A183534 Square array of generalized Ulam numbers U(n,k), n>=1, k>=2, read by antidiagonals: U(n,k) = n if n<=k; for n>k, U(n,k) = least number > U(n-1,k) which is a unique sum of k distinct terms U(i,k) with i

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 6, 6, 1, 2, 3, 4, 9, 8, 1, 2, 3, 4, 10, 10, 11, 1, 2, 3, 4, 5, 16, 11, 13, 1, 2, 3, 4, 5, 15, 17, 12, 16, 1, 2, 3, 4, 5, 6, 25, 18, 28, 18, 1, 2, 3, 4, 5, 6, 21, 26, 19, 29, 26, 1, 2, 3, 4, 5, 6, 7, 36, 27, 22, 30, 28, 1, 2, 3, 4, 5, 6, 7, 28, 37, 28, 64, 53, 36

Offset: 1

Author

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

Keywords

• nonn
• tabl
• look

Comments

The columns are Ulam-type sequences - see A002858 for further information. Some of these sequences - but not all - seem to have quite simple generating functions.

U(k+1,k) = k*(k+1)/2.

U(k+2+j,k) = k^2+j for k>=3 and 0<=j

U(2*k+2,k) = k*(3*k-1)/2 for k>=3.

Examples

			Square array U(n,k) begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  3,  3,  3,  3,  3,  3,  ...
  4,  6,  4,  4,  4,  4,  ...
  6,  9, 10,  5,  5,  5,  ...
  8, 10, 16, 15,  6,  6,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..87
• Index entries for Ulam numbers

Crossrefs

Columns k=2-10 give: A002858, A007086, A183527, A183528, A183529, A183530, A183531, A183532, A183533.

Cf. A135737.

Programs

Maple

b:= proc(n,i,k,h) option remember;
      local t;
      if n<0 or h<0 then 0
    elif n=0 then `if`(h=0, 1, 0)
    elif i=0 or h=0 then 0
    elif h=1 then t:= v(n, k);
                  `if`(t>0 and t<=i, 1, 0)
             else t:= b(n -U(i, k), i-1, k, h-1);
                  t+ `if`(t>1, 0, b(n, i-1, k, h))
      fi
    end:
v:= proc() 0 end:
U:= proc(n,k) option remember;
      local m;
      if n<=k then v(n,k):= n
              else for m from U(n-1, k)+1
                     while b(m, n-1, k, k)<>1 do od;
                   v(m,k):= n; m
      fi
    end:
seq(seq(U(n, 2+d-n), n=1..d), d=1..12);

Mathematica

b[n_, i_, k_, h_] := b[n, i, k, h] = Module[{t}, Which[n < 0 || h < 0, 0, n == 0, If[h == 0, 1, 0], i == 0 || h == 0, 0, h == 1, t = v[n, k]; If[t > 0 && t <= i, 1, 0], True, t = b[n-U[i, k], i-1, k, h-1]; t+If[t > 1, 0, b[n, i-1, k, h]] ] ]; v[, ] = 0; U[n_, k_] := U[n, k] = Module[{m}, If[n <= k, v[n, k] = n, For[m = U[n-1, k]+1, b[m, n-1, k, k] != 1, m++]; v[m, k] = n; m] ]; Table[Table[U[n, 2+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

PARI

Ulam(N,k=2,v=0)={ my( a=vector(k,i,i), c );
for( n=k,N-1, for( t=1+a[n],9e9, c=0;
  forvec(v=vector(k,i,[i,n]),sum(j=1,k,a[v[j]])==t & c++>1 & next(2),2);
  c||next; v&print1(t","); a=concat(a,t); break;
)); a}
/* M. F. Hasler */

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