A196416 -id:A196416 - cp's OEIS frontend

A024206 Expansion of x^2(1+x-x^2)/((1-x^2)(1-x)^2).

0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 63, 71, 80, 89, 99, 109, 120, 131, 143, 155, 168, 181, 195, 209, 224, 239, 255, 271, 288, 305, 323, 341, 360, 379, 399, 419, 440, 461, 483, 505, 528, 551, 575, 599, 624, 649, 675, 701, 728, 755, 783, 811, 840

Offset: 1

Clark Kimberling

a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.
[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.
First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .
Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002
a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.
Starting (1, 3, 5, 8, 11, ...), = row sums of triangle A135841. - Gary W. Adamson, Dec 01 2007
Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010
Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010
Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015
Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016
Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020
a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2]  [0 2 2]  [1 0 2]  [0 0 2]  [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021

			There are five 2 X 3 binary matrices with no zero rows or columns up to row and column permutation:
   [1 0 0]  [1 0 0]  [1 1 0]  [1 1 0]  [1 1 1]
   [0 1 1]  [1 1 1]  [0 1 1]  [1 1 1]  [1 1 1].

O. Giering, Vorlesungen über höhere Geometrie, Vieweg, Braunschweig, 1982. See p. 59.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Tricia Muldoon Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See p. 15.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
William F. Lunnon, A postage stamp problem, Comput. J. 12 (1969), 377-380.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, Vol. 8 (2008), pp. 45-52.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A014616, A135841, A034856, A005744 (partial sums), A008619 (1st differences).
A row or column of the array A196416 (possibly with 1 subtracted from it).
Cf. A008619.
Second column of A232206.

a:=[0,1,3,5];; for n in [5..65] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; a; # Muniru A Asiru, Oct 23 2018

a024206 n = (n - 1) * (n + 3) `div` 4
a024206_list = scanl (+) 0 $ tail a008619_list
-- Reinhard Zumkeller, Dec 18 2013

[(2*n^2+4*n-7-(-1)^n)/8 : n in [1..100]]; // Wesley Ivan Hurt, Jul 22 2014

A024206:=n->(2*n^2+4*n-7-(-1)^n)/8: seq(A024206(n), n=1..100);

f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ Floor[ f[2, n^2 + 2 n - 2] /2], {n, 57}] (* Robert G. Wilson v, Aug 11 2010 *)
LinearRecurrence[{2,0,-2,1},{0,1,3,5},60] (* Harvey P. Dale, Jun 14 2013 *)
Rest[CoefficientList[Series[x^2 (1 + x - x^2)/((1 - x^2) (1 - x)^2), {x, 0, 70}], x]] (* Vincenzo Librandi, Oct 02 2015 *)

a(n)=(n-1)*(n+3)\4 \\ Charles R Greathouse IV, Jun 26 2013

x='x+O('x^99); concat(0, Vec(x^2*(1+x-x^2)/ ((1-x^2)*(1-x)^2))) \\ Altug Alkan, Apr 05 2016

def A024206(n): return (n+1)**2//4 - 1 # Ya-Ping Lu, Jan 01 2024

G.f.: x^2*(1+x-x^2)/((1-x^2)*(1-x)^2) = x^2*(1+x-x^2) / ( (1+x)*(1-x)^3 ).
a(n+1) = A002623(n) - A002623(n-1) - 1.
a(n) = A002620(n+1) - 1 = A014616(n-2) + 1.
a(n+1) = A002620(n) + n, n >= 0. - Philippe Deléham, Feb 27 2004
a(0)=0, a(n) = floor(a(n-1) + sqrt(a(n-1)) + 1) for n > 0. - Gerald McGarvey, Jul 30 2004
a(n) = floor((n+1)^2/4) - 1. - Franz Vrabec, Feb 22 2008
a(n) = A005744(n-1) - A005744(n-2). - R. J. Mathar, Nov 04 2008
a(n) = a(n-1) + [side length of the least square > a(n-1) ], that is a(n) = a(n-1) + ceiling(sqrt(a(n-1) + 1)). - Ctibor O. Zizka, Oct 06 2009
For a(1)=0, a(2)=1, a(n) = 2*a(n-1) - a(n-2) + 1 if n is odd; a(n) = 2*a(n-1) - a(n-2) if n is even. - Vincenzo Librandi, Dec 23 2010
a(n) = A181971(n, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4); a(1)=0, a(2)=1, a(3)=3, a(4)=5. - Harvey P. Dale, Jun 14 2013
a(n) = floor( (n-1)*(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = (2*n^2 + 4*n - 7 - (-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = a(-n-2) = n-1 + floor( (n-1)^2/4 ). - Bruno Berselli, Feb 03 2015
a(n) = (1/4)*(n+3)^2 - (1/8)*(1 + (-1)^n) - 1. - Serkan Sonel, Aug 26 2020
a(n) + a(n+1) = A034856(n). - R. J. Mathar, Mar 13 2021
a(2*n) = n^2 + n - 1, a(2*n+1) = n^2 + 2*n. - Greg Dresden and Zijie He, Jun 28 2022
Sum_{n>=2} 1/a(n) = 7/4 + tan(sqrt(5)*Pi/2)*Pi/sqrt(5). - Amiram Eldar, Dec 10 2022
E.g.f.: (4 + (x^2 + 3*x - 4)*cosh(x) + (x^2 + 3*x - 3)*sinh(x))/4. - Stefano Spezia, Aug 06 2024

Corrected and extended by Vladeta Jovovic, Jun 02 2000

A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps.

