A084221 - OEIS

%I #50 Sep 08 2022 08:45:11

%S 1,3,4,12,16,48,64,192,256,768,1024,3072,4096,12288,16384,49152,65536,

%T 196608,262144,786432,1048576,3145728,4194304,12582912,16777216,

%U 50331648,67108864,201326592,268435456,805306368,1073741824,3221225472,4294967296,12884901888

%N a(n+2) = 4*a(n), with a(0)=1, a(1)=3.

%C Binomial transform is A060925. Binomial transform of A084222.

%C Sequences with similar recurrence rules: A016116 (multiplier 2), A038754 (multiplier 3), A133632 (multiplier 5). See A133632 for general formulas. - _Hieronymus Fischer_, Sep 19 2007

%C Equals A133080 * A000079. A122756 is a companion sequence. - _Gary W. Adamson_, Sep 19 2007

%H Vincenzo Librandi, <a href="/A084221/b084221.txt">Table of n, a(n) for n = 0..1000</a>

%H Hester Graves, <a href="https://arxiv.org/abs/1802.08281">The Minimal Euclidean Function on the Gaussian Integers</a>, arXiv:1802.08281 [math.NT], 2018. See Definition 2.3 p.3.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4)

%F a(n) = (5*2^n-(-2)^n)/4.

%F G.f.: (1+3*x)/((1-2*x)(1+2*x)).

%F E.g.f.: (5*exp(2*x) - exp(-2*x))/4.

%F a(n) = A133628(n) - A133628(n-1) for n>1. - _Hieronymus Fischer_, Sep 19 2007

%F Equals A133080 * [1, 2, 4, 8, ...]. Row sums of triangle A133087. - _Gary W. Adamson_, Sep 08 2007

%F a(n+1)-2a(n) = A000079 signed. a(n)+a(n+2)=5*a(n). First differences give A135520. - _Paul Curtz_, Apr 22 2008

%F a(n) = A074323(n+1)*A016116(n). - _R. J. Mathar_, Jul 08 2009

%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011

%F a(n) = Sum_{k=0..n+1} A181650(n+1,k)*2^k. - _Philippe Deléham_, Nov 19 2011

%F a(2*n) = A000302(n); a(2*n+1) = A164346(n). - _Philippe Deléham_, Mar 21 2014

%e Binary...............Decimal

%e 1..........................1

%e 11.........................3

%e 100........................4

%e 1100......................12

%e 10000.....................16

%e 110000....................48

%e 1000000...................64

%e 11000000.................192

%e 100000000................256

%e 1100000000...............768

%e 10000000000.............1024

%e 110000000000............3072, etc. - _Philippe Deléham_, Mar 21 2014

%t CoefficientList[Series[(-3*x - 1)/(4*x^2 - 1), {x, 0, 200}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)

%o (Magma) [(5*2^n-(-2)^n)/4: n in [0..40]]; // _Vincenzo Librandi_, Aug 13 2011

%o (PARI) a(n)=([0,1; 4,0]^n*[1;3])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016

%Y For partial sums see A133628. Partial sums for other multipliers p: A027383(p=2), A087503(p=3), A133629(p=5).

%Y Other related sequences: A132666, A132667, A132668, A132669.

%Y Cf. A000302, A133080, A133087, A164346.

%K nonn,easy

%O 0,2

%A _Paul Barry_, May 21 2003

%E Edited by _N. J. A. Sloane_, Dec 14 2007