A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.
1, 14, 124, 1016, 8176, 65504, 524224, 4194176, 33554176, 268434944, 2147482624, 17179867136, 137438949376, 1099511619584, 8796093005824, 70368744144896, 562949953355776, 4503599627239424, 36028797018701824, 288230376151187456
Offset: 1
Examples
1_10 is 1_2;
14_10 is 1110_2;
124_10 is 1111100_2;
1016_10 is 1111111000_2.
Links
- Nathaniel Johnston, Table of n, a(n) for n = 1..500
- Index entries for linear recurrences with constant coefficients, signature (10,-16).
Programs
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GAP
List([1..25], n-> 2^(n-2)*(4^n-2)); # G. C. Greubel, Jul 27 2019
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Magma
[8^n/4-2^(n-1): n in [1..25]]; // Vincenzo Librandi, Jul 27 2019
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Maple
seq(8^n/4-2^(n-1),n=1..25); # Nathaniel Johnston, Apr 30 2011
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Mathematica
LinearRecurrence[{10,-16},{1,14},30] (* Harvey P. Dale, Oct 10 2014 *) Table[8^n / 4 - 2^(n - 1), {n, 25}] (* Vincenzo Librandi, Jul 27 2019 *) -
PARI
vector(25, n, 2^(n-2)*(4^n-2)) \\ G. C. Greubel, Jul 27 2019
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Sage
[2^(n-2)*(4^n-2) for n in (1..25)] # G. C. Greubel, Jul 27 2019
Formula
a(n) = A147537(n)/2.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = 8^n/4 - 2^(n-1) = A083332(2n-2).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: x*(1+4*x)/((1-2*x)*(1-8*x)). (End)
From César Aguilera, Jul 26 2019: (Start)
Lim_{n->infinity} a(n)/a(n-1) = 8;
a(n)/a(n-1) = 8 + 6/A083420(n). (End)
E.g.f.: (1/4)*(exp(2*x)*(-2 + exp(6*x)) + 1). - Stefano Spezia, Aug 05 2019
Extensions
More terms from R. J. Mathar, Jul 13 2009
Typo in a(12) corrected by Omar E. Pol, Jul 20 2009
Comments