Original entry on oeis.org

2, 4, 7, 10, 14, 18, 23, 28, 34, 40, 47, 54, 62, 70, 79, 88, 98, 108, 119, 130, 142, 154, 167, 180, 194, 208, 223, 238, 254, 270, 287, 304, 322, 340, 359, 378, 398, 418, 439, 460, 482, 504, 527, 550, 574, 598, 623, 648, 674, 700, 727, 754, 782, 810, 839, 868

Offset: 1

Eric W. Weisstein

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(n-2), for n >= 3, is the number of independent entries of a bisymmetric n X n matrix B_n with B_n[1,1] and B_n[n,n] fixed. Hence a(n-2) = A002620(n+1) - 2. See the Jul 07 2015 comment on A002620. For n=1 and n=2 this matrix B_n is fixed. Bisymmetric matrices B_n, with B_n[1,1] and B_n[n,n] fixed, are, for n >= 3, determined by giving the a(n-2) entries for [1,2], ...., [1,n-1]; [2,2], ..., [2,n-1]; [3,3], ..., [3,n-2]; ..., [ceiling(n/2),n-(ceiling(n/2)-1)]. - Wolfdieter Lang, Aug 16 2015
a(n-1) is the largest possible n-th element in an additive basis of order 2. - Charles R Greathouse IV, May 05 2020

			Bisymmetric matrix B_5, with B_5[1,1] and B_5[5,5] fixed, have a(3) free entries: for rows 1 and 2: each 3, row 3:  1, altogether 3 + 3 + 1 = 7 = a(5-2). Mark the corresponding matrix entries with x, and obtain a pattern symmetric around the central vertical. - _Wolfdieter Lang_, Aug 16 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section C12, pp. 185-190.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Mario Bravo, Thierry Champion, and Roberto Cominetti, Universal bounds for fixed point iterations via optimal transport metrics, arXiv:2108.00300 [math.OC], 2021.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Alfred Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I., Journal für die reine und angewandte Mathematik (1955), Volume: 194, page 40-65. See p. 47.
Hugh Thomas and Stephanie van Willigenburg, Compact symmetric solutions to the postage stamp problem, arXiv:0706.3250 [math.NT], 2007-2008.
Amitabha Tripathi, A Note on the Postage Stamp Problem, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.3.
Eric Weisstein's World of Mathematics, Postage stamp problem.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index to sequences related to the Postage Stamp Problem.

A row or column of the array A196416 (possibly with 1 subtracted from it).

Cf. A002620, A004116, A024206, A028884, A046691.

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

[(2*n*(n+6)-(-1)^n+1)/8: n in [1..60]]; // Vincenzo Librandi, Jul 09 2011

a[n_?OddQ] := (n^2 + 6*n + 1)/4; a[n_?EvenQ] := n*(n + 6)/4; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Dec 14 2011, after first formula *)
LinearRecurrence[{2,0,-2,1},{2,4,7,10},60] (* Harvey P. Dale, Oct 04 2015 *)

a(n)=(n^2 + 6*n + 1)\4 \\ Charles R Greathouse IV, Feb 06 2017

a(n) = floor((n^2 + 6*n + 1)/4).
a(n) = A002620(n+3) - 2 = A024206(n+2) - 1 = (2*n*(n+6)-(-1)^n+1)/8.
G.f.: x*(-2 + x^2)/((1 + x)*(x - 1)^3). - R. J. Mathar, Jul 09 2011
a(n) = floor(A028884(n+1)/4). - Reinhard Zumkeller, Apr 07 2013
a(n)+a(n+1) = A046691(n+1). - R. J. Mathar, Mar 13 2021
a(n) = 2*n + A002620(n-1). - Michael Chu, Apr 28 2022
a(n) = A004116(n) + 1. - Michael Chu, May 02 2022
E.g.f.: (x*(7 + x)*cosh(x) + (1 + 7*x + x^2)*sinh(x))/4. - Stefano Spezia, Nov 09 2022
Sum_{n>=1} 1/a(n) = 67/36 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - Amiram Eldar, Dec 10 2022

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

More terms from John W. Layman, Apr 13 1999

A001212 a(n) = solution to the postage stamp problem with n denominations and 2 stamps.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 26, 32, 40, 46, 54, 64, 72, 80, 92, 104, 116, 128, 140, 152, 164, 180, 196, 212

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.

