A005318 Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).

0, 1, 2, 4, 7, 13, 24, 44, 84, 161, 309, 594, 1164, 2284, 4484, 8807, 17305, 34301, 68008, 134852, 267420, 530356, 1051905, 2095003, 4172701, 8311101, 16554194, 32973536, 65679652, 130828948, 261127540, 521203175, 1040311347, 2076449993, 4144588885, 8272623576

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Conway and Guy conjecture that the set of k numbers {s_i = a(k) - a(k-i) : 1 <= i <= k} has the property that all its subsets have distinct sums - see Guy's book. These k-sets are the rows of A096858. [This conjecture has apparently now been proved by Bohman. - I. Halupczok (integerSequences(AT)karimmi.de), Feb 20 2006]

References

• J. H. Conway and R. K. Guy, Solution of a problem of Erdos, Colloq. Math. 20 (1969), p. 307.
• R. K. Guy, Unsolved Problems in Number Theory, C8.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• M. Wald, Problem 1192, Unequal sums, J. Rec. Math., 15 (No. 2, 1983-1984), pp. 148-149.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3324 (first 301 terms from T. D. Noe)
• Tom Bohman, A sum packing problem of Erdos and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.
• P. Borwein and M. J. Mossinghoff, Newman Polynomials with Prescribed Vanishing and Integer Sets with Distinct Subset Sums, Math. Comp., 72 (2003), 787-800.
• J. H. Conway & R. K. Guy, Sets of natural numbers with distinct sums, Manuscript.
• R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
• R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
• R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982. (Annotated scanned copy)
• G. Kreweras, Sur quelques problèmes relatifs au vote pondéré [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
• G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights] C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 8, 345-347.
• G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights], C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 8, 345-347. (Annotated scanned copy)
• W. F. Lunnon, Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.
• M. Wald & N. J. A. Sloane, Correspondence and Attachment, 1987

Crossrefs

A276661 is the main entry for the distinct subset sums problem.

Cf. A037254, A096858, A096796, A096824, A205744, A206239, A083920, A003056.

Programs

Haskell

a005318 n = a005318_list !! n
a005318_list = 0 : 1 : zipWith (-)
   (map (* 2) $ tail a005318_list) (map a005318 a083920_list)
-- Reinhard Zumkeller, Feb 12 2012

Mathematica

a[n_] := a[n] = 2*a[n-1] - a[n - Floor[Sqrt[2]*Sqrt[n-1] + 1/2] - 1]; a[0]=0; a[1]=1; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 15 2013 *)

PARI

a(n)=if(n<=1,n==1,2*a(n-1)-a(n-1-(sqrtint(8*n-15)+1)\2))

PARI

A=[]; /* This is the program above with memoization. */
a(n)=if(n<3, return(n)); if(n>#A, A=concat(A,vector(n-#A)), if(A[n], return(A[n]))); A[n]=2*a(n-1)-a(n-1-(sqrtint(8*n-15)+1)\2) \\ Charles R Greathouse IV, Sep 09 2016

Python

from sympy import sqrt, floor
def a(n): return n if n<2 else 2*a(n - 1) - a(n - floor(sqrt(2)*sqrt(n - 1) + 1/2) - 1) # Indranil Ghosh, Jun 03 2017

Formula

a(n+1) = 2*a(n)-A205744(n), A205744(n) = a(A083920(n)), A083920(n) = n - A002024(n). - N. J. A. Sloane, Feb 11 2012

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 21 2000