A099965
First column (also row sums) of triangle in A099964.
Original entry on oeis.org
1, 1, 1, 2, 3, 8, 17, 39, 126, 352, 1026, 2943, 11429, 39138, 138903, 486490, 1711141, 7720713, 31383260, 131236182, 543056039, 2254646235, 9349845765, 48137959377, 226341090088, 1090094720010, 5203693239201, 24909594299924
Offset: 0
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a099965 = flip a099964 0
a099965_list = map head a099964_tabf
-- 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 31 do if member(n,tr)=false then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: seq(R[n][1],n=0..31); # 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]); Table[row[n][[1]], {n, 0, 27}] (* Jean-François Alcover, Feb 10 2018 *)
A099966
Sequence of the next-to-last entries in the rows of A099964.
Original entry on oeis.org
1, 1, 3, 6, 14, 39, 109, 313, 900, 2943, 10403, 36195, 127474, 447352, 1711141, 7234223, 29672119, 123515469, 511672779, 2123410053, 9349845765, 45883313142, 216991244323, 1041956760633, 4977352149113, 23819499579914
Offset: 1
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a099966 n = a099966_list !! (n-1)
a099966_list = map (last . init) $ tail a099964_tabf
-- 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 30 do if member(n,tr)=false 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=1..30); # Emeric Deutsch, Nov 16 2004
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triQ[n_] := Module[{k}, 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]); a[n_] := row[n][[-2]]; Array[a, 26] (* Jean-François Alcover, Feb 13 2018 *)
A099968
Second column of triangle in A099964.
Original entry on oeis.org
1, 2, 3, 6, 14, 31, 78, 235, 665, 1926, 5886, 21832, 75333, 266377, 933842, 3422282, 14954936, 61055379, 254751651, 1054728818, 4378056288, 18699691530, 94021272519, 443332334411, 2132051480643, 10181045388314, 48729093879838
Offset: 1
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a099968 = flip a099964 1 -- 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 30 do if member(n,tr)=false then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: seq(R[n][2],n=1..30); # Emeric Deutsch, Nov 16 2004
Showing 1-3 of 3 results.