A007300 -id:A007300 - cp's OEIS frontend

A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

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

Offset: 1

N. J. A. Sloane

Ulam conjectured that this sequence has density 0. However, calculations up to 6.759*10^8 (Jud McCranie) indicate that the density hovers near 0.074.
A plot of the first 3 million terms shows that they lie very close to the straight line 13.51*n, so even if we cannot prove it, we believe we now know how this sequence grows (see the plots in the links below). - N. J. A. Sloane, Sep 27 2006
After a few initial terms, the sequence settles into a regular pattern of dense clumps separated by sparse gaps, with period 21.601584+. This pattern continues up to at least a(n) = 5*10^6. (This comment is just a qualitative statement about the wavelike distribution of Ulam numbers, not meant to imply that every period includes Ulam numbers.) - David W. Wilson
_Don Knuth_ (Sep 26 2006) remarks that a(4952)=64420 and a(4953)=64682 (a gap of more than ten "dense clumps"); and there is a gap of 315 between a(18857) and a(18858).
1,2,3,47 are the only values of x < 6.759*10^8 such that x and x+1 are both Ulam numbers. - Jud McCranie, Jun 08 2001.  This holds through the first 28 billion Ulam numbers - Jud McCranie, Jan 07 2016.
From Jud McCranie on David W. Wilson's illustration, Jun 20 2008: (Start)
The integers are shown from left to right, top to bottom, with a dot where there is an Ulam number. I think his plot is 216 wide. The local density of Ulam numbers goes in waves with a period of 21.6+, so his plot shows ten cycles.
When they are arranged that way you can see the waves. The crests of the density waves don't always have Ulam numbers there but the troughs are practically void of Ulam numbers. I noticed that the ratio of that period (21.6+) to the frequency of Ulam numbers (1 in 13.52) is very close to 8/5. (End)
a(50000000) = 675904508. - Jud McCranie, Feb 29 2012
a(100000000) = 1351856726. - Jud McCranie, Jul 31 2012
a(1000000000) = 13517664323. - Jud McCranie, Aug 28 2015
a(28000000000) = 378485625853 - Philip Gibbs & Jud McCranie, Sep 09 2015
3 (=1+2) and 131 (=62+69) are the only two Ulam numbers in the first 28 billion Ulam numbers that are the sum of two consecutive Ulam numbers. - Jud McCranie, Jan 09 2016
Named after the Polish-American scientist Stanislaw Ulam (1909-1984). - Amiram Eldar, Jun 08 2021

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.2.
Richard K. Guy, Unsolved Problems in Number Theory, C4.
Donald E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.3.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 116.
Marvin C. Wunderlich, The improbable behavior of Ulam's summation sequence, pp. 249-257 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
David Zeitlin, Ulam's sequence {U_n}, U_1=1, U_2=2, is a complete sequence, Notices Amer. Math. Soc., 22 (No. 7, 1975), Abstract 75T-A267, p. A-707.

