A005347 -id:A005347 - cp's OEIS frontend

A001011 Number of ways to fold a strip of n blank stamps.

1, 1, 2, 5, 14, 38, 120, 353, 1148, 3527, 11622, 36627, 121622, 389560, 1301140, 4215748, 14146335, 46235800, 155741571, 512559195, 1732007938, 5732533570, 19423092113, 64590165281, 219349187968, 732358098471, 2492051377341, 8349072895553, 28459491475593

Offset: 1

N. J. A. Sloane, Stéphane Legendre

M. Gardner, Mathematical Games, Sci. Amer. Vol. 209 (No. 3, Mar. 1963), p. 262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence - see entry 576, Fig. 17, and the front cover).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..45 [from S. Legendre, 2013]
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) 12 pages.
S. P. Castell, Computer Puzzles, Computer Bulletin, March 1975, pages 3, 33, 34. [Annotated scanned copy]
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings.
Santo Diano, Letter to N. J. A. Sloane, circa Dec 01 1979.
R. Dickau, Stamp Folding.
R. Dickau, Stamp Folding. [Cached copy, PDF format, with permission]
R. Dickau, Unlabeled Stamp Foldings.
R. Dickau, Unlabeled Stamp Foldings. [Cached copy, PDF format, with permission]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152. [Annotated, corrected, scanned copy]
S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014.
David Orden, In how many ways can you fold a strip of stamps?, 2014.
Frank Ruskey, Information on Stamp Foldings.
J. Sawada and R. Li, Stamp foldings, semi-meanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).
N. J. A. Sloane, Illustration of initial terms. (Fig. 17 of the 1973 Handbook of Integer Sequences. The initial terms are also embossed on the front cover.)
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 2.
Eric Weisstein's World of Mathematics, Stamp Folding.
Index entries for sequences obtained by enumerating foldings

Cf. A000682, A086441.

A000136 = Import["https://oeis.org/A000136/b000136.txt", "Table"][[All, 2]];
A001010 = Cases[Import["https://oeis.org/A001010/b001010.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 1, 1, (A001010[[n]] + A000136[[n]])/4];
Array[a, 45] (* Jean-François Alcover, Sep 04 2019 *)

a(n) = (A001010(n) + A000136(n)) / 4 for n >= 2. - Andrew Howroyd, Dec 07 2015

a(17) and a(20) corrected by Sean A. Irvine, Mar 17 2013

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

N. J. A. Sloane, R. K. Guy

nonn

Laatsch (1986) proved that for n >= 2, a(n) gives the smallest number of distinct prime factors in even numbers having an abundancy index > n.
The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006
The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - T. D. Noe, May 08 2006
Equal to A256968 except for n = 2 and n = 3.  See comment in A256968. - Chai Wah Wu, Apr 17 2015

			The products Product_{i=1..k} prime(i)/(prime(i)-1) for k >= 0 start with 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..44
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

A001611 is similar but strictly different.

Cf. A000720, A001611, A005580, A060753, A023199, A038110, A072986, A091440, A256968.

(* For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic. *)
prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n,0,25}] (* T. D. Noe, May 08 2006 *)
prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1),30]]; If[Min[prod]>n, k, "too few digits"], {n,0,38}]

a(n)=my(s=1,k); forprime(p=2,, s*=p/(p-1); k++; if(s>n, return(k))) \\ Charles R Greathouse IV, Aug 20 2015

from sympy import nextprime
def a_list(upto: int) -> list[int]:
    L: list[int] = [0]
    count = 1; bn = 1; bd = 1; p = 2
    for k in range(1, upto + 1):
        bn *= p
        bd *= p - 1
        while bn > count * bd:
            L.append(k)
            count += 1
        p = nextprime(p)
    return L
print(a_list(1000))  # Chai Wah Wu, Apr 17 2015, adapted by Peter Luschny, Jan 25 2025

a(n) = smallest k such that A002110(k)/A005867(k) > n. - Artur Jasinski, Nov 06 2008

a(n) = PrimePi(A091440(n)) = A000720(A091440(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

Edited by T. D. Noe, May 08 2006
a(26) added by T. D. Noe, Sep 18 2008
Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
Comment corrected by T. D. Noe, Apr 04 2010
a(37)-a(39) from T. D. Noe, Nov 16 2010
Edited and terms a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025

