The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A139011 Real part of (2 + i)^n, where i = sqrt(-1). 8
 1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Imaginary part of (2 + i)^n gives A099456. (a(n))^2 + (A099456(n))^2 = 5^n. Example: (a(5))^2 + (A099456(5))^2 = 3125 = 5^5 = (-38)^2 + 41^2 = 1444 + 1681. Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..2862 (first 201 terms from Vincenzo Librandi) Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54. Index entries for linear recurrences with constant coefficients, signature (4,-5). FORMULA Real part of (2 + i)^n, i^2 = -1. Term (1,1) of matrix [2,-1; 1,2]^n. Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495. From R. J. Mathar, Apr 06 2008: (Start) O.g.f.: (1-2x) /(1-4x+5x^2). a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End) a(n) = (1/2)*((2-i)^n + (2+i)^n) where i=sqrt(-1). - Vim Wenders, Apr 12 2008; Paolo P. Lava, Jul 14 2008 E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009 a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010 a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008 2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011 a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011 EXAMPLE 1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ... a(5) = -38 since (2 + i)^5 = (-38 + 41*i). a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5). a(5) = -38 = A006496(5). MAPLE restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=1..29 ); # Zerinvary Lajos, Apr 06 2009 PROG (Sage) [lucas_number2(n, 4, 5)/2 for n in range(0, 31)] # Zerinvary Lajos, Jul 08 2008 (PARI) a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */ (PARI) Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017 (MAGMA) [ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011 CROSSREFS Cf. A099456, A006495, A006496, A056594, A146559. Sequence in context: A014784 A048601 A008317 * A338770 A063708 A096488 Adjacent sequences:  A139008 A139009 A139010 * A139012 A139013 A139014 KEYWORD sign,easy AUTHOR Gary W. Adamson, Apr 05 2008 EXTENSIONS Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009 Added a(0)=1 by Michael Somos, Dec 26 2010 Edited by Franklin T. Adams-Watters, Apr 10 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 01:37 EST 2021. Contains 349426 sequences. (Running on oeis4.)