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A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps. 32
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) = floor(A028884(n+1)/4). - Reinhard Zumkeller, Apr 07 2013

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

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Erich Friedman, Postage stamp problem

W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

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)

FORMULA

a(n) = floor((n^2 + 6*n + 1)/4).

a(n) = A002620(n+3)-2 = (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

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-Fran├žois 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

Equals A024206 - 1.

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).

Cf. A002620.

Sequence in context: A214048 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 July 21 18:45 EDT 2017. Contains 289643 sequences.