A026552 - OEIS

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A026552

Irregular triangular array T read by rows: T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2 + 1), for even n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), otherwise T(n, k) = T(n-1, k-2) + T(n-1, k).

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)|<=1 if i is even or i = 1, |s(i)-s(i-1)| = 1 if i is odd and i >= 3.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..10200 [Offset changed to 0 by Georg Fischer, Mar 01 2022]

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

Sum_{k=0..2*n} T(n,k) = A026565(n). - G. C. Greubel, Dec 17 2021

EXAMPLE

First 5 rows:

1, 1, 1;

1, 2, 3, 2, 1;

1, 2, 4, 4, 4, 2, 1;

1, 3, 7, 10, 12, 10, 7, 3, 1;

MATHEMATICA

z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2 + 1]; t[n_, k_] := Floor[n/2 + 1] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];

TableForm[u] (* A026552 array *)

v = Flatten[u] (* A026552 sequence *)

PROG

(Sage)

@CachedFunction

def T(n, k): # T = A026552

if (k==0 or k==2*n): return 1

elif (k==1 or k==2*n-1): return (n+2)//2

elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)

else: return T(n-1, k) + T(n-1, k-2)

flatten([[T(n, k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021

CROSSREFS