A014616 -id:A014616 - cp's OEIS frontend

A001216 a(n) = solution to the postage stamp problem with n denominations and 6 stamps.

6, 18, 52, 114, 216, 388, 638, 1007, 1545, 2287

Offset: 1

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.

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).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
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]
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

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).

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(10) from Friedman by Robert Price, Jul 19 2013

A028884 a(n) = (n + 3)^2 - 8.

Original entry on oeis.org

1, 8, 17, 28, 41, 56, 73, 92, 113, 136, 161, 188, 217, 248, 281, 316, 353, 392, 433, 476, 521, 568, 617, 668, 721, 776, 833, 892, 953, 1016, 1081, 1148, 1217, 1288, 1361, 1436, 1513, 1592, 1673, 1756, 1841, 1928, 2017, 2108, 2201, 2296, 2393

Offset: 0

Patrick De Geest

From Klaus Purath, Jan 04 2023: (Start)
The product of two consecutive terms belongs to the sequence: a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-n-1) + 1.
a(n) is never divisible by primes given in A003629.
Each odd prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -6 (mod p).
The prime factors are listed in A038873 and the primes in A028886.
For n > 0, this is a proper subsequence of A079896.
Conjecture: a(n) = A079896(A265284(n-1)). -
(End)

			From _Stefano Spezia_, Nov 08 2022: (Start)
Illustrations for n = 0..4:
          *       * * *     * * * * *
      a(0) = 1    *   *     *       *
                  * * *     *   *   *
                a(1) = 8    *       *
                            * * * * *
                            a(2) = 17
.
   * * * * * * *    * * * * * * * * *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   *   *   *   *    *   *   *   *   *
   *           *    *               *
   * * * * * * *    *   *   *   *   *
     a(3) = 28      *               *
                    * * * * * * * * *
                        a(4) = 41
(End)

Altug Alkan, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005563, A014616, A028560, A082111, A190576.

a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013

Range[3, 50]^2 - 8 (* Alonso del Arte, Aug 15 2016 *)

a(n)=(n+3)^2-8 \\ Charles R Greathouse IV, Oct 07 2015

(3 to 49).map(n => n * n - 8) // Alonso del Arte, May 07 2020

a(n) = a(n-1) + 2*n + 5 (with a(0) = 1). - Vincenzo Librandi, Aug 05 2010
a(n) = A028560(n) + 1; A014616(n) = floor(a(n+1)/4). - Reinhard Zumkeller, Apr 07 2013
G.f.: (-1 - 5*x + 4*x^2)/(x - 1)^3. - R. J. Mathar, Mar 24 2013
Sum_{n >= 0} 1/a(n) = 51/112 - Pi*cot(2*Pi*sqrt(2))/(4*sqrt(2)) = 1.3839174974448... . - Vaclav Kotesovec, Apr 10 2016
E.g.f.: (1 + 7*x + x^2)*exp(x). - G. C. Greubel, Aug 19 2017
Sum_{n >= 0} (-1)^n/a(n) = (-19 + 14*sqrt(2)*Pi*cosec(2*sqrt(2)*Pi))/112. - Amiram Eldar, Nov 04 2020
From Klaus Purath, Jan 04 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2, n >= 2.
a(n) = A082111(n) + n.
a(n) = A190576(n+1) - n. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 7*Pi/(45*sqrt(2)*sin(2*sqrt(2)*Pi)).
Product_{n>=0} (1 + 1/a(n)) = (4*sqrt(14)/9)*sin(sqrt(7)*Pi)/sin(2*sqrt(2)*Pi). (End)

Definition corrected by Omar E. Pol, Jul 27 2009

A005342 a(n) = solution to the postage stamp problem with n denominations and 7 stamps.

Original entry on oeis.org

7, 23, 69, 165, 345, 664, 1137, 1911

Offset: 1

N. J. A. Sloane

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.

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).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(8) from Challis and Robinson by Robert Price, Jul 19 2013

A005343 a(n) = solution to the postage stamp problem with n denominations and 8 stamps.

Original entry on oeis.org

8, 28, 89, 234, 512, 1045, 2001, 3485

Offset: 1

N. J. A. Sloane

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.

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).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
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]
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(8) from Challis and Robinson. John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010

A005344 a(n) = solution to the postage stamp problem with n denominations and 9 stamps.

Original entry on oeis.org

9, 34, 112, 326, 797, 1617, 3191

Offset: 1

N. J. A. Sloane

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.

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).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(7) from Challis and Robinson by Robert Price, Jul 19 2013

A053346 a(n) = solution to the postage stamp problem with 7 denominations and n stamps.

Original entry on oeis.org

7, 26, 70, 162, 336, 638, 1137, 2001, 3191, 5047, 7820, 11568, 17178

Offset: 1

N. J. A. Sloane, Jun 20 2003

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.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis, Two new techniques for computing extremal h-bases A_kComp. J. 36(2) (1993) 117-126
M. F. Challis, J. P. Robinson, Some extremal postage stamp bases, JIS 13 (2010) #10.2.3.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

a(9) from Challis by R. J. Mathar, Apr 01 2006
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(10)-a(13) from Challis and Robinson by Robert Price, Jul 19 2013

A053348 a(n) = solution to the postage stamp problem with 8 denominations and n stamps.

