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A060692 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.

%I A060692 #58 Jan 07 2025 05:38:56
%S A060692 2,3,6,6,26,36,28,186,265,738,1105,3186,5269,15516,29728,55761,35228,
%T A060692 235278,441475,272526,1861166,3478866,6231073,1899171,5672262,
%U A060692 50533341,17325482,186108951,21328109,63792576,1264831925,3794064336,7086578554
%N A060692 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 A060692 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 A060692 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 A060692 a(n) = A002379(n) + A002380(n) = floor(3^n/2^n) + (3^n mod 2^n).
%F A060692 For n > 2, a(n) = 3^n mod (2^n-1). - _Alex Ratushnyak_, Jul 22 2012
%e A060692 3^4 = 81 = 16 + 16 + 16 + 16 + 16 + 1, so a(4) = 5 + 1 = 6;
%e A060692 3^5 = 243 = 32 + 32 + 32 + 32 + 32 + 32 + 32 + 19*1, so a(5) = 7 + 19 = 26.
%t A060692 Table[3^n - (-1 + 2^n) Floor[(3/2)^n], {n, 33}] (* _Fred Daniel Kline_, Nov 01 2017 *)
%t A060692 x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi;
%t A060692 y[n_] := 3^n - 2^n * x[n]; yplusx[n_] := y[n] + x[n];
%t A060692 Array[yplusx, 33] (* _Fred Daniel Kline_, Dec 21 2017 *)
%t A060692 f[n_] := Floor[3^n/2^n] + PowerMod[3, n, 2^n]; Array[f, 33] (* _Robert G. Wilson v_, Dec 27 2017 *)
%o A060692 (PARI) a(n) = { my(d=divrem(3^n,2^n)); d[1]+d[2] }
%o A060692 (PARI) a(n) = { (3^n\2^n) + (3^n%2^n) } \\ _Harry J. Smith_, Jul 09 2009
%o A060692 (Haskell)
%o A060692 a060692 n = uncurry (+) $ divMod (3 ^ n) (2 ^ n)
%o A060692 -- _Reinhard Zumkeller_, Jul 11 2014
%Y A060692 Cf. A002379, A002380, A064464, A064630.
%Y A060692 Cf. A000079, A000244.
%K A060692 nonn,easy
%O A060692 1,1
%A A060692 _Labos Elemer_, Apr 20 2001
%E A060692 Edited by _Klaus Brockhaus_, May 24 2003