A099962
First column (also row sums) of triangle in A099961.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 5, 13, 28, 64, 207, 578, 1685, 6518, 22361, 79319, 357180, 1453177, 6075215, 31216968, 146906126, 707344776, 4084254624, 21823224623, 118932148555, 762346849634, 4559139897594, 27742486678915, 195472113041924
Offset: 0
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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 35 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: seq(R[n][1],n=0..35); # Emeric Deutsch, Nov 16 2004
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r[n_] := r[n] = If[n == 0, {1}, Module[{a}, Join[a = Accumulate[Reverse[r[n-1]]], If[Mod[n, 3] == 2, {Last[a]}, {}]]]];
a[n_] := If[n == 0, 1, r[n][[1]]];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 10 2023 *)
A099963
Sequence of the next-to-last entries in the rows of A099961.
Original entry on oeis.org
1, 1, 2, 5, 10, 23, 64, 179, 514, 1685, 5940, 20676, 79319, 334819, 1373858, 6075215, 29763791, 140830911, 707344776, 3937348498, 21115879847, 118932148555, 740523625011, 4440207749039, 27742486678915, 190912973144330
Offset: 0
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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 30 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: seq(R[n][vectdim(R[n])-1],n=2..30); # Emeric Deutsch, Nov 16 2004
A099967
Second column of triangle in A099961.
Original entry on oeis.org
1, 2, 3, 5, 10, 23, 51, 128, 386, 1092, 3370, 12458, 43037, 158638, 691999, 2827035, 12150430, 60980759, 287737037, 1414689552, 8021603122, 42939104470, 237864297110, 1502870474645, 8999347646633, 55484973357830
Offset: 0
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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 31 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: seq(R[n][2],n=2..31); # Emeric Deutsch, Nov 16 2004
A099959
Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every other row repeating the final term.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 6, 8, 8, 14, 17, 17, 17, 34, 48, 56, 56, 104, 138, 155, 155, 155, 310, 448, 552, 608, 608, 1160, 1608, 1918, 2073, 2073, 2073, 4146, 6064, 7672, 8832, 9440, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 38227, 76454, 112608
Offset: 0
Triangle begins
1;
1,
1, 1;
1, 2,
2, 3, 3;
3, 6, 8,
8, 14, 17, 17;
17, 34, 48, 56,
56, 104, 138, 155, 155;
First column (and row sums) gives
A099960.
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a099959 n k = a099959_tabl !! n !! k
a099959_row n = a099959_tabl !! n
a099959_tabl = map snd $ iterate f (False,[1]) where
f (s,xs) = (not s, if s then zs ++ [last zs] else zs)
where zs = scanl1 (+) (reverse xs)
-- Reinhard Zumkeller, Dec 28 2011
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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 18 do if n mod 2 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 18 do evalm(R[n]) od; # program yields the successive rows # Emeric Deutsch, Nov 16 2004
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row[0] = row[1] = {1}; row[n_?OddQ] := Accumulate[ Reverse[ row[n-1] ] ]; row[n_?EvenQ] := (r = Accumulate[ Reverse[ row[n-1] ] ]; AppendTo[r, Last[r] ]); Flatten[ Table[ row[n], {n, 0, 13}]] (* Jean-François Alcover, Dec 16 2011 *)
A099964
Triangle read by rows: The n-th row is constructed by forming the partial sums of the previous row, reading from the right and if n is a triangular number repeating the final term.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 3, 3, 3, 6, 8, 8, 14, 17, 17, 31, 39, 39, 39, 78, 109, 126, 126, 235, 313, 352, 352, 665, 900, 1026, 1026, 1926, 2591, 2943, 2943, 2943, 5886, 8477, 10403, 11429, 11429, 21832, 30309, 36195, 39138, 39138, 75333, 105642, 127474, 138903
Offset: 0
Triangle begins
1;
1, 1;
1, 2,
2, 3, 3;
3, 6, 8,
8, 14, 17,
17, 31, 39, 39;
39, 78, 109, 126,
126, 235, 313, 352,
352, 665, 900, 1026,
1026, 1926, 2591, 2943, 2943;
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a099964 n k = a099964_tabf !! n !! k
a099964_row n = a099964_tabf !! n
a099964_tabf = scanl f [1] $ tail a010054_list where
f row t = if t == 1 then row' ++ [last row'] else row'
where row' = scanl1 (+) $ reverse row
-- Reinhard Zumkeller, May 02 2012
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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: tr:={seq(n*(n+1)/2,n=1..30)}: R[0]:=vector(1,1): for n from 1 to 15 do if member(n,tr)=false then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 15 do evalm(R[n]) od; # Emeric Deutsch, Nov 16 2004
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triQ[n_] := Reduce[ n == k(k+1)/2, k, Integers] =!= False; row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = (ro = Accumulate[ Reverse[ row[n-1]]]; If[triQ[n], Append[ ro, Last[ro] ], ro]); Flatten[ Table[ row[n], {n, 0, 13}]](* Jean-François Alcover, Nov 24 2011 *)
Showing 1-5 of 5 results.
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