Original entry on oeis.org

8, 32, 93, 228, 524, 1007, 1911, 3485

Offset: 1

N. J. A. Sloane, Jun 20 2003

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.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis, Two new techniques for computing extremal h-bases A_kComp. J. 36(2) (1993) 117-126
M. F. Challis, J. P. Robinson, Some extremal postage stamp bases, JIS 13 (2010) #10.2.3.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

a(6) from Challis by R. J. Mathar, Apr 01 2006
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(7)-a(8) from Challis and Robinson by Robert Price, Jul 19 2013

A075060 a(n) = solution to the postage stamp problem with n denominations and 10 stamps.

Original entry on oeis.org

10, 40, 146, 427, 1055, 2510, 5047

Offset: 1

N. J. A. Sloane, Jun 20 2003

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.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
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]
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(7) from Challis and Robinson by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010

A084192 Array read by antidiagonals: T(n,k) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 8, 7, 4, 5, 12, 15, 10, 5, 6, 16, 24, 26, 14, 6, 7, 20, 36, 44, 35, 18, 7, 8, 26, 52, 70, 71, 52, 23, 8, 9, 32, 70, 108, 126, 114, 69, 28, 9, 10, 40, 93, 162, 211, 216, 165, 89, 34, 10, 11, 46, 121, 228, 336, 388, 345, 234, 112, 40, 11, 12, 54, 154, 310, 524

Offset: 0

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given in this sequence and in A001208, A001209, A001210, A001211, A001212, ... are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

			Array begins:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11, ...
   2,   4,   7,  10,  14,  18,  23,  28,  34,  40, ...
   3,   8,  15,  26,  35,  52,  69,  89, 112, ...
   4,  12,  24,  44,  71, 114, 165, 234, ...
   5,  16,  36,  70, 126, 216, 345, ...
   6,  20,  52, 108, 211, 388, ...
   7,  26,  70, 162, 336, ...
   8,  32,  93, 228, ...
   9,  40, 121, ...
  10,  46, ...
  11, ...
  ...

Postage stamp sequences: A001208 A001209 A001210 A001211 A001212 A001213 A001214 A001215 A001216 A005342 A005343 A005344 A014616 A053346 A053348 A075060 A084192 A084193

A084193 gives transposed array. Rows and columns give rise to A014616, A001208, A001209, A001210, A001211, A053346, A053348, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A075060.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003
Comments corrected by Shawn Pedersen, Apr 17 2012

A084193 Array read by antidiagonals: T(k,n) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 7, 8, 4, 5, 10, 15, 12, 5, 6, 14, 26, 24, 16, 6, 7, 18, 35, 44, 36, 20, 7, 8, 23, 52, 71, 70, 52, 26, 8, 9, 28, 69, 114, 126, 108, 70, 32, 9, 10, 34, 89, 165, 216, 211, 162, 93, 40, 10, 11, 40, 112, 234, 345, 388, 336, 228, 121, 46, 11, 12, 47, 146, 326, 512

Offset: 0

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given in this sequence and in A001208, A001209, A001210, A001211, A001212, ... are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

			Array begins:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11, ...
   2,   4,   8,  12,  16,  20,  26,  32,  40,  46, ...
   3,   7,  15,  24,  36,  52,  70,  93, 121, ...
   4,  10,  26,  44,  70, 108, 162, 228, ...
   5,  14,  35,  71, 126, 211, 336, ...
   6,  18,  52, 114, 216, 388, ...
   7,  23,  69, 165, 345, ...
   8,  28,  89, 234, ...
   9,  34, 112, ...
  10,  40, ...
  11, ...
  ...

Index to sequences related to the Postage Stamp Problem

A084192 gives transposed array. Rows and columns give rise to A014616, A001208, A001209, A001210, A001211, A053346, A053348, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A075060.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003
Comments corrected by Shawn Pedersen, Apr 17 2012

cp's OEIS Frontend

A001216 a(n) = solution to the postage stamp problem with n denominations and 6 stamps.

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A028884 a(n) = (n + 3)^2 - 8.

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A005342 a(n) = solution to the postage stamp problem with n denominations and 7 stamps.

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A005343 a(n) = solution to the postage stamp problem with n denominations and 8 stamps.

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A005344 a(n) = solution to the postage stamp problem with n denominations and 9 stamps.

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A053346 a(n) = solution to the postage stamp problem with 7 denominations and n stamps.

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A053348 a(n) = solution to the postage stamp problem with 8 denominations and n stamps.

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A075060 a(n) = solution to the postage stamp problem with n denominations and 10 stamps.

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A084192 Array read by antidiagonals: T(n,k) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).