Jud McCranie, Table of n, a(n) for n = 1..10000
Richard A. Becker, Plot of residuals a(n) - 13.5167*n for n <= 3000000, postscript file [uses Jud McCranie's values of a(n)].
Richard A. Becker, Plot of residuals a(n) - 13.5167*n for n <= 3000000, pdf file [uses Jud McCranie's values of a(n)].
Thomas Bloom, Problem 342, Erdős Problems.
Henning Fernau and Kshitij Gajjar, The Space Complexity of Sum Labelling, arXiv:2107.12973 [cs.DS], 2021.
Steven R. Finch, Ulam s-Additive Sequences. [From Steven Finch, Apr 20 2019]
Steven R. Finch, Stolarsky-Harborth Constant.
Steven R. Finch, Stolarsky-Harborth Constant. [From the Wayback machine]
Philip Gibbs, An Efficient Way to Compute Ulam Numbers.
Philip Gibbs, A Conjecture for Ulam Sequences.
Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
Alois P. Heinz, Ulam spiral of Ulam numbers.
Donald E. Knuth, Downloadable programs
Noah Kravitz and Stefan Steinerberger, Ulam Sequences and Ulam Sets, arXiv:1705.01883 [math.CO], 2017.
Borys Kuca, Structures in Additive Sequences, arXiv:1804.09594 [math.NT], 2018. See p. 3.
Jud McCranie, Density 10^2 to 10^12
Jud McCranie, Density 10^4 to 10^12
Jud McCranie, Density 10^6 to 10^12
J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973.
Ed Pegg, Jr., Graph of 10^6 terms of a(n) - 13.5*n.
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #8. [Annotated and scanned copy]
R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.
Bernardo Recamán, Questions on a sequence of Ulam, Amer. Math. Monthly, Vol. 80, No. 8 (1973), pp. 919-920.
Ivano Salvo and Agnese Pacifico, Computing Integer Sequences: Filtering vs Generation (Functional Pearl), arXiv:1807.11792 [cs.PL], 2018.
James Schmerl and Eugene Spiegel, The regularity of some 1-additive sequences, J. Combin. Theory. Ser. A, Vol. 66, No. 1 (1994), pp. 172-175. Math. Rev. 95h:11010.
Arseniy Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, arXiv:1910.00109 [math.NT], 2019.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
Arseniy (Senia) Sheydvasser, The Ulam Sequence of Linear Integer Polynomials, J. Int. Seq., Vol. 24 (2021), Article 21.10.8.
Stefan Steinerberger, A hidden signal in the Ulam sequence, Research Report YALEU/DCS/TR-1508, Yale University, 2015.
Stefan Steinerberger, A hidden signal in the Ulam sequence, Figure 2 from Research Report YALEU/DCS/TR-1508, Yale University, 2015.
Stefan Steinerberger, The Ulam Sequence, blog post, April 12, 2016.
Stanislaw M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]
Stanislaw Ulam, Combinatorial analysis in infinite sets and some physical theories, SIAM Rev., Vol. 6, No. 4 (1964), pp. 343-355. Reprinted in Selected Papers, MIT Press, see p. 393.
Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.
David W. Wilson, Plot of initial terms, showing their quasiperiodicity as vertical bars. The image width was chosen to include approximately 10 periods. For an explanation of this picture, see Comments above.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
David Zeitlin, Ulam's sequence {U_n}, U_1=1, U_2=2, is a complete sequence, Notices Amer. Math. Soc., 22 (No. 7, 1975), Abstract 75T-A267, p. A-707. (Annotated scanned copy)
Index entries for Ulam numbers

Cf. A002859 (version beginning 1,3), A054540, A003667, A001857, A007300, A117140, A214603.
First differences: A072832, A072540.
Cf. A080287, A080288, A004280 (if distinct removed from definition).
Cf. A199016, A199017, A033629, A274522.
See also the density plots in A080573 and A285884.

a002858 n = a002858_list !! (n-1)
a002858_list = 1 : 2 : ulam 2 2 a002858_list
ulam :: Int -> Integer -> [Integer] -> [Integer]
ulam n u us = u' : ulam (n + 1) u' us where
   u' = f 0 (u+1) us'
   f 2 z _                         = f 0 (z + 1) us'
   f e z (v:vs) | z - v <= v       = if e == 1 then z else f 0 (z + 1) us'
                | z - v `elem` us' = f (e + 1) z vs
                | otherwise        = f e z vs
   us' = take n us
-- Reinhard Zumkeller, Nov 03 2011

function isUlam(u, n, h, i, r)
    h == 2 && return false
    ur = u[r]; ui = u[i]
    ur <= ui && return h == 1
    if ur + ui > n
        r -= 1
    elseif ur + ui < n
        i += 1
    else
        h += 1; i += 1; r -= 1
    end
    isUlam(u, n, h, i, r)
end
function UlamList(len)
    u = Array{Int, 1}(undef, len)
    u[1] = 1; u[2] = 2; i = 2; n = 2
    while i < len
        n += 1
        if isUlam(u, n, 0, 1, i)
            i += 1
            u[i] = n
        end
    end
    return u
end
println(UlamList(59)) # Peter Luschny, Apr 07 2019

UlamList := proc(len) local isUlam, nextUlam, behead; behead := u -> u[2..numelems(u)]; isUlam := proc(n, h, u, r) local hu, tu, hr, tr; hu := u[1]; hr := r[1]; if h = 2 then return false fi; if hr <= hu then return evalb(h = 1) fi; if (hr + hu) = n then tu := behead(u); tr := behead(r); return isUlam(n, h+1, tu, tr) fi; if (hr + hu) < n then tu := behead(u): return isUlam(n, h, tu, r) fi; tr := behead(r); isUlam(n, h, u, tr) end: nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len-1 then return [op(u), n] fi; nextUlam(n+1, [op(u), n], [n, op(r)]) else nextUlam(n+1, u, r) fi end: nextUlam(3, [1, 2], [2, 1]) end:
UlamList(59); # Peter Luschny, Apr 05 2019

