cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A193735 Mirror of the triangle A193734.

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 32, 32, 10, 1, 128, 160, 72, 14, 1, 512, 768, 448, 128, 18, 1, 2048, 3584, 2560, 960, 200, 22, 1, 8192, 16384, 13824, 6400, 1760, 288, 26, 1, 32768, 73728, 71680, 39424, 13440, 2912, 392, 30, 1, 131072, 327680, 360448, 229376, 93184, 25088, 4480, 512, 34, 1
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

A193735 is obtained by reversing the rows of the triangle A193734.
Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
    1;
    2,   1;
    8,   6,   1;
   32,  32,  10,   1;
  128, 160,  72,  14,  1;
  512, 768, 448, 128, 18, 1;

Links

Crossrefs

Cf. A084938, A193722, A193734.
Cf. A001075, A001077, A005053, A081294, A133494.

Programs

  • Magma
    function T(n, k) // T = A193735
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2023
  • Mathematica
    (* First program *)
z = 8; a = 2; b = 1; c = 1; d = 2;
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}]] (* A193734 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]      (* A193735 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1, k] + T[n -1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2023 *)
  • SageMath
    def T(n, k): # T = A193735
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 19 2023

Formula

T(n,k) = A193734(n,n-k).
T(n,k) = T(n-1,k-1) + 4*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)/(1-4*x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 19 2023: (Start)
T(n, 0) = A081294(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001077(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001075(n). (End)

A120027 Triangle, generated from (3^(n-k) * 5^k) table.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 27, 45, 75, 125, 81, 135, 225, 375, 625, 243, 405, 675, 1125, 1875, 3125, 729, 1215, 2025, 3375, 5625, 9375, 15625, 2187, 3645, 6075, 10125, 16875, 28125, 46875, 78125, 6561, 10935, 18225, 30375, 50625, 84375, 140625, 234375
Offset: 0

Views

Author

Gary W. Adamson, Jun 04 2006

Keywords

Comments

Row 1 of the array (3, 15, 75, 375, ...) = A005053, (3 * 5^n), deleting the "1".

Examples

			First few rows of the array:
  1,  5,  25,  125, ...
  3, 15,  75,  375, ...
  9, 45, 225, 1125, ...
First few rows of the triangle are:
   1;
   3,  5;
   9, 15, 25;
  27, 45, 75, 125;
  ...
Example: a(17) = 675 = (3,2) in the array, = 3^3 * 5^2.

Links

Crossrefs

Cf. A005053, A036561, A036565, A036566.

Programs

  • Mathematica
    Table[3^(n - k)*5^k, {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jun 06 2006 *)

Formula

Antidiagonals of the (3^i * 5^j) multiplication table, as an array.
From Boris Putievskiy, Jan 09 2013: (Start)
T(n,k) = 3^(k-1)*5^(n-1) n, k >0 read by antidiagonals.
a(n) = 3^(A004736(n)-1) * 5^(A002260(n)-1), n > 0, or
a(n) = 3^(j-1) * 5^(i-1), n > 0,
where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). (End)
G.f.: 1/((1 - 3*x)(1 - 5*x*y)). - Ilya Gutkovskiy, Jun 03 2017

Extensions

More terms from Robert G. Wilson v, Jun 06 2006

A097111 Expansion of (1 + 3x - 2x^2 - 12x^3)/(1 - 9x^2 + 20x^4).

Original entry on oeis.org

1, 3, 7, 15, 43, 75, 247, 375, 1363, 1875, 7327, 9375, 38683, 46875, 201607, 234375, 1040803, 1171875, 5335087, 5859375, 27199723, 29296875, 138095767, 146484375, 698867443, 732421875, 3527891647, 3662109375, 17773675963, 18310546875
Offset: 0

Views

Author

Paul Barry, Jul 25 2004

Keywords

Links

Crossrefs

Cf. A005053 (bisection), A193656 (bisection?).

Formula

G.f.: 3*(1+x)/(1-5x^2) - 2/(1-4x^2);
a(n) = 9*a(n-2) - 20*a(n-4);
a(n) = (3/2 + 3*sqrt(5)/10)*(sqrt(5))^n + (3/2 - 3*sqrt(5)/10)*(-sqrt(5))^n - 2^(n+1)*(1+(-1)^n)/2;
a(n) = Sum_{k=0..n} binomial(floor(n/2), floor(k/2))*2^k.

A174971 Periodic sequence: Repeat 3, -3.

Original entry on oeis.org

3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3
Offset: 0

Views

Author

Klaus Brockhaus, Apr 04 2010

Keywords

Comments

Interleaving of A010701 and -A010701; signed version of A010701.
Essentially first differences of A010674.
Inverse binomial transform of 3 followed by A000004.
Second inverse binomial transform of A010701.
Third inverse binomial transform of A007283.
Fourth inverse binomial transform of A000244 without initial term 1.
Fifth inverse binomial transform of A164346.
Sixth inverse binomial transform of A005053 without initial term 1.
Seventh inverse binomial transform of A169604.
Eighth inverse binomial transform of A169634.
Ninth inverse binomial transform of A103333 without initial term 1.
Tenth inverse binomial transform of A013708.
Eleventh inverse binomial transform of A093138 without initial term 1.

Links

Crossrefs

Cf. A010701 (all 3's sequence), A000004 (all zeros sequence), A007283 (3*2^n), A000244 (powers of 3), A164346 (3*4^n), A005053 (expand (1-2x)/(1-5x)), A169604 (3*6^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2n+1)), A093138 (expand (1-7x)/(1-10x)).

Programs

  • Magma
    &cat[ [3, -3]: n in [0..41] ];
[ 3*(-1)^n: n in [0..83] ];
  • Mathematica
    PadRight[{},120,{3,-3}] (* or *) NestList[-1#&,3,120] (* Harvey P. Dale, Dec 30 2023 *)
  • PARI
    a(n)=3*(-1)^n \\ Charles R Greathouse IV, Jun 13 2013

Formula

a(n) = 3*(-1)^n.
a(n) = -a(n-1) for n > 0; a(0) = 3.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = -3.
G.f.: 3/(1+x).
Previous Showing 11-14 of 14 results.

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