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%I #97 Dec 10 2022 10:45:52
%S 2,4,7,10,14,18,23,28,34,40,47,54,62,70,79,88,98,108,119,130,142,154,
%T 167,180,194,208,223,238,254,270,287,304,322,340,359,378,398,418,439,
%U 460,482,504,527,550,574,598,623,648,674,700,727,754,782,810,839,868
%N a(n) = solution to the postage stamp problem with 2 denominations and n stamps.
%C _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.
%C 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
%C a(n-1) is the largest possible n-th element in an additive basis of order 2. - _Charles R Greathouse IV_, May 05 2020
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section C12, pp. 185-190.
%H Vincenzo Librandi, <a href="/A014616/b014616.txt">Table of n, a(n) for n = 1..10000</a>
%H Mario Bravo, Thierry Champion, and Roberto Cominetti, <a href="https://arxiv.org/abs/2108.00300">Universal bounds for fixed point iterations via optimal transport metrics</a>, arXiv:2108.00300 [math.OC], 2021.
%H Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/0403.html">Postage stamp problem</a>
%H W. F. Lunnon, <a href="http://dx.doi.org/10.1093/comjnl/12.4.377">A postage stamp problem</a>, Comput. J. 12 (1969) 377-380.
%H Alfred Stöhr, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002177099&physid=PHYS_0051">Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I.</a>, Journal für die reine und angewandte Mathematik (1955), Volume: 194, page 40-65. See p. 47.
%H Hugh Thomas and Stephanie van Willigenburg, <a href="http://arXiv.org/abs/0706.3250">Compact symmetric solutions to the postage stamp problem</a>, arXiv:0706.3250 [math.NT], 2007-2008.
%H Amitabha Tripathi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Tripathi/tripathi14.html">A Note on the Postage Stamp Problem</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PostageStampProblem.html">Postage stamp problem</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%H <a href="/index/Pos#postage_stamp_problem">Index to sequences related to the Postage Stamp Problem</a>.
%F a(n) = floor((n^2 + 6*n + 1)/4).
%F a(n) = A002620(n+3) - 2 = A024206(n+2) - 1 = (2*n*(n+6)-(-1)^n+1)/8.
%F G.f.: x*(-2 + x^2)/((1 + x)*(x - 1)^3). - _R. J. Mathar_, Jul 09 2011
%F a(n) = floor(A028884(n+1)/4). - _Reinhard Zumkeller_, Apr 07 2013
%F a(n)+a(n+1) = A046691(n+1). - _R. J. Mathar_, Mar 13 2021
%F a(n) = 2*n + A002620(n-1). - _Michael Chu_, Apr 28 2022
%F a(n) = A004116(n) + 1. - _Michael Chu_, May 02 2022
%F E.g.f.: (x*(7 + x)*cosh(x) + (1 + 7*x + x^2)*sinh(x))/4. - _Stefano Spezia_, Nov 09 2022
%F Sum_{n>=1} 1/a(n) = 67/36 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - _Amiram Eldar_, Dec 10 2022
%e 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
%t 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 *)
%t LinearRecurrence[{2,0,-2,1},{2,4,7,10},60] (* _Harvey P. Dale_, Oct 04 2015 *)
%o (Magma) [(2*n*(n+6)-(-1)^n+1)/8: n in [1..60]]; // _Vincenzo Librandi_, Jul 09 2011
%o (Haskell)
%o a014616 n = (n * (n + 6) + 1) `div` 4 -- _Reinhard Zumkeller_, Apr 07 2013
%o (PARI) a(n)=(n^2 + 6*n + 1)\4 \\ _Charles R Greathouse IV_, Feb 06 2017
%Y A row or column of the array A196416 (possibly with 1 subtracted from it).
%Y Cf. A002620, A004116, A024206, A028884, A046691.
%K nonn,nice,easy
%O 1,1
%A _Eric W. Weisstein_
%E Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
%E More terms from _John W. Layman_, Apr 13 1999
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