Ulam4Compiled = Compile[{{nmax, _Integer}, {init, _Integer, 1}, {s, _Integer}}, Module[{ulamhash = Table[0, {nmax}], ulam = init}, ulamhash[[ulam]] = 1; Do[ If[Quotient[Plus @@ ulamhash[[i - ulam]], 2] == s, AppendTo[ulam, i]; ulamhash[[i]] = 1], {i, Last[init] + 1, nmax}]; ulam]]; ulams = Ulam4Compiled[355, {1, 2}, 1]
(* Second program: *)
ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {100}]; ulams (* Jean-François Alcover, Sep 08 2011 *)
findUlams[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ Count[ss, k] != 1, k++]; Append[s, k]]; ulams = Nest[findUlams[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)

aupto(N)= my(seen=vector(N), U=[]); seen[1]=seen[2]=1; for(i=1,N, if(1==seen[i], for(j=1,#U, my(sum=i+U[j]); if(sum>N, break); seen[sum]++); U=concat(U,i))); U \\ Ruud H.G. van Tol, Dec 29 2022

def isUlam(n, h, u, r):
    if h == 2: return False
    hu = u[0]; hr = r[0]
    if hr <= hu: return h == 1
    if hr + hu > n: r = r[1:]
    elif hr + hu < n: u = u[1:]
    else: h += 1; r = r[1:]; u = u[1:]
    return isUlam(n, h, u, r)
def UlamList(length):
    u = [1, 2]; r = [2, 1]; n = 2
    while len(u) < length:
        n += 1
        if isUlam(n, 0, u[:], r[:]):
            u.append(n); r.insert(0, n)
    return u
print(UlamList(59)) # Peter Luschny, Apr 06 2019

More terms from Jud McCranie

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

Original entry on oeis.org

1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680

Offset: 0

N. J. A. Sloane

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003
Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005
This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008
For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010
a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011
Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Bill Sands, Problem 3257, Crux Math. 33 (2007), No.5, p. 298.
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001019, A001045, A014551, A078008, A098158.

Cf. A034494, A081340-A081342, A034659.

List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019

[2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017

CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x,0,30}], x] (* or *)
LinearRecurrence[{2,8}, {1,1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

PARI
```
a(n)=2^(n-1)*( 2^n + (-1)^n );
```

[2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

From Paul Barry, Mar 01 2003: (Start)
a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.
a(n) = (4^n + (-2)^n)/2.
G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)
From Paul Barry, Apr 05 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.
E.g.f. exp(x)*cosh(3*x). (End)
a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004
Given a(0)=1, b(0)=1 then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005
a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
If p[1]=1, and p[i]=9, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

Entry revised by N. J. A. Sloane, Nov 22 2006

A135737 Ulam type (1-additive) sequences u[1]=2, u[2]=2n+1, u[k+1] is least unique sum u[i]+u[j]>u[k], 1<=i

Original entry on oeis.org

2, 3, 2, 5, 5, 2, 7, 7, 7, 2, 8, 9, 9, 9, 2, 9, 11, 11, 11, 11, 2, 13, 12, 13, 13, 13, 13, 2, 14, 13, 15, 15, 15, 15, 15, 2, 18, 15, 16, 17, 17, 17, 17, 17, 2, 19, 19, 17, 19, 19, 19, 19, 19, 19, 2, 24, 23, 19, 20, 21, 21, 21, 21, 21, 21, 2, 25, 27, 21, 21, 23, 23, 23, 23, 23, 23, 23

Offset: 1

M. F. Hasler, Nov 26 2007

Any of the sequences u=U(2,2n+1) has u[1]=2 and u[n+4]=4n+4; in between these there are the odd numbers 2n+1,...,4n-3. For n>1 there are no other even terms and the sequence of first differences becomes periodic for k>=t (transient phase), such that u[k] = u[k-floor((k-t)/p)*p] + floor((k-t)/p)*d, where p is the period (cf. A100729) and d the fundamental difference (cf. A100730). See the cross-references, especially A002858, for more information.