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

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).

R. Alter, Letter to N. J. A. Sloane, Mar 25 1977
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_k, Comp J 36(2) (1993) 117-126
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
F. H. Kierstead, Jr.,, The Stamp Problem, J. Rec. Math., Vol. ?, Year ?, page 298. [Annotated and scanned copy]
J. Kohonen, A meet-in-the-middle algorithm for finding extremal restricted additive 2-bases, arXiv preprint arXiv:1403.5945, 2014 and J. Int. Seq. 17 (2014) # 14.6.8.
J. Kohonen, J. Corander, Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications, J. Integer Seq., 17 (2014), Article 14.3.4.
J. Kohonen, Early Pruning in the Restricted Postage Stamp Problem, arXiv preprint arXiv:1503.03416, 2015
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
W. F. Lunnon, A postage stamp problem [Annotated scanned copy]
J. P. Robinson, Some postage stamp 2-bases, JIS 12 (2009) 09.1.1.
E. S. Selmer, Letter to N. J. A. Sloane, Sep 10 1991
Eric Weisstein's World of Mathematics, Postage stamp problem
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Postage stamp sequences: A001208 A001209 A001210 A001211 A001212 A001213 A001214 A001215 A001216 A005342 A005343 A005344 A014616 A053346 A053348 A075060 A084192 A084193
Cf. A006638, A123509.
Equals A196094(n) - 1 and A234941(n+1)-2.
A row or column of the array A196416 (possibly with 1 subtracted from it).

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

A001209 a(n) is the solution to the postage stamp problem with 4 denominations and n stamps.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

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).

Robert Price, Table of n, a(n) for n = 1..54
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_k, Comp. J. 36(2) (1993) 117-126
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
S. Mossige, Algorithms for Computing the h-Range of the Postage Stamp Problem, Math. Comp. 36 (1981) 575-582.
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.
Equals A196069 - 1.
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
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

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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_k, Comp. J. 36(2) (1993) 117-126.
Erich Friedman, Postage stamp problem
F. H. Kierstead, Jr.,, The Stamp Problem, J. Rec. Math., Vol. ?, Year ?, page 298. [Annotated and scanned copy]
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.
Equals A195618 - 1.
A row or column of the array A196416 (possibly with 1 subtracted from it).

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

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 *)

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

A001213 a(n) is the solution to the postage stamp problem with n denominations and 3 stamps.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
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 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
More terms from Al Zimmermann, Feb 20 2002
Further terms from Friedman web site, Jun 20 2003
Incorrect value of a(17) removed by Al Zimmermann, Nov 08 2009
a(17)-a(25) from Friedman added by Robert Price, Jul 19 2013

A001210 a(n) is the solution to the postage stamp problem with 5 denominations and n stamps.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

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).

Robert Price, Table of n, a(n) for n = 1..67
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_k, Comp. J. 36(2) (1993) 117-126.
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.

A row or column of the array A196416 (possibly with 1 subtracted from it).

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

A001211 a(n) is the solution to the postage stamp problem with 6 denominations and n stamps.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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, Two new techniques for computing extremal h-bases A_k, Comp. J. 36(2) (1993) 117-126.
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 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 row or column of the array A196416 (possibly with 1 subtracted from it).

a(11)-a(15) 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(16)-a(25) from Challis and Robinson added by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010

A001214 a(n) is the solution to the postage stamp problem with n denominations and 4 stamps.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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, Two new techniques for computing extremal h-bases A_k, Comp. J. 36(2) (1993) 117-126
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, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
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).

a(10) 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(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

A001215 a(n) is the solution to the postage stamp problem with n denominations and 5 stamps.

Original entry on oeis.org

5, 14, 35, 71, 126, 211, 336, 524, 726, 1016, 1393, 1871, 2494, 3196, 4063, 5113, 6511, 7949, 9865, 11589

Offset: 1

N. J. A. Sloane

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, Two new techniques for computing extremal h-bases A_k, Comp. J. 36(2) (1993) 117-126
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3.
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).

a(9) 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(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

cp's OEIS Frontend

A024206 Expansion of x^2*(1+x-x^2)/((1-x^2)*(1-x)^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

Extensions

A001212 a(n) = solution to the postage stamp problem with n denominations and 2 stamps.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A001209 a(n) is the solution to the postage stamp problem with 4 denominations and n stamps.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A001213 a(n) is the solution to the postage stamp problem with n denominations and 3 stamps.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A001210 a(n) is the solution to the postage stamp problem with 5 denominations and n stamps.

Views

Author

Keywords

A024206 Expansion of x^2(1+x-x^2)/((1-x^2)(1-x)^2).