A176483 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 1, where b(n) = 5*b(n-1) - 4*b(n-2) + 3*b(n-3) - 2*b(n-4) - b(n-5) and b(0) = 0, b(1) = 1, b(2) = 5, b(3) = 21, b(4) = 88.

1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 67, 79, 67, 1, 1, 281, 344, 344, 281, 1, 1, 1176, 1453, 1504, 1453, 1176, 1, 1, 4921, 6093, 6358, 6358, 6093, 4921, 1, 1, 20594, 25511, 26671, 26885, 26671, 25511, 20594, 1, 1, 86185, 106775, 111680, 112789, 112789, 111680, 106775, 86185, 1

Offset

0,5

Comments

Row sums are {1, 2, 6, 34, 215, 1252, 6764, 34746, 172439, 834860, 3967727, ...}.

Links

Indranil Ghosh, Rows 0..120, flattened

Indranil Ghosh, Python Program to generate the b-file

Formula

Let b(n) = 5*b(n-1) - 4*b(n-2) + 3*b(n-3) - 2*b(n-4) - b(n-5), with b(0) = 0, b(1) = 1, b(2) = 5, b(3) = 21, b(4) = 88, then T(n, k) = b(n) - b(k) - b(n-k) + 1.

Example

Triangle begins as:
  1;
  1,     1;
  1,     4,      1;
  1,    16,     16,      1;
  1,    67,     79,     67,      1;
  1,   281,    344,    344,    281,      1;
  1,  1176,   1453,   1504,   1453,   1176,      1;
  1,  4921,   6093,   6358,   6358,   6093,   4921,      1;
  1, 20594,  25511,  26671,  26885,  26671,  25511,  20594,     1;
  1, 86185, 106775, 111680, 112789, 112789, 111680, 106775, 86185, 1;
...
T(3,2) = b(3) - b(2) - b(3 - 2) + 1 = 21 - 5 - 1 + 1 = 16 [b(1) = 1, b(2) = 5, b(3) = 21]. - Indranil Ghosh, Feb 17 2017

Mathematica

b[0]:=0; b[1]:=1; b[2]:=5; b[3]:=21; b[4]:=88;
b[n_]:= 5*b[n-1] -4*b[n-2] +3*b[n-3] -2*b[n-4] -b[n-5];
T[n_, m_]:= b[n] -b[m] -b[n-m] +1;
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, May 06 2019 *)

Prog

(PARI)
{b(n) = if(n==0, 0, if(n==1, 1, if(n==2, 5, if(n==3, 21, if(n==4, 88, 5*b(n-1) -4*b(n-2) +3*b(n-3) -2*b(n-4) -b(n-5))))))};
{T(n, k) = b(n) -b(k) -b(n-k) +1};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 06 2019
(Sage)
def b(n):
    if (n==0): return 0
    elif (n==1): return 1
    elif (n==2): return 5
    elif (n==3): return 21
    elif (n==4): return 88
    else: return 5*b(n-1) -4*b(n-2) +3*b(n-3) -2*b(n-4) -b(n-5)
def T(n, k): return b(n) - b(k) - b(n-k) + 1
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 06 2019

Crossrefs

Cf. A095263.

Sequence in context: A008304 A203846 A118185 * A174639 A173814 A176467

Adjacent sequences: A176480 A176481 A176482 * A176484 A176485 A176486

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 18 2010

Extensions

Edited by G. C. Greubel, May 06 2019

Status

approved