Basic Examples

Let
u=1
and
v=2
. Call the sequence
s
n
. Then
s
1
=1
and
s
2
=2
:

○
UlamSequence
[2,{1,2}]
In[]:=
{1,2}
Out[]=

For term 3, there is only one sum,
3=1+2
, so
s
3
=3
:

○
UlamSequence
[3,{1,2}]
In[]:=
{1,2,3}
Out[]=

For term 4, only the sum
1+3
gives 4, so
s
4
=4
. The sum
2+2
is not a sum of unique previous terms:

○
UlamSequence
[4,{1,2}]
In[]:=
{1,2,3,4}
Out[]=

For term 5, the two sums
1+4
and
2+3
give 5, so
s
5
≠5
. The sum
2+4
is 6; the sum
1+2+3=6
has more than one summand. So
s
5
=6
:

○
UlamSequence
[6,{1,2}]
In[]:=
{1,2,3,4,6}
Out[]=

Here are the terms up to 1000:

○
UlamSequence
[1000,{1,2}]
In[]:=
{1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,106,114,126,131,138,145,148,155,175,177,180,182,189,197,206,209,219,221,236,238,241,243,253,258,260,273,282,309,316,319,324,339,341,356,358,363,370,382,390,400,402,409,412,414,429,431,434,441,451,456,483,485,497,502,522,524,544,546,566,568,585,602,605,607,612,624,627,646,668,673,685,688,690,695,720,722,732,734,739,751,781,783,798,800,820,847,849,861,864,866,891,893,905,927,949,983,986,991}
Out[]=

The plot is wrinkly:

ListLinePlot[%]
In[]:=
Out[]=

Scope

This sequence starts with 1, 3 instead of 1, 2:

○
UlamSequence
[1000,{1,3}]
In[]:=
{1,3,4,5,6,8,10,12,17,21,23,28,32,34,39,43,48,52,54,59,63,68,72,74,79,83,98,99,101,110,114,121,125,132,136,139,143,145,152,161,165,172,176,187,192,196,201,205,212,216,223,227,232,234,236,243,247,252,256,258,274,278,283,285,287,294,298,314,318,325,329,340,343,347,349,356,374,376,378,385,389,396,400,405,409,416,420,425,427,429,438,447,451,456,460,462,467,471,482,493,507,511,518,522,547,549,551,558,560,562,569,573,578,580,582,591,609,613,620,624,660,664,666,686,691,695,711,715,722,726,731,735,737,742,746,762,764,766,784,788,813,817,819,835,839,864,868,886,915,917,919,937,941,946,950,966,970}
Out[]=

More wrinkles:

ListLinePlot@%
In[]:=
Out[]=