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A001208 a(n) = solution to the postage stamp problem with 3 denominations and n stamps.

%I A001208 M2721 N1351 #50 Feb 16 2025 08:32:22
%S A001208 3,8,15,26,35,52,69,89,112,146,172,212,259,302,354,418,476,548,633,
%T A001208 714,805,902,1012,1127,1254,1382,1524,1678,1841,2010,2188,2382,2584,
%U A001208 2801,3020,3256,3508,3772,4043,4326,4628,4941,5272,5606,5960,6334,6723,7120
%N A001208 a(n) = solution to the postage stamp problem with 3 denominations and n stamps.
%C A001208 _Fred Lunnon_ [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.
%D A001208 R. K. Guy, Unsolved Problems in Number Theory, C12.
%D A001208 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001208 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001208 Vincenzo Librandi, <a href="/A001208/b001208.txt">Table of n, a(n) for n = 1..1000</a>
%H A001208 R. Alter and J. A. Barnett, <a href="http://www.jstor.org/stable/2321610">A postage stamp problem</a>, Amer. Math. Monthly, 87 (1980), 206-210.
%H A001208 M. F. Challis, <a href="http://dx.doi.org/10.1093/comjnl/36.2.117">Two new techniques for computing extremal h-bases A_k</a>, Comp. J. 36(2) (1993) 117-126.
%H A001208 Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/0403.html">Postage stamp problem</a>
%H A001208 F. H. Kierstead, Jr.,, <a href="/A001208/a001208.pdf">The Stamp Problem</a>, J. Rec. Math., Vol. ?, Year ?, page 298. [Annotated and scanned copy]
%H A001208 W. F. Lunnon, <a href="https://doi.org/10.1093/comjnl/12.4.377">A postage stamp problem</a>, Comput. J. 12 (1969) 377-380.
%H A001208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PostageStampProblem.html">Postage stamp problem</a>
%p A001208 c2 :=array(0..8,[3,3,5,5,7,6,8,8,10]) ; c3 :=array(0..8,1..2,[[1,1],[1,1],[2,1],[2,1],[3,1],[2,2],[3,2],[3,2],[4,2]]); c4 :=array(0..8,1..3,[[0,0,0],[0,0,1],[1,0,1],[1,0,2],[2,0,2],[2,1,2],[3,1,2],[3,1,3],[4,1,3]]) ; for n from 23 to 100 do r := n mod 9 ; t := iquo(n,9) ; a2 := 6*t+c2[r] ; a3 := (2*t+c3[r,1])+(2*t+c3[r,2])*a2 ; printf("%a,",4*t+c4[r,1]+(2*t+c4[r,2])*a2+(3*t+c4[r,3])*a3) ; end: # _R. J. Mathar_, Apr 01 2006
%t A001208 ClearAll[c2, c3, c4, a]; Evaluate[ Array[c2, 9, 0]] = {3, 3, 5, 5, 7, 6, 8, 8, 10}; Evaluate[ Array[c3, {9, 2}, {0, 1}]] = {{1, 1}, {1, 1}, {2, 1}, {2,1}, {3, 1}, {2, 2}, {3, 2}, {3, 2}, {4, 2}}; Evaluate[ Array[c4, {9, 3}, {0, 1}]] = {{0, 0, 0}, {0, 0, 1}, {1, 0,1}, {1, 0, 2}, {2, 0, 2}, {2, 1, 2}, {3, 1, 2}, {3, 1, 3}, {4, 1,3}}; Evaluate[ Array[a, 19]] = {3, 8, 15, 26, 35, 52, 69, 89, 112, 146, 172, 212, 259, 302, 354, 418, 476, 548, 633}; a[n_] := (r = Mod[n, 9]; t = Quotient[n, 9]; a2 = 6t + c2[r]; a3 = (2t + c3[r, 1]) + (2t + c3[r, 2])*a2; 4t + c4[r, 1] + (2t + c4[r, 2])*a2 + (3t + c4[r, 3])*a3); Table[a[n], {n, 1, 48}] (* _Jean-François Alcover_, Dec 19 2011, after _R. J. Mathar_'s Maple program *)
%Y A001208 Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.
%Y A001208 Equals A195618 - 1.
%Y A001208 A row or column of the array A196416 (possibly with 1 subtracted from it).
%K A001208 nonn,nice
%O A001208 1,1
%A A001208 _N. J. A. Sloane_
%E A001208 Maple recursion program valid for n>=23 from Challis added by _R. J. Mathar_, Apr 01 2006
%E A001208 At least 64 terms are known, see Friedman link.
%E A001208 Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
%E A001208 More terms from Jean Gaumont (jeangaum87(AT)yahoo.com), Apr 16 2006