#JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Side note: finding the h-Stöhr sequence (verifying the sum of up to h numbers instead of just 2). The R function tries all possibile sums of up a given number of list elements.

S=(n,h=2,s=[],v=1,R=(t,v,l,i=0,r=t,w)=>{
  for(;r&&l&&v[i];i++)
    w=[...v],r=!R(t-w.splice(i,1),w,l-1)
  return!r;
})=>R(v,s,h)||(s.push(v),--n)?S(n,h,s,v+1):s

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console. Simple function:

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

Advanced function:

S(10,5)

[1, 2, 4, 8, 16, 32, 63, 94, 125, 156]

JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Side note: finding the h-Stöhr sequence (verifying the sum of up to h numbers instead of just 2). The R function tries all possibile sums of up a given number of list elements.

S=(n,h=2,s=[],v=1,R=(t,v,l,i=0,r=t,w)=>{
  for(;r&&l&&v[i];i++)
    w=[...v],r=!R(t-w.splice(i,1),w,l-1)
  return!r;
})=>R(v,s,h)||(s.push(v),--n)?S(n,h,s,v+1):s

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console. Simple function:

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

Advanced function:

S(10,5)

[1, 2, 4, 8, 16, 32, 63, 94, 125, 156]

#JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Side note: finding the h-Stöhr sequence (verifying the sum of up to h numbers instead of just 2). The R function tries all possibile sums of up a certain numer of list elements.

S=(n,h=2,s=[],v=1,R=(t,v,l,i=0,r=t,w)=>{
  for(;r&&l&&v[i];i++)
    w=[...v],r=!R(t-w.splice(i,1)[0],w,l-1)
  return!r;
})=>R(v,s,h)||(s.push(v),--n)?S(n,h,s,v+1):s

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console. Simple function:

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

Advanced function:

S(10,5)

[1, 2, 4, 8, 16, 32, 63, 94, 125, 156]

#JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Side note: finding the h-Stöhr sequence (verifying the sum of up to h numbers instead of just 2). The R function tries all possibile sums of up a given number of list elements.

S=(n,h=2,s=[],v=1,R=(t,v,l,i=0,r=t,w)=>{
  for(;r&&l&&v[i];i++)
    w=[...v],r=!R(t-w.splice(i,1),w,l-1)
  return!r;
})=>R(v,s,h)||(s.push(v),--n)?S(n,h,s,v+1):s

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console. Simple function:

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

Advanced function:

S(10,5)

[1, 2, 4, 8, 16, 32, 63, 94, 125, 156]

#JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

#JavaScript (ES6) 92

As a recursive function based upon the problem definition

S=(n,v=1,s=[],r=0)=>[for(a of s)for(b of s)r+=(a-b&&a+b==v)]|r||(s.push(v),--n)?S(n,v+1,s):s

Using the pattern 1,2, 1+3*k : 58

S=(n)=>(i=>{for(t=1;n>r.push(t+=i);i+=(i<3));})(0,r=[])||r

Side note: finding the h-Stöhr sequence (verifying the sum of up to h numbers instead of just 2). The R function tries all possibile sums of up a certain numer of list elements.

S=(n,h=2,s=[],v=1,R=(t,v,l,i=0,r=t,w)=>{
  for(;r&&l&&v[i];i++)
    w=[...v],r=!R(t-w.splice(i,1)[0],w,l-1)
  return!r;
})=>R(v,s,h)||(s.push(v),--n)?S(n,h,s,v+1):s

Ungolfed roughly equivalent (and ES5 compatible)

function S(n, v, s)
{
  var r=0,a,b
  v = v||1
  s = s||[]
  for(a of s)
    for(b of s)
    {
      if (a != b && a+b == v) 
        r++;
    }
  if (r == 0) 
  {
    s.push(v);
    --n;
  }
  if (n != 0)
     return S(n,v+1,s)
  else
     return s
}

Test In FireFox/FireBug console. Simple function:

S(20)

[1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55]

Advanced function:

S(10,5)

[1, 2, 4, 8, 16, 32, 63, 94, 125, 156]