A099961 Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every third row repeating the final term.

1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 5, 10, 13, 13, 23, 28, 28, 51, 64, 64, 64, 128, 179, 207, 207, 386, 514, 578, 578, 1092, 1478, 1685, 1685, 1685, 3370, 4848, 5940, 6518, 6518, 12458, 17306, 20676, 22361, 22361, 43037, 60343, 72801, 79319, 79319, 79319

Offset: 0

Author

N. J. A. Sloane, Nov 13 2004

Keywords

• nonn
• tabf
• nice
• easy

Comments

...

Examples

			Triangle begins
   1;
   1,
   1,  1;
   1,  2,
   2,  3,
   3,  5,  5;
   5, 10, 13,
  13, 23, 28,
  28, 51, 64, 64;
		

Links

• Reinhard Zumkeller, Rows n=0..120 of triangle, flattened

Crossrefs

First column (and row sums) gives A099962. Cf. A099963, A099967.

If an extra term is added to /every/ row we get A008282. Cf. A099959.

Programs

Haskell

a099961 n k = a099961_tabl !! n !! k
a099961_row n = a099961_tabl !! n
a099961_tabl = map snd $ iterate f (0,[1]) where
   f (s,xs) = (s+1, if s `mod` 3 == 1 then zs ++ [last zs] else zs)
     where zs = scanl1 (+) (reverse xs)
-- Reinhard Zumkeller, Dec 28 2011

Maple

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n,p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j],j=1..i): vector(n,q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j],j=1..i) else sum(a[j],j=1..n) fi end: vector(n+1,q) end: R[0]:=vector(1,1): for n from 1 to 19 do if n mod 3 = 0 or n mod 3 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 19 do evalm(R[n]) od; # program yields the successive rows # Emeric Deutsch, Nov 16 2004

Mathematica

r[0] = {1}; r[n_] := r[n] = Join[ a = Accumulate[ Reverse[r[n-1]]], If[Mod[n, 3] == 2, {Last[a]}, {}]]; Flatten[ Table[r[n], {n, 0, 15}]](* Jean-François Alcover, Mar 13 2012 *)

Extensions

More terms from Emeric Deutsch, Nov 16 2004