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A014616
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a(n) = solution to the postage stamp problem with 2 denominations and n stamps.
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37
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section C12, pp. 185-190.
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LINKS
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FORMULA
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a(n) = floor((n^2 + 6*n + 1)/4).
G.f.: x*(-2 + x^2)/((1 + x)*(x - 1)^3). - R. J. Mathar, Jul 09 2011
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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
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CROSSREFS
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A row or column of the array A196416 (possibly with 1 subtracted from it).
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
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STATUS
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approved
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