A026552 -id:A026552 - cp's OEIS frontend

A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098

Offset: 0

Clark Kimberling

a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.

			Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   2;
  1, 3,  5,   6,   4;
  1, 4,  9,  14,  15,  10;
  1, 5, 14,  27,  38,  39,   25;
  1, 6, 20,  46,  79, 104,  102,   64;
  1, 7, 27,  72, 145, 229,  285,  270,  166;
  1, 8, 35, 106, 244, 446,  659,  784,  721,  436;
  1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;

Clark Kimberling, Rows 0..100, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A026269, A026270, A026288, A026289, A026290, A026291, A026292, A026293.
Cf. A026294, A026295, A026296, A026297, A026299, A026519, A026536, A026552.
Cf. A026584, A027926, A212342.

f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n,k) // T = A026268
  if k eq 0 or n lt 3 then return 1;
  elif k eq 1 then return n-1;
  elif k eq 2 then return f(n);
  elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
  else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022

T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)

def T(n,k): # T = A026268
    if n<3 or k==0: return 1
    elif k==1: return n-1
    elif k==2: return (n^2 -n -2)//2 + int(n==2)
    elif k==n: return T(n-1, n-2) + T(n-1, n-1)
    else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022

From G. C. Greubel, Sep 24 2022: (Start)
T(n, 1) = A000027(n-1), n >= 1.
T(n, 2) = A212342(n-1), n >= 2.
T(n, n-1) = A026270(n), n >= 2.
T(n, n-2) = A026288(n), n >= 2.
T(n, n-3) = A026289(n), n >= 3.
T(n, n-4) = A026290(n), n >= 4.
T(n, n) = A026269(n), n >= 2.
T(n, floor(n/2)) = A026297(n), n >= 0.
T(2*n, n) = A026292(n).
T(2*n, n-1) = A026295(n), n >= 1.
T(2*n, n+1) = A026296(n), n >= 1.
T(2*n-1, n-1) = A026291(n), n >= 2.
T(3*n, n) = A026293(n), n >= 0.
T(4*n, n) = A026294(n), n >= 0.
Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)

Updated by Clark Kimberling, Aug 29 2014

Indices of b-file corrected by Sidney Cadot, Jan 06 2023.

A026565 a(n) = 6*a(n-2), starting with 1, 3, 9.

Original entry on oeis.org

1, 3, 9, 18, 54, 108, 324, 648, 1944, 3888, 11664, 23328, 69984, 139968, 419904, 839808, 2519424, 5038848, 15116544, 30233088, 90699264, 181398528, 544195584, 1088391168, 3265173504, 6530347008, 19591041024, 39182082048

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A026532, A026534, A026551, A026552, A026567.

[1] cat [n le 2 select 3^n else 6*Self(n-2): n in [1..35]]; // G. C. Greubel, Dec 17 2021

Table[(1/4)*6^(n/2)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)) - (1/2)*Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Dec 17 2021 *)

def A026565(n): return ( (3/2)*6^(n/2) if (n%2==0) else 3*6^((n-1)/2) ) - bool(n==0)/2
[A026565(n) for n in (0..30)] # G. C. Greubel, Dec 17 2021

a(n) = Sum_{j=0..2*n} A026552(n, j).
G.f.: (1+3*x+3*x^2)/(1-6*x^2). - Ralf Stephan, Feb 03 2004
a(0)=1, a(1)=3; a(n) = 3*a(n-1) if n is even, a(n) = 2*a(n-1) if n is odd. - Vincenzo Librandi, Nov 19 2010
a(n) = (1/4)*6^(n/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) - (1/2)*[n=0]. - G. C. Greubel, Dec 17 2021

Better name from Ralf Stephan, Jul 17 2013

cp's OEIS Frontend

A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

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