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
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.
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
Keywords
Comments
An Ulam-type sequence - see A002858 for further information.
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
- Index entries for Ulam numbers
Programs
-
Maple
# see A183534 for programs.
Formula
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.)
Comments
Examples
Links
Crossrefs
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 */