A135737 -id:A135737 - cp's OEIS frontend

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

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

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

			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,  ...

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

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

Cf. A135737.

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);

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 *)

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 */

A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 64, 65, 66, 68, 69, 128, 132, 188, 190, 191, 194, 252, 253, 255, 313, 314, 318, 374, 376, 377, 436, 441, 496, 497, 499, 500, 502, 560, 561, 563, 621, 622, 626, 682, 684, 685, 687, 745, 746, 805, 811

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(5) = 10 = 1 + 2 + 3 + 4 = 4*5/2, because it is the least number >4 with a unique sum of 4 distinct earlier terms.
a(6) = 16 = 1 + 2 + 3 + 10 = 4^2, because it is the least number >10 with a unique sum of 4 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=4 of A183534.

Cf. A002858, A007086, A183528-A183533, A007300, A135737.

Maple
```
# see A183534 for programs.
```

Conjectured G.f.: (-61*x^124-56*x^118+53*x^117+3*x^116 -x^115+2*x^114-65*x^113-58*x^112+57*x^111 -56*x^110-x^109-57*x^108+54*x^106 +3*x^105+58*x^104-52*x^102+50*x^101-55*x^100 +56*x^95-53*x^94-3*x^93+x^92-2*x^91 +2*x^90+x^87-x^86 +41*x^84-51*x^83-66*x^82-58*x^81 -52*x^79+4*x^78-58*x^77 -x^74-x^73-x^72-54*x^71 -6*x^70-59*x^69-x^68-58*x^67-2*x^66 -x^65-2*x^64-56*x^63-4*x^62-x^61 -58*x^60-2*x^59-59*x^58-x^57-x^56 -2*x^55-x^54-x^53-54*x^52-6*x^51 -59*x^50-x^49-58*x^48-2*x^47-x^46 -2*x^45-56*x^44-4*x^43-x^42-58*x^41 -2*x^40-x^39-58*x^38-2*x^37-x^36 -2*x^35-x^34-55*x^33-5*x^32-59*x^31 -x^30-2*x^29-56*x^28-4*x^27-x^26 -58*x^25-2*x^24-x^23-58*x^22-3*x^21 -x^20-2*x^19-56*x^18-4*x^17-59*x^16 -x^15-2*x^14-x^13-x^12-42*x^11 -3*x^10-x^9-x^8-x^7-6*x^6 -6*x^5-x^4-x^3-x^2-x) / (-x^74+x^73+x-1). (This has been verified for n up to 1000.)

A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 55, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 145, 163, 190, 217, 235, 271, 280, 1740, 1741, 1744, 1745, 1799, 1804, 1805, 1824, 1825, 1831, 1859, 1869, 1913, 1914, 3554, 10521, 10522, 10526, 10527, 10537, 10563, 10564

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(11) = 55 = 1 + ... + 10 = 10*11/2, because it is the least number >10 with a unique sum of 10 distinct earlier terms.
a(12) = 100 = 1 + ... + 9 + 55 = 10^2, because it is the least number >55 with a unique sum of 10 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..120
Index entries for Ulam numbers

Column k=10 of A183534.

Cf. A002858, A007086, A183527-A183532, A135737.

Maple
```
# see A183534 for programs.
```

A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 45, 81, 82, 83, 84, 85, 86, 87, 88, 89, 117, 133, 153, 177, 189, 221, 225, 1325, 1326, 1328, 1329, 1373, 1378, 1379, 1391, 1392, 1398, 1423, 1427, 2717, 2718, 4031, 4032, 4035, 4037, 4039, 5316, 5319, 5346, 5352, 5353, 5354, 5361

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(10) = 45 = 1 + ... + 9 = 9*10/2, because it is the least number >9 with a unique sum of 9 distinct earlier terms.
a(11) = 81 = 1 + ... + 8 + 45 = 9^2, because it is the least number >45 with a unique sum of 9 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Index entries for Ulam numbers