A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 5, 13, 38, 116, 382, 1310, 4748, 17848, 70076, 284252, 1195240, 5174768, 23103368, 105899656, 498656912, 2404850720, 11879332048, 59976346448, 309442319456, 1628921941312, 8746095288800, 47840221880288, 266492604100288, 1510338372987776

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of set partitions of [n] in which the block containing 1 is of length <= 3 and all other blocks are of length <= 2. Example: a(4)=13 counts all 15 partitions of [4] except 1234 and 1/234. - David Callan, Jul 22 2008

Empirical: a(n) is the sum of the entries in the second-last row of the lower-triangular matrix of coefficients giving the expansion of degree-(n+1) complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

			G.f. = x + 2*x + 5*x^2 + 13*x^3 + 38*x^4 + 116*x^5 + 382*x^6 + 1310*x^7 + ... - _Michael Somos_, Jan 23 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86 (divided by 2).
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).

John Cerkan, Table of n, a(n) for n = 1..795
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.

Cf. A000085, A001189, A013989, A076276, A248475.

a:=[1, 2];; for n in [3..10^2] do a[n] := a[n-1] + n*a[n-2]; od; a;  # Muniru A Asiru, Jan 25 2018

I:=[1,2]; [n le 2 select I[n] else Self(n-1)+n*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 31 2018

a := proc(n) option remember: if n = 1 then 1 elif n = 2 then 2 elif  n >= 3 then procname(n-1) +n*procname(n-2) fi; end:
seq(a(n), n = 1..100); # Muniru A Asiru, Jan 25 2018

RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1]+n a[n-2]},a,{n,30}] (* Harvey P. Dale, Apr 21 2012 *)
(* Programs from Michael Somos, Jan 23 2018 *)
a[n_]:= With[{m=n+1}, If[m<2, 0, Sum[(2 k-1)!! Binomial[m, 2 k], {k, 0, m/2}]/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricU[-m/2, 1/2, -1/2] / (-1/2)^(m/2)/2]];
a[n_]:= With[{m=n+1}, If[m<2, 0, HypergeometricPFQ[{-m/2, (1-m)/2}, {}, 2]/2]];
a[n_]:= If[ n<1, 0, n! SeriesCoefficient[Exp[x+x^2/2]*(1+x)/2, {x, 0, n}]]; (* End *)
Fold[Append[#1, #1[[-1]] + #2 #1[[-2]]] &, {1, 2}, Range[3, 26]] (* Michael De Vlieger, Jan 23 2018 *)

{a(n) = if( n<1, 0, n! * polcoeff( exp( x + x^2/2 + x * O(x^n)) * (1 + x) / 2, n))}; /* Michael Somos, Jan 23 2018 */

my(N=30,x='x+O('x^N)); Vec(serlaplace((1/2)*( (1+x)*exp(x + x^2/2) - 1))) \\ Joerg Arndt, Sep 04 2023

def A001475_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( ((1+x)*exp(x+x^2/2) -1)/2 ).egf_to_ogf().list()
a=A001475_list(40); a[1:] # G. C. Greubel, Sep 03 2023

a(n) = (1/2)*A000085(n+1).
E.g.f.: (1/2)*( (1+x)*exp(x + x^2/2) - 1). - Vladeta Jovovic, Nov 04 2003
Given e.g.f. y(x), then 0 = y'(x) * (1+x) - (y(x)+1/2) * (2+2*x+x^2) = 1 - y''(x) + y'(x)*(1 + x) + 2*y(x). - Michael Somos, Jan 23 2018
0 = +a(n)*(+a(n+1) +a(n+2) -a(n+3)) +a(n+1)*(-a(n+1) +a(n+2)) for all n>0. - Michael Somos, Jan 23 2018
a(n) ~ n^((n+1)/2) / (2^(3/2) * exp(n/2 - sqrt(n) + 1/4)) * (1 + 19/(24*sqrt(n))). - Vaclav Kotesovec, Apr 01 2018

More terms from Harvey P. Dale, Apr 21 2012

A005350 a(1) = a(2) = a(3) = 1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n >= 4.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 25, 26, 27, 27, 28

Offset: 1

N. J. A. Sloane, R. K. Guy

a(n) - a(n-1) = 0 or 1 (see the 1991 Monthly reference). - Emeric Deutsch, Jun 06 2005

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..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.

