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%I #50 Jan 22 2018 07:11:29
%S 2,3,6,6,26,36,28,186,265,738,1105,3186,5269,15516,29728,55761,35228,
%T 235278,441475,272526,1861166,3478866,6231073,1899171,5672262,
%U 50533341,17325482,186108951,21328109,63792576,1264831925,3794064336,7086578554
%N Number of parts if 3^n is partitioned into parts of size 2^n as far as possible and into parts of size 1^n.
%C Corresponds to the only solution of the Diophantine equation 3^n = x*2^n + y*1^n with constraint 0 <= y < 2^n. [Since 3^n is odd, of course y cannot be zero.)
%H Iain Fox, <a href="/A060692/b060692.txt">Table of n, a(n) for n = 1..3322</a> (first 500 terms from Harry J. Smith)
%F a(n) = A002379(n) + A002380(n) = floor(3^n/2^n) + (3^n mod 2^n).
%F For n > 2, a(n) = 3^n mod (2^n-1). [_Alex Ratushnyak_, Jul 22 2012]
%e 3^4 = 81 = 16 + 16 + 16 + 16 + 16 + 1, so a(4) = 5 + 1 = 6;
%e 3^5 = 243 = 32 + 32 + 32 + 32 + 32 + 32 + 32 + 19*1, so a(5) = 7 + 19 = 26.
%t Table[3^n - (-1 + 2^n) Floor[(3/2)^n], {n, 33}] (* _Fred Daniel Kline_, Nov 01 2017 *)
%t x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi;
%t y[n_] := 3^n - 2^n * x[n]; yplusx[n_] := y[n] + x[n];
%t Array[yplusx, 33] (* _Fred Daniel Kline_, Dec 21 2017 *)
%t f[n_] := Floor[3^n/2^n] + PowerMod[3, n, 2^n]; Array[f, 33] (* _Robert G. Wilson v_, Dec 27 2017 *)
%o (PARI) {for(n=1,33,d=divrem(3^n,2^n); print1(d[1]+d[2],","))}
%o (PARI) { for (n=1, 500, write("b060692.txt", n, " ", floor(3^n/2^n) + (3^n%2^n)); ) } \\ _Harry J. Smith_, Jul 09 2009
%o (Haskell)
%o a060692 n = uncurry (+) $ divMod (3 ^ n) (2 ^ n)
%o -- _Reinhard Zumkeller_, Jul 11 2014
%Y Cf. A002379, A002380, A064464, A064630.
%Y Cf. A000079, A000244.
%K nonn
%O 1,1
%A _Labos Elemer_, Apr 20 2001
%E Edited by _Klaus Brockhaus_, May 24 2003
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