			The sequence contains the terms of the table T[n,k] = U(2,2n+1)[k], read by antidiagonals: a[1]=T[1,1]=2, a[2]=T[1,2]=3, a[3]=T[2,1]=2, a[4]=T[1,3]=5,...
n=1: U(2,3)= 2, 3, 5, 7, 8, 9,13,14...
n=2: U(2,5)= 2, 5, 7, 9,11,12,...
n=3: U(2,7)= 2, 7, 9,11,13,...
n=4: U(2,9)= 2, 9,11,...

M. F. Hasler, Table of n, a(n) for n = 1..1000, Apr 10 2025
Project Euler, Problem 167: Investigating Ulam sequences

Cf. A001857 = U(2, 3) = row 1, A007300 = U(2, 5) = row 2, A003668 = U(2, 7) = row 3; A100729-A100730 (period).
Cf. A002858 = U(1, 2): this would be row 0, with u[1], u[2] exchanged.
See also: A002859 = U(1, 3), A003666 = U(1, 4), A003667 = U(1, 5).

ulam(a,b,Nmax=30,i)=a=[a,b]; b=[a[1]+b]; for( k=3,Nmax, i=1; while(( i<#b && b[i]==b[i+1] && i+=2 ) || ( i>1 && b[i]==b[i-1] && i++),); a=concat(a,b[i]); b=vecsort(concat(vecextract(b,Str("^..",i)),vector(k-1,j,a[k]+a[j]))); i=0; for(j=1,#b-2, if( b[j]==b[j+2], i+=1<A135737(Nmax=100)=local(T=vector(sqrtint(Nmax*2)+1,n, ulam(2,2*n+1, sqrtint(Nmax*2)+2-n)),i,j); vector(Nmax,k,if(j>1,T[i++ ][j-- ],j=i+1;T[i=1][j]))

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.)

A003666 a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n and a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 10, 16, 18, 19, 21, 31, 32, 33, 42, 46, 56, 57, 66, 70, 79, 82, 91, 96, 104, 105, 107, 116, 129, 130, 131, 141, 158, 165, 168, 179, 180, 182, 191, 204, 205, 206, 216, 217, 218, 219, 229, 230, 244, 256, 266, 267, 268, 281, 290, 315, 316, 317, 318, 328

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

An Ulam-type sequence - see A002858 for many further references, comments, etc. - T. D. Noe, Jan 21 2008

R. K. Guy, "s-Additive sequences", preprint, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A003662, A199120, A199121.

a003666 n = a003666_list !! (n-1)
a003666_list = 1 : 4 : ulam 2 4 a003666_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Total, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {1, 4}, 58] (* Michael De Vlieger, Nov 16 2017 *)

A003662 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 8, 11, 13, 16, 18, 23, 25, 28, 30, 35, 37, 40, 42, 47, 49, 52, 54, 59, 61, 64, 66, 71, 73, 76, 78, 83, 85, 88, 90, 95, 97, 100, 102, 107, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 143, 145, 148, 150, 155, 157, 160, 162, 167, 169, 172, 174, 179, 181, 184

Offset: 1

N. J. A. Sloane, Mira Bernstein

R. K. Guy, "s-Additive sequences", preprint, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A060469.

Cf. A003666.

Sort[Join[{8},Select[Range[200],MemberQ[{1,4,6,11},Mod[#,12]]&]]] (* Harvey P. Dale, Apr 26 2011 *)

Vec(x*(2*x^8+x^6-x^5+2*x^4+2*x^3+2*x^2+3*x+1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 27 2015

Numbers congruent to {1, 4, 6, 11} mod 12 plus the number 8.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7. - Colin Barker, Feb 27 2015
G.f.: x*(2*x^8 + x^6 - x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 3*x + 1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 27 2015

Name clarified by David A. Corneth, Mar 13 2023

A100729 Period of the first difference of Ulam 1-additive sequence U(2,2n+1).

Original entry on oeis.org

32, 26, 444, 1628, 5906, 80, 126960, 380882, 2097152, 1047588, 148814, 8951040, 5406720, 242, 127842440, 11419626400, 12885001946, 160159528116, 687195466408, 6390911336402, 11728121233408, 20104735604736

Offset: 2

Ralf Stephan, Dec 03 2004

nonn

It was proved by Akeran that a(2^k-1) = 3^(k+1) - 1.

Note that a(n)=2^(2n+1) as soon as A100730(n)=2^(2n+3)-2, that happens for n=(m-2)/2 with m>=6 being an even element of A073639.

			For k=2, we have a(3)=3^3-1=26.