Cf. A004001, A005185, A005707.

a005350 n = a005350_list !! (n-1)
a005350_list = 1 : 1 : 1 : h 4 1 where
   h x y = z : h (x + 1) z where z = a005350 y + a005350 (x - y)
-- Reinhard Zumkeller, Jul 20 2012

A005350 := proc(n) option remember; if n<=3 then 1 else procname(procname(n-1)) + procname(n-procname(n-1)); end if; end proc:
seq(A005350(n),n=1..64) ;

a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[a[n-1]] + a[n-a[n-1]]; Table[a[n], {n, 1, 64}] (* Jean-François Alcover, Feb 11 2014 *)

@CachedFunction
def a(n): return 1 if (n<4) else a(a(n-1)) + a(n-a(n-1))
[a(n) for n in range(1,100)]  # G. C. Greubel, Nov 14 2022

A005181 a(n) = ceiling(exp((n-1)/2)).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 91, 149, 245, 404, 666, 1097, 1809, 2981, 4915, 8104, 13360, 22027, 36316, 59875, 98716, 162755, 268338, 442414, 729417, 1202605, 1982760, 3269018, 5389699, 8886111, 14650720, 24154953, 39824785, 65659970, 108254988, 178482301

Offset: 0

N. J. A. Sloane, R. K. Guy

nonn

This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - Alonso del Arte, Mar 18 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Cf. A000045, A019774.

seq(round(ceil(exp((n-1)/2))), n=0..50); # Vladimir Pletser, Sep 15 2013

Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* Alonso del Arte, Mar 18 2013 *)

import math
for n in range(99):
    print(str(int(math.ceil(math.e**((n-1)*0.5)))), end=', ')
# Alex Ratushnyak, Mar 18 2013

Limit_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - Alois P. Heinz, Feb 19 2019

A few more terms from Alonso del Arte, Mar 18 2013

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

N. J. A. Sloane

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

R. K. Guy, Unsolved Problems in Number Theory, E25.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
R. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Cf. A054540, A004978, A247556.

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000

A005348 Number of ways to add n ordinals.

Original entry on oeis.org

1, 2, 5, 13, 33, 81, 193, 449, 1089, 2673, 6561, 15633, 37249, 88209, 216513, 531441, 1266273, 3017169, 7189057, 17537553, 43046721, 102568113, 244390689, 582313617, 1420541793, 3486784401, 8308017153, 19795645809, 47167402977, 115063885233, 282429536481

Offset: 1

N. J. A. Sloane, R. K. Guy, Bill Sands and Tommy Kucera

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 270-271.
W. Sierpiński, Cardinal and Ordinal Numbers, 2nd ed. p 275.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Erdős, Some remarks on set theory, Proc. Am. Math. Soc. 1 (1950) 127-141.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Bill Sands and Tommy Kucera, Letter to N. J. A. Sloane, Jun 10 1975.
A. Wakulicz, Sur la somme d'un nombre fini de nombres ordinaux, Fund. Math. 36 (1949), 254-266.
Eric Weisstein's World of Mathematics, Ordinal Number
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,81).

I:=[1,2,5,13,33,81,193,449,1089,2673,6561,15633,37249, 88209,216513,531441,1266273,3017169,7189057,17537553, 43046721]; [n le 21 select I[n] else 81*Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 17 2015

Join[{1, 2, 5, 13, 33, 81, 193, 449, 1089, 2673, 6561, 15633, 37249, 88209}, LinearRecurrence[ {0, 0, 0, 0, 81}, {216513, 531441, 1266273, 3017169, 7189057}, 20]] (* Harvey P. Dale, Dec 15 2014 *)

a(n) = 81*a(n-5) for n >= 21.

A005675 Deficit in peeling rinds.

Original entry on oeis.org

1, 2, 8, 18, 55, 138, 470, 1164, 4055, 10140, 35609, 89782, 316513, 803040

Offset: 1

N. J. A. Sloane, R. K. Guy

R. K. Guy, Monthly unsolved problems 1969-1985, Amer. Math. Monthly, 92:10 (1985), 717-725. See p. 720.
Allen J. Schwenk, How many rinds can a finite sequence of pairs have?, pp. 713-739 of Y. Alavi et al., eds., Graph Theory, Combinatorics and Applications. Wiley, NY, 2 vols., 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letters to N. J. A. Sloane, 1986-88

cp's OEIS Frontend

A001011 Number of ways to fold a strip of n blank stamps.

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A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

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A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2.

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A005350 a(1) = a(2) = a(3) = 1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n >= 4.

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Haskell

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A005181 a(n) = ceiling(exp((n-1)/2)).

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Maple

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A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

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