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Cf. A001020 (powers of 11).

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A row sum 11^n triangular recursion sequence:Prime[j]=11=scale; ATriangle T(n, k) = AT(n - 1, k - 1) + AT(n - 1, k-1) + (j+4)*Prime[prime(j])*AT(n - 2, k - 1) with j=5, read by rows.

2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1529, 26220, 1529, 2, 2, 1729, 159320, 159320, 1729, 2, 2, 1929, 312420, 2914420, 312420, 1929, 2, 2, 2129, 485520, 18999520, 18999520, 485520, 2129, 2, 2, 2329, 678620, 50414620, 326526620, 50414620, 678620, 2329, 2

Row sums are:

{2, 22, 242, 2662, 29282, 322102, 3543122, 38974342, 428717762, 4715895382,...}.

Plot of the lowest level of the fractal is:

a = Table[Table[If[m <= n, If[Mod[A[n, m], 11] == 0, 0, 1], 0], {m, 1, 10}], {n, 1, 10}] ;

ListDensityPlot[a, Mesh -> False, Axes -> False]

G. C. Greubel, <a href="/A153650/b153650.txt">Rows n = 1..50 of the triangle, flattened</a>

AT(n, k) = AT(n - 1, k - 1) + AT(n - 1, k-1) + (j+4)*Prime[prime(j])*AT(n - 2, k - 1) with j=5.

From G. C. Greubel, Mar 04 2021: (Start)

T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (1,4,5).

Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=5 = 2*A001020(n-1). (End)

{2},

Triangle begins as:

2;

{ 11, 11},;

{ 2, 238, 2},;

{ 2, 1329, 1329, 2},;

{ 2, 1529, 26220, 1529, 2},;

{ 2, 1729, 159320, 159320, 1729, 2},;

{ 2, 1929, 312420, 2914420, 312420, 1929, 2},;

{ 2, 2129, 485520, 18999520, 18999520, 485520, 2129, 2},;

{ 2, 2329, 678620, 50414620, 326526620, 50414620, 678620, 2329, 2},;

{ 2, 2529, 891720, 99159720, 2257893720, 2257893720, 99159720, 891720, 2529, 2};

ClearT[n_, k_, p_, q_, j_]:= T[n, k, p, q, j]= If[t, n, m, A, a==2, Prime[j]; , If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j ]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k== 5n, 2, T[n-1, k, p, q, j] + T[n-1, k-1, p, q, j] + (p*j+q)*Prime[j]*T[n-2, k-1, p, q, j] ]]];

A[2, 1] := A[2, 2] = Prime[j];

A[3, 2] = 2*Prime[j]^2 - 4;

A[4, 2] = A[4, 3] = Prime[j]^3 - 2;

A[n_, 1] := 2; A[n_, n_] := 2;

A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + (j+4)*Prime[j]*A[n - 2, k - 1];

Table[Table[A[n, m], {m, 1, n}], {n, 1, 10}] ;

Flatten[%] Table[Sum[A[n, m], {m, 1, n}], {n, 1, 10}] ;

Table[Sum[A[n, m], {m, 1, n}]/(2*Prime[j]^(n - 1)), {n, 1, 10}]

Table[T[n, k, 1, 4, 5], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)

(Sage)

@CachedFunction

def f(n, j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)

def T(n, k, p, q, j):

if (n==2): return nth_prime(j)

elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n, j)

elif (k==1 or k==n): return 2

else: return T(n-1, k, p, q, j) + T(n-1, k-1, p, q, j) + (p*j+q)*nth_prime(j)*T(n-2, k-1, p, q, j)

flatten([[T(n, k, 1, 4, 5) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021

(Magma)

f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;

function T(n, k, p, q, j)

if n eq 2 then return NthPrime(j);

elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n, j);

elif (k eq 1 or k eq n) then return 2;

else return T(n-1, k, p, q, j) + T(n-1, k-1, p, q, j) + (p*j+q)*NthPrime(j)*T(n-2, k-1, p, q, j);

end if; return T;

end function;

[T(n, k, 1, 4, 5): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), this sequence (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A001020.

nonn,uned,tabl

Edited by G. C. Greubel, Mar 04 2021

approved

editing

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Dec 30 2008

A row sum 11^n triangular recursion sequence:Prime[j]=11=scale; A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + (j+4)*Prime[j]*A(n - 2, k - 1).

2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1529, 26220, 1529, 2, 2, 1729, 159320, 159320, 1729, 2, 2, 1929, 312420, 2914420, 312420, 1929, 2, 2, 2129, 485520, 18999520, 18999520, 485520, 2129, 2, 2, 2329, 678620, 50414620, 326526620, 50414620

1,1

Row sums are:

{2, 22, 242, 2662, 29282, 322102, 3543122, 38974342, 428717762, 4715895382,...}.

Plot of the lowest level of the fractal is:

a = Table[Table[If[m <= n, If[Mod[A[n, m], 11] == 0, 0, 1], 0], {m, 1, 10}], {n, 1, 10}] ;

ListDensityPlot[a, Mesh -> False, Axes -> False]

A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + (j+4)*Prime[j]*A(n - 2, k - 1).

{2},

{11, 11},

{2, 238, 2},

{2, 1329, 1329, 2},

{2, 1529, 26220, 1529, 2},

{2, 1729, 159320, 159320, 1729, 2},

{2, 1929, 312420, 2914420, 312420, 1929, 2},

{2, 2129, 485520, 18999520, 18999520, 485520, 2129, 2},

{2, 2329, 678620, 50414620, 326526620, 50414620, 678620, 2329, 2},

{2, 2529, 891720, 99159720, 2257893720, 2257893720, 99159720, 891720, 2529, 2}

Clear[t, n, m, A, a]; j = 5;

A[2, 1] := A[2, 2] = Prime[j];

A[3, 2] = 2*Prime[j]^2 - 4;

A[4, 2] = A[4, 3] = Prime[j]^3 - 2;

A[n_, 1] := 2; A[n_, n_] := 2;

A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + (j+4)*Prime[j]*A[n - 2, k - 1];

Table[Table[A[n, m], {m, 1, n}], {n, 1, 10}] ;

Flatten[%] Table[Sum[A[n, m], {m, 1, n}], {n, 1, 10}] ;

Table[Sum[A[n, m], {m, 1, n}]/(2*Prime[j]^(n - 1)), {n, 1, 10}]

nonn,uned,tabl,new

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 30 2008

approved