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URL: https://oeis.org/A084221

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A084221
a(n+2) = 4*a(n), with a(0)=1, a(1)=3.
17
1, 3, 4, 12, 16, 48, 64, 192, 256, 768, 1024, 3072, 4096, 12288, 16384, 49152, 65536, 196608, 262144, 786432, 1048576, 3145728, 4194304, 12582912, 16777216, 50331648, 67108864, 201326592, 268435456, 805306368, 1073741824, 3221225472, 4294967296, 12884901888
OFFSET
0,2
COMMENTS
Binomial transform is A060925.
Binomial transform of A084222.
Sequences with similar recurrence rules: A016116 (multiplier 2), A038754 (multiplier 3), A133632 (multiplier 5). See A133632 for general formulas. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * A000079. A122756 is a companion sequence. - Gary W. Adamson, Sep 19 2007
LINKS
Hester Graves, The Minimal Euclidean Function on the Gaussian Integers, arXiv:1802.08281 [math.NT], 2018. See Definition 2.3, p.3.
FORMULA
a(n) = (5*2^n-(-2)^n)/4.
G.f.: (1+3*x)/((1-2*x)(1+2*x)).
E.g.f.: (5*exp(2*x) - exp(-2*x))/4.
a(n) = A133628(n) - A133628(n-1) for n>1. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * [1, 2, 4, 8, ...]. Row sums of triangle A133087. - Gary W. Adamson, Sep 08 2007
a(n+1)-2a(n) = A000079 signed. a(n)+a(n+2)=5*a(n). First differences give A135520. - Paul Curtz, Apr 22 2008
a(n) = A074323(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = Sum_{k=0..n+1} A181650(n+1,k)*2^k. - Philippe Deléham, Nov 19 2011
a(2*n) = A000302(n); a(2*n+1) = A164346(n). - Philippe Deléham, Mar 21 2014
From Amiram Eldar, Feb 12 2026: (Start)
Sum_{n>=0} 1/a(n) = 16/9.
Sum_{n>=0} (-1)^n/a(n) = 8/9. (End)
EXAMPLE
Binary...............Decimal
1..........................1
11.........................3
100........................4
1100......................12
10000.....................16
110000....................48
1000000...................64
11000000.................192
100000000................256
1100000000...............768
10000000000.............1024
110000000000............3072, etc. - Philippe Deléham, Mar 21 2014
MATHEMATICA
CoefficientList[Series[(-3*x - 1)/(4*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
PROG
(Magma) [(5*2^n-(-2)^n)/4: n in [0..40]]; // Vincenzo Librandi, Aug 13 2011
(PARI) a(n)=([0, 1; 4, 0]^n*[1; 3])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
CROSSREFS
For partial sums see A133628. Partial sums for other multipliers p: A027383 (p=2), A087503 (p=3), A133629 (p=5).
Other related sequences: A132666, A132667, A132668, A132669.
Sequence in context: A282458 A360755 A348949 * A142866 A274224 A280525
KEYWORD
nonn,easy
AUTHOR
Paul Barry, May 21 2003
EXTENSIONS
Edited by N. J. A. Sloane, Dec 14 2007
STATUS
approved