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A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps. 34
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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(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

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

(Haskell)

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

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

CROSSREFS

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

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

Sequence in context: A328659 A088236 A194244 * A184674 A227353 A183136

Adjacent sequences: A014613 A014614 A014615 * A014617 A014618 A014619

KEYWORD

nonn,nice,easy

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

More terms from John W. Layman, Apr 13 1999

STATUS

approved

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Last modified December 5 20:36 EST 2022. Contains 358589 sequences. (Running on oeis4.)