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A005061
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a(n) = 4^n - 3^n.
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82
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0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837, 281192547174175, 1125052618233181
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OFFSET
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0,3
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COMMENTS
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Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - R. H. Hardin, Mar 21 2002
Number of binary vectors (x_1, x_2, ..., x_{2n}) such that in at least one of the disjoint pairs (x_1, x_2), (x_3, x_4), ..., (x_{2n-1}, x_{2n}) both x_{2i-1} and x_{2i} are both 1. Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_3*x_4 + x_5*x_6 + ... +x_{2n-1}*x_{2n} = 1 in base-2 lunar arithmetic. - N. J. A. Sloane, Apr 23 2011
a(n)/4^n is the probability that two randomly selected (with replacement) subsets of [n] will have at least one element in common if the probability of selection is equal for all subsets. - Geoffrey Critzer, May 09 2009
This sequence is also the second column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). (See the e.g.f. given below.) - Wolfdieter Lang, Oct 08 2011
Also, the number of numbers with at most n digits whose largest digit equals 3. See A255463 for the first differences (i.e., ...with exactly n digits...). - M. F. Hasler, May 03 2015
a(n) is the number of ordered n-tuples with elements from {0,1,2,3} in which any of these elements, say 0, appears at least once. For example, a(2)=7 since 01,10,02,20,03,30,00 are the ordered 2-tuples that contain 0. - Enrique Navarrete, Apr 05 2021
a(n) is the number of n-digit numbers whose smallest decimal digit is 6. - Stefano Spezia, Nov 15 2023
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LINKS
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D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, 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]
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FORMULA
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a(n) = 4*a(n-1) + 3^(n-1) for n>=1. - Xavier Acloque, Oct 20 2003
G.f.: 1/(1-4*x)-1/(1-3*x).
E.g.f.: exp(4*x)-exp(3*x). (End)
a(n) = 2^n * Sum_{i=0...n} binomial(n,i)*(2^i-1)/2^i. - Geoffrey Critzer, May 09 2009
a(n) = 3*a(n-1) + 4^(n-1) for n>=0.
a(n+1) = Sum_{k=0..n} 4^(n-k) * 3^k. (End)
a(n) = -a(-n) * 12^n for all n in Z. - Michael Somos, Jan 22 2017
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EXAMPLE
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G.f. = x + 7*x^2 + 37*x^3 + 175*x^4 + 781*x^5 + 3367*x^6 + 14197*x^7 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{7, -12}, {0, 1}, 30] (* Harvey P. Dale, May 04 2012 *)
Table[Numerator[1-(3/4)^n], {n, 0, 20}] (* see link Wolfram Mathworld - Fred Daniel Kline, Feb 05 2018 *)
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PROG
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(Python)
def a(n): return 4**n - 3**n
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CROSSREFS
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Array column A047969(n-1, 3), or triangle's sudiagonal A047969(n+2, n-1), for n >= 1.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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