 - cp's OEIS frontend

A231349 Number of triangles added at n-th stage to the structure of A231348.

1, 2, 4, 4, 4, 8, 10, 8, 4, 8, 12, 16, 10, 20, 22, 16, 4, 8, 12, 16, 12, 24, 28, 32, 10, 20, 28, 40, 22, 44, 46, 32, 4, 8, 12, 16, 12, 24, 28, 32, 12, 24, 32, 48, 28, 56, 60, 64, 10, 20, 28, 40, 28, 56, 64, 80, 22, 44, 60, 88, 46, 92, 94, 64

Offset: 1

Omar E. Pol, Dec 15 2013

First differences of A231348.
Observation: the row sums of the first seven rows coincide with the first seven elements of A199923.
Is A199923 the row sums of this triangle?

			Written as an irregular triangle in which row lengths is A011782 the sequence begins:
1;
2;
4,4;
4,8,10,8;
4,8,12,16,10,20,22,16;
4,8,12,16,12,24,28,32,10,20,28,40,22,44,46,32;
4,8,12,16,12,24,28,32,12,24,32,48,28,56,60,64,10,20,28,40, 28,56,64,80,22,44,60,88,46,92,94,64;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Right border gives A000079.

Cf. A001316, A011782, A231348.

A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.

Original entry on oeis.org

1, 3, 11, 41, 149, 527, 1823, 6197, 20777, 68891, 226355, 738113, 2391485, 7705895, 24712007, 78918989, 251105873, 796364339, 2518233179, 7942120025, 24988621541, 78452649023, 245818300271, 768835960421, 2400651060089

Offset: 0

Paul Barry, Jul 26 2003

Binomial transform of A057711 (without leading zero). Second binomial transform of (1,1,3,3,5,5,7,7,9,9,11,11,...).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A007051, A027471, A057711, A081038.

Partial sums of A199923.

[n*3^(n-1) + (3^n+1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

Table[((2*n+3)*3^(n-1) +1)/2, {n,0,30}] (* G. C. Greubel, Nov 24 2023 *)

Vec((1-4*x+5*x^2)/((1-x)*(1-3*x)^2) + O(x^40)) \\ Michel Marcus, Mar 08 2016

[((2*n+3)*3^(n-1) +1)//2 for n in range(31)] # G. C. Greubel, Nov 24 2023

a(n) = (1/2)*(A081038(n) + 1).
G.f.: (1-4*x+5*x^2)/((1-x)*(1-3*x)^2).
a(n) = A027471(n) + A007051(n).
E.g.f.: (1/2)*( exp(x) + (2*x+1)*exp(3*x) ). - G. C. Greubel, Nov 24 2023

A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

Original entry on oeis.org

1, 2, 1, 4, 8, 3, 8, 28, 30, 9, 16, 80, 144, 108, 27, 32, 208, 528, 648, 378, 81, 64, 512, 1680, 2880, 2700, 1296, 243, 128, 1216, 4896, 10800, 14040, 10692, 4374, 729, 256, 2816, 13440, 36288, 60480, 63504, 40824, 14580, 2187, 512, 6400, 35328, 112896, 229824, 308448, 272160, 151632, 48114, 6561

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   2,   1;
   4,   8,   3;
   8,  28,  30,   9;
  16,  80, 144, 108,  27;
  32, 208, 528, 648, 378, 81;

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

Cf. A000079, A005053, A084938, A130129, A133494, A165326.

Cf. A193722, A193731, A199923.

function T(n, k) // T = A193730
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 2*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023

(* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]     (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)

def T(n, k): # T = A193730
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 2*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x*y)/(1-2*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A000079(n).
T(n, 1) = A130129(n-1).
T(n, n) = A133494(n).
T(n, n-1) = A199923(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)

A209998 Triangle of coefficients of polynomials v(n,x) jointly generated with A209996; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 8, 9, 2, 10, 30, 27, 2, 10, 46, 108, 81, 2, 10, 50, 198, 378, 243, 2, 10, 50, 242, 810, 1296, 729, 2, 10, 50, 250, 1122, 3186, 4374, 2187, 2, 10, 50, 250, 1234, 4986, 12150, 14580, 6561, 2, 10, 50, 250, 1250, 5946, 21330, 45198, 48114, 19683

Offset: 1

Clark Kimberling, Mar 23 2012

Row n starts 2, 2*5, 2*5^2,... ; ends with 3^(n-1).
Conjecture: penultimate term in row n is A199923(n).
Alternating row sums: A077925
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...3
2...8....9
2...10...30...27
2...10...46...108...81
First three polynomials v(n,x): 1, 2 + 3x , 2 + 8x + 9x^2.

Cf. A209996, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*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[%]    (* A209996 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209998 *)

u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

a(55) corrected by Georg Fischer, Sep 03 2021

A199922 Table read by rows, T(0,0) = 1 and for n>0, 0<=k<=3^(n-1) T(n,k) = gcd(k,3^(n-1)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 81, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81

Offset: 0

Peter Luschny, Nov 12 2011

			             1
            1, 1
         3, 1, 1, 3
9, 1, 1, 3, 1, 1, 3, 1, 1, 9

G. C. Greubel, Rows n = 0..8 of the irregular triangle, flattened

Cf. A000007, A198069, A199923.

[1] cat [Gcd(k, 3^(n-1)): k in [0..3^(n-1)], n in [1..6]]; // G. C. Greubel, Nov 24 2023

seq(print(seq(gcd(k,3^(n-1)), k=0..3^(n-1))),n=0..4);

T[n_, k_]:= If[n==0, 1, GCD[k, 3^(n-1)]];
Table[T[n, k], {n,0,6}, {k,0,3^(n-1)}]//Flatten (* G. C. Greubel, Nov 24 2023 *)

def A199922(n,k): return gcd(k, 3^(n-1)) + (2/3)*int(n==0)
flatten([[A199922(n,k) for k in range(int(3^(n-1))+1)] for n in range(7)]) # G. C. Greubel, Nov 24 2023

From G. C. Greubel, Nov 24 2023: (Start)
T(n, 3^(n-1) - k) = T(n, k).
Sum_{k=0..3^(n-1)} T(n, k) = A199923(n).
Sum_{k=0..3^(n-1)} (-1)^k * T(n, k) = A000007(n). (End)

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A231349 Number of triangles added at n-th stage to the structure of A231348.

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A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.

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A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

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A209998 Triangle of coefficients of polynomials v(n,x) jointly generated with A209996; see the Formula section.

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A199922 Table read by rows, T(0,0) = 1 and for n>0, 0<=k<=3^(n-1) T(n,k) = gcd(k,3^(n-1)).

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