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%I #41 Aug 22 2021 13:53:02
%S 1,1,2,4,8,16,32,64,128,256,1,5,22,92,376,1520,6112,24512,98176,
%T 392960,2,22,200,1696,13952,113152,911360,7315456,58621952,469368832,
%U 4,92,1696,28928,477184,7749632,124911616,2005925888,32153534464,514926313472,8
%N Number of lunar partitions of n: number of ways of writing n as a lunar sum of distinct terms, ignoring order.
%C Without the condition that the numbers are distinct the answers are infinite because 1+1+1+...+1 = 1 in lunar arithmetic - see A087061.
%H D Applegate and N. J. A. Sloane, <a href="/A087079/b087079.txt">Table of n, a(n) for n = 0..2000</a>
%H D. Applegate, <a href="/A087061/a087061.txt">C program for lunar arithmetic and number theory</a> [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="http://arxiv.org/abs/1107.1130">Dismal Arithmetic</a>, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Sloane/carry2.html">Dismal Arithmetic</a>, J. Int. Seq. 14 (2011) # 11.9.8.
%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>
%F For 1 <= a < 10 and 0 <= b < 10, a(10a+b) = 2^(ab+a+b-1)+(2^a-1)(2^b-1)2^(ab-1). - _David Wasserman_, Apr 14 2005
%e a(5) = 16: we can write 5 = 5 + any subset of {4, 3, 2, 1} (16 ways).
%e a(12) = 22: we can write 12 = 12 + any subset of {11, 10, 2, 1} (16 ways), 12 = 2 + 11 + 10 = 2 + 11 = 2 + 10 and those three with 1 added (6 ways).
%o (PARI) A087079(n) = { my(v, r = 0, i, j, b); v = select(x -> x != 0, digits(n)); for (i = 0, 2^#v - 1, b = Vecrev(binary(i)); b = vector(#v, i, if (i <= #b, b[i], 0)); r += (-1)^vecsum(b) * 2^prod(j = 1, #v, if (b[j] == 1, v[j], v[j] + 1)); ); r/2;} /* _Jerome Raulin_, Feb 15 2017 */
%Y Cf. A010036.
%Y The subsequence a(n) where n = 111..11 is A003465. - _N. J. A. Sloane_, May 21 2011
%K nonn
%O 0,3
%A _Marc LeBrun_, Oct 09 2003
%E More terms from _David Wasserman_, Apr 14 2005
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