This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle: 1; 1, 2; 2, 3, 4; 3, 5, 6, 7; 6, 9, 11, 12, 13; ...
a037254 n k = a037254_tabl !! (n-1) !! (k-1) a037254_row n = a037254_tabl !! (n-1) a037254_tabl = map fst $ iterate f ([1], drop 2 a002083_list) where f (row, (x:xs)) = (map (+ x) (0 : row), xs) -- Reinhard Zumkeller, Nov 18 2012
a[1, 1] = 1; a[n_, 1] := a[n, 1] = a[n - 1, Floor[(n + 1)/2]]; a[n_, k_ /; k > 1] := a[n, k] = a[n, 1] + a[n - 1, k - 1]; A037254 = Flatten[ Table[ a[n, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Apr 03 2012, after given recurrence *)
def T(n, k):
if k==1:
if n==1: return 1
else: return T(n - 1, (n + 1)//2)
return T(n, 1) + T(n - 1, k - 1)
for n in range(1, 12): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 03 2017
Numbers n and an example of a subset of {1..n} exhibiting the maximum cardinality c=a(n):
1, {1}
2, {1, 2}
3, {1, 2}
4, {1, 2, 4}
5, {1, 2, 4}
6, {1, 2, 4}
7, {3, 5, 6, 7}
8, {1, 2, 4, 8}
9, {1, 2, 4, 8}
10, {1, 2, 4, 8}
11, {1, 2, 4, 8}
12, {1, 2, 4, 8}
13, {3, 6, 11, 12, 13}
14, {1, 6, 10, 12, 14}
15, {1, 6, 10, 12, 14}
16, {1, 2, 4, 8, 16}
17, {1, 2, 4, 8, 16}
18, {1, 2, 4, 8, 16}
For examples of maximum-cardinality subsets at values of n where a(n) > a(n-1), see A096858. - _Jon E. Schoenfield_, Nov 28 2013
# is any subset of L uniquely determined by its total weight?
iswts := proc(L)
local wtset,s,c,subL,thiswt ;
# the weight sums are to be unique, so sufficient to remember the set
wtset := {} ;
# loop over all subsets of weights generated by L
for s from 1 to nops(L) do
c := combinat[choose](L,s) ;
for subL in c do
# compute the weight sum in this subset
thiswt := add(i,i=subL) ;
# if this weight sum already appeared: not a candidate
if thiswt in wtset then
return false;
else
wtset := wtset union {thiswt} ;
end if;
end do:
end do:
# All different subset weights were different: success
return true;
end proc:
# main sequence: given grams 1 to n, determine a subset L
# such that each subset of this subset has a different sum.
wts := proc(n)
local s,c,L ;
# select sizes from n (largest size first) down to 1,
# so the largest is detected first as required by the puzzle.
for s from n to 1 by -1 do
# all combinations of subsets of s different grams
c := combinat[choose]([seq(i,i=1..n)],s) ;
for L in c do
# check if any of these meets the requir, print if yes
# and return
if iswts(L) then
print(n,L) ;
return nops(L) ;
end if;
end do:
end do:
print(n,"-") ;
end proc:
# loop for weights with maximum n
for n from 1 do
wts(n) ;
end do: # R. J. Mathar, Aug 24 2010
a096796 n = a096796_list !! n a096796_list = 0 : 1 : zipWith (-) (map (* 2) $ tail a096796_list) (map a096796 $ tail a083920_list) -- Reinhard Zumkeller, Feb 12 2012
a[n_] := a[n] = If[n < 3, a[n] = n, 2a[n - 1] - a[n - Floor[1/2 + Sqrt[2n]] ]]; Table[ a[n], {n, 0, 35}] (* Robert G. Wilson v *)
{m=35;v=vector(m+1);for(n=0,m,if(n<=2,a=n,k=n-floor(1/2+sqrt(2*n));a=2*v[n]-v[k+1]);v[n+1]=a;print1(a,","))} \\ Klaus Brockhaus, Aug 20 2004
a096824 n = a096824_list !! n a096824_list = 0 : 1 : 2 : zipWith (-) (map (* 2) $ drop 2 a096824_list) (map a096824 $ tail a122797_list) -- Reinhard Zumkeller, Feb 12 2012
a[n_] := a[n] = If[n < 3, a[n] = n, 2a[n - 1] - a[n - Floor[1/2 + Sqrt[2(n - 1)]]]]; Table[ a[n], {n, 0, 35}] (* Robert G. Wilson v, Aug 20 2004 *)
{m=36;v=vector(m+1);for(n=0,m,if(n<=2,a=n,k=n-floor(1/2+sqrt(2*(n-1)));a=2*v[n]-v[k+1]);v[n+1]=a;print1(a,","))} \\ Klaus Brockhaus, Aug 20 2004
a={};k=1;Do[While[(t=1;While[t<=4&&DuplicateFreeQ[Total/@Subsets[Join[a,{k}],{t}]],t++];t)<=4,k++];AppendTo[a,k];Print@k,30] (* Giorgos Kalogeropoulos, Dec 02 2021 *)
# See links.
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