Column k=9 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 15, 25, 26, 27, 28, 29, 35, 43, 45, 165, 171, 172, 174, 180, 181, 328, 333, 338, 339, 340, 341, 493, 499, 500, 647, 652, 657, 658, 659, 660, 661, 662, 663, 815, 818, 819, 971, 1127, 1137, 1138, 1139, 1140, 1141, 1142

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(6) = 15 = 1 + 2 + 3 + 4 + 5 = 5*6/2, because it is the least number >5 with a unique sum of 5 distinct earlier terms.
a(7) = 25 = 1 + 2 + 3 + 4 + 15 = 5^2, because it is the least number >15 with a unique sum of 5 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=5 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183529 An Ulam-type sequence: a(n) = n if n<=6; for n>6, a(n) = least number > a(n-1) which is a unique sum of 6 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 21, 36, 37, 38, 39, 40, 41, 51, 61, 66, 284, 285, 289, 290, 297, 298, 299, 310, 312, 559, 561, 562, 570, 571, 574, 575, 834, 836, 837, 838, 839, 840, 841, 849, 850, 1109, 1124, 1125, 1126, 1127, 1386, 1401, 1402, 1661, 1676, 1677, 1936, 1951

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(7) = 21 = 1 + ... + 6 = 6*7/2, because it is the least number >6 with a unique sum of 6 distinct earlier terms.
a(8) = 36 = 1 + ... + 5 + 21 = 6^2, because it is the least number >21 with a unique sum of 6 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..700
Index entries for Ulam numbers

Column k=6 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183530 An Ulam-type sequence: a(n) = n if n<=7; for n>7, a(n) = least number > a(n-1) which is a unique sum of 7 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 28, 49, 50, 51, 52, 53, 54, 55, 70, 82, 91, 109, 112, 555, 556, 563, 564, 572, 573, 576, 583, 584, 591, 593, 1103, 1104, 1105, 1111, 1112, 1124, 1632, 1637, 1642, 1643, 1648, 1653, 1654, 1655, 1656, 1657, 1660, 1661, 1662, 1671, 1672, 2184

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(8) = 28 = 1 + ... + 7 = 7*8/2, because it is the least number >7 with a unique sum of 7 distinct earlier terms.
a(9) = 49 = 1 + ... + 6 + 28 = 7^2, because it is the least number >28 with a unique sum of 7 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Index entries for Ulam numbers

Column k=7 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A183531 An Ulam-type sequence: a(n) = n if n<=8; for n>8, a(n) = least number > a(n-1) which is a unique sum of 8 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 36, 64, 65, 66, 67, 68, 69, 70, 71, 92, 106, 120, 141, 148, 792, 793, 802, 803, 816, 817, 820, 828, 830, 838, 844, 1591, 1592, 1606, 1607, 2356, 2357, 2358, 2359, 2360, 2384, 2395, 2402, 3149, 3150, 3169, 3919, 3920, 3921, 3922, 3923

Offset: 1

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

nonn

An Ulam-type sequence - see A002858 for further information.

			a(9) = 36 = 1 + ... + 8 = 8*9/2, because it is the least number >8 with a unique sum of 8 distinct earlier terms.
a(10) = 64 = 1 + ... + 7 + 36 = 8^2, because it is the least number >36 with a unique sum of 8 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Index entries for Ulam numbers

Column k=8 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

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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

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A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

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A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

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A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

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A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

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A183529 An Ulam-type sequence: a(n) = n if n<=6; for n>6, a(n) = least number > a(n-1) which is a unique sum of 6 distinct earlier terms.

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A183530 An Ulam-type sequence: a(n) = n if n<=7; for n>7, a(n) = least number > a(n-1) which is a unique sum of 7 distinct earlier terms.

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A183531 An Ulam-type sequence: a(n) = n if n<=8; for n>8, a(n) = least number > a(n-1) which is a unique sum of 8 distinct earlier terms.

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