Max Alekseyev, Table of n, a(n) for n = 2..31
M. Akeran, On some 1-additive sequences
J. Cassaigne and S. R. Finch, A class of 1-additive sequences and additive recurrences
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.

Cf. A100730 for the fundamental difference, A001857 for U(2, 3), A007300 for U(2, 5), A003668 for U(2, 7).

Cf. also A006844.

a(3) corrected from 25 to 26 by Hugo van der Sanden and Bertram Felgenhauer (int-e(AT)gmx.de), Nov 11 2007
More terms from Balakrishnan V (balaji.iitm1(AT)gmail.com), Nov 15 2007
a(21..31) and b-file from Max Alekseyev, Dec 01 2007

A003663 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 8, 10, 12, 15, 17, 19, 24, 26, 28, 33, 35, 37, 42, 44, 46, 51, 53, 55, 60, 62, 64, 69, 71, 73, 78, 80, 82, 87, 89, 91, 96, 98, 100, 105, 107, 109, 114, 116, 118, 123, 125, 127, 132, 134, 136, 141, 143, 145, 150, 152, 154, 159, 161, 163, 168, 170, 172, 177, 179

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Numbers congruent to {1, 6, 8} mod 9 plus the number 12.

R. K. Guy, "s-Additive sequences", preprint, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
S. R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A199162, A060469, A060470, A060471, A060472.

Cf. A003662, A005408, A026471, A026474, A033627, A051039, A051040, A244151.

I:=[1,6,8,10,12,15,17,19,24]; [n le 9 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Feb 22 2018

f[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {j}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#, 2] &, {1, 6}, 65] (* Robert G. Wilson v, Jul 05 2014 *)
LinearRecurrence[{1,0,1,-1},{1,6,8,10,12,15,17,19,24},70] (* Harvey P. Dale, Jul 25 2018 *)

From Chai Wah Wu, Feb 21 2018: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 9.
G.f.: x*(2*x^8 + x^5 - 3*x^4 + x^3 + 2*x^2 + 5*x + 1)/(x^4 - x^3 - x + 1). (End)

Name clarified by David A. Corneth, Mar 13 2023

A003667 a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n and a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 10, 12, 20, 22, 23, 24, 26, 38, 39, 40, 41, 52, 57, 69, 70, 71, 82, 87, 98, 102, 113, 119, 129, 130, 133, 144, 160, 161, 162, 163, 175, 196, 205, 208, 209, 222, 223, 224, 226, 237, 253, 254, 255, 256, 268, 269, 270, 271, 272, 284, 285, 286, 303, 318

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

An Ulam-type sequence - see A002858 for many further references, comments, etc.

R. K. Guy, "s-Additive sequences", preprint, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Ulam Sequence

a003667 n = a003667_list !! (n-1)
a003667_list = 1 : 5 : ulam 2 5 a003667_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Total, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {1, 5}, 58] (* Michael De Vlieger, Nov 16 2017 *)

Name clarfied by David A. Corneth, Mar 13 2023

A100730 Fundamental difference of Ulam 1-additive sequence starting U(2,2n+1).

Original entry on oeis.org

126, 126, 1778, 6510, 23622, 510, 507842, 1523526, 8388606, 4194302, 597870, 35791394, 21691754, 2046, 511305630, 45678505642, 51539607546, 640638112422, 2748779069430, 25563645345606, 46912496118442, 80418967640942

Offset: 2

Ralf Stephan, Dec 03 2004

nonn

Max Alekseyev, Table of n, a(n) for n = 2..610
M. Akeran, On some 1-additive sequences
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.

Cf. A100729 for the period, A001857 for U(2, 3), A007300 for U(2, 5), A003668 for U(2, 7).

a(n) = 2 * A046932(2*n+2)

2 more terms from Balakrishnan V (balaji.iitm1(AT)gmail.com), Nov 15 2007
Further new terms and b-file from Max Alekseyev, Dec 01 2007
b-file extended by Max Alekseyev, Aug 17 2015

cp's OEIS Frontend

A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

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A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

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A135737 Ulam type (1-additive) sequences u[1]=2, u[2]=2n+1, u[k+1] is least unique sum u[i]+u[j]>u[k], 1<=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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A003666 a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n and a(1) = 1, a(2) = 4.

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A003662 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 4.

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A100729 Period of the first difference of Ulam 1-additive sequence U(2,2n+1).

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