Displaying 1-10 of 18 results found.
a(n) = solution to the postage stamp problem with n denominations and 2 stamps.
(Formerly M1089 N0972)
+10
28
2, 4, 8, 12, 16, 20, 26, 32, 40, 46, 54, 64, 72, 80, 92, 104, 116, 128, 140, 152, 164, 180, 196, 212
COMMENTS
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.
a(20)=152: There is only one set of 20 denominations covering all sums through 152: {1, 3, 4, 5, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 71, 72, 73, 75, 76}. - Tim Peters (tim.one(AT)comcast.net), Oct 04 2006
REFERENCES
Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 115 (Coins of the Realm), 1984.
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
LINKS
F. H. Kierstead, Jr.,, The Stamp Problem, J. Rec. Math., Vol. ?, Year ?, page 298. [Annotated and scanned copy]
CROSSREFS
A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Corrected a(17). Added a(18) and a(19) from Challis. - R. J. Mathar, Apr 01 2006 Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(20) from Tim Peters (tim.one(AT)comcast.net), Oct 04 2006
Added terms a(21) and a(22) from Challis and Robinson. John P Robinson (john-robinson(AT)uiowa.edu), Feb 19 2010
Added term a(23) from Challis and Robinson's July 2013 addendum, by Jukka Kohonen, Oct 25 2013 Added a(24) from Kohonen and Corander (2013). - N. J. A. Sloane, Jan 08 2014
a(n) is the solution to the postage stamp problem with 4 denominations and n stamps.
(Formerly M3432 N1568)
+10
24
4, 12, 24, 44, 71, 114, 165, 234, 326, 427, 547, 708, 873, 1094, 1383, 1650, 1935, 2304, 2782, 3324, 3812, 4368, 5130, 5892, 6745, 7880, 8913, 9919, 11081, 12376, 13932, 15657, 17242, 18892, 21061, 23445, 25553, 27978, 31347, 33981, 36806, 39914, 43592
COMMENTS
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.
Challis lists up to a(54) and provides recursions up to a(157). - R. J. Mathar, Apr 01 2006 Additional terms a(29) through a(254) can be computed using 3 sets of equations and a table of 10 coefficients available on line at Challis and Robinson. - John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(15) to a(28) from Table 1 of Mossige reference added by R. J. Mathar, Mar 29 2006 a(29)-a(54) from Challis and Robinson added by Robert Price, Jul 19 2013
a(n) = solution to the postage stamp problem with 3 denominations and n stamps.
(Formerly M2721 N1351)
+10
21
3, 8, 15, 26, 35, 52, 69, 89, 112, 146, 172, 212, 259, 302, 354, 418, 476, 548, 633, 714, 805, 902, 1012, 1127, 1254, 1382, 1524, 1678, 1841, 2010, 2188, 2382, 2584, 2801, 3020, 3256, 3508, 3772, 4043, 4326, 4628, 4941, 5272, 5606, 5960, 6334, 6723, 7120
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
LINKS
F. H. Kierstead, Jr.,, The Stamp Problem, J. Rec. Math., Vol. ?, Year ?, page 298. [Annotated and scanned copy]
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Maple recursion program valid for n>=23 from Challis added by R. J. Mathar, Apr 01 2006 At least 64 terms are known, see Friedman link.
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
More terms from Jean Gaumont (jeangaum87(AT)yahoo.com), Apr 16 2006
a(n) is the solution to the postage stamp problem with n denominations and 3 stamps.
(Formerly M2647 N1340)
+10
21
3, 7, 15, 24, 36, 52, 70, 93, 121, 154, 186, 225, 271, 323, 385, 450, 515, 606, 684, 788, 865, 977, 1091, 1201, 1361
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
Further terms from Friedman web site, Jun 20 2003
a(17)-a(25) from Friedman added by Robert Price, Jul 19 2013
a(n) is the solution to the postage stamp problem with 5 denominations and n stamps.
(Formerly M3864 N1707)
+10
20
5, 16, 36, 70, 126, 216, 345, 512, 797, 1055, 1475, 2047, 2659, 3403, 4422, 5629, 6865, 8669, 10835, 12903, 15785, 18801, 22456, 26469, 31108, 36949, 42744, 49436, 57033, 66771, 75558, 86303, 96852, 110253, 123954, 140688, 158389, 178811, 197293, 223580
COMMENTS
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.
Additional terms a(30) through a(67) are available on line at Challis and Robinson. - John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
LINKS
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3. [From John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010]
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Terms up to a(29) from Challis added by R. J. Mathar, Apr 01 2006 Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(30)-a(67) from Challis and Robinson added by Robert Price, Jul 19 2013
a(n) is the solution to the postage stamp problem with 6 denominations and n stamps.
(Formerly M4136 N1836)
+10
20
6, 20, 52, 108, 211, 388, 664, 1045, 1617, 2510, 3607, 5118, 7066, 9748, 12793, 17061, 22342, 28874, 36560, 45745, 57814, 72997, 87555, 106888, 129783
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(16)-a(25) from Challis and Robinson added by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
a(n) is the solution to the postage stamp problem with n denominations and 4 stamps.
(Formerly M3391 N1559)
+10
20
4, 10, 26, 44, 70, 108, 162, 228, 310, 422, 550, 700, 878, 1079, 1344, 1606, 1944, 2337, 2766, 3195, 3668, 4251, 4923, 5631, 6429
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
LINKS
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3. [From John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010]
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(11) from Challis & Robinson added by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
a(12)-a(25) from Friedman added by Robert Price, Jul 19 2013
a(n) is the solution to the postage stamp problem with n denominations and 5 stamps.
(Formerly M3845 N1706)
+10
20
5, 14, 35, 71, 126, 211, 336, 524, 726, 1016, 1393, 1871, 2494, 3196, 4063, 5113, 6511, 7949, 9865, 11589
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(10) from Challis and Robinson added by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
a(11)-a(20) from Friedman added by Robert Price, Jul 19 2013
a(n) = solution to the postage stamp problem with n denominations and 6 stamps.
(Formerly M4120 N1831)
+10
20
6, 18, 52, 114, 216, 388, 638, 1007, 1545, 2287
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
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).
LINKS
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3. [From John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010]
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193. A row or column of the array A196416 (possibly with 1 subtracted from it).
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
Added terms a(8) and a(9) from Challis and Robinson. John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010
a(n) = solution to the postage stamp problem with n denominations and 7 stamps.
(Formerly M4380)
+10
19
7, 23, 69, 165, 345, 664, 1137, 1911
COMMENTS
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.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, C12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
CROSSREFS
Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.
EXTENSIONS
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
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