0,3

For n>0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3)=21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman (layman(AT)math.vt.edu), Dec 18 2001

a(n) is composite for all n > 2 and has factors x, (3x+2(-1)^n) where x belongs to A001045. In binary the terms are 1, 101, 10101, 1010101, etc. - John McNamara (mistermac39(AT)yahoo.com), Jan 16 2002

Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - Ron Hardin (rhhardin(AT)att.net), Mar 16 2002

Collatz-function iteration started at with a[n] will surely ended by 1 in exactly 2n steps. - Labos E. (labos(AT)ana.sote.hu), Sep 30 2002

Also sum of squares of divisors of 2^n: a(n)=A001157[A000079(n)] - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003

All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast (ghodges14(AT)comcast.net), Apr 10 2003

Binomial transform of A000244 (with leading zero) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003

Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n=2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004

Also number of walks of length 2n+1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba (kociemba(AT)t-online.de), Jul 05 2004

a(n+1) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005

a(n+1)=sum of square divisors of 4^n. - Paul Barry (pbarry(AT)wit.ie), Oct 13 2005

a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric Weisstein (eric(AT)weisstein.com), Apr 08 2006

a(k)=[M^k]_2,1, where M is the 3 by 3 matrix defined as follows: M = [1,1,1;1,3,1;1,1,1]. - Simone Severini (ss54(AT)york.ac.uk), Jun 11 2006

a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006

n is in the sequence if and only if C(4n+1,n) (A052203) is odd; - Paul Barry (pbarry(AT)wit.ie), Mar 26 2007

This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n-1). - Keith Briggs (keith.briggs(AT)bt.com), Jun 19 2007

All numbers of the form n*4^n+(4^n-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity n*4^n+(4^n-1)/3=4(4(..4(4n+1)+1)+1)+1..)+1. - Artur Jasinski (grafix(AT)csl.pl), Nov 12 2007

Successive numbers contain only the digit 1 in base 4 positional system: 1, 11, 111, 1111 etc. [From Artur Jasinski (grafix(AT)csl.pl), Sep 30 2008]

a(n) for n >= 1 written in base 2: 1,101,10101,1010101,101010101, ...(see A094028(n-1)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 02 2009]

Except for the first term, a(n)=4*a(n-1)+1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 29 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to linear recurrences with constant coefficients

H. Bottomley, Illustration of initial terms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 373

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Rule 250

a(n+1)= sum(A060921(n, m), m=0..n). G.f.: x/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001

a(n)= sum{k=0..n-1, 4^k} a(n)= A001045(2n). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003

Second binomial transform of A001045. - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003

E.g.f. (exp(4x)-exp(x))/3 - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003

a(0) = 0, a(n+1) = 4*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004

a(n)=Sum(C(2n-1-i, i)2^i, i=0, .., n-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004

a(n+1)=sum{k=0..n, binomial(n+1, k+1)3^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004

a(n) = center term in M^n * [1 0 0], where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g. a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004

a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)A001045(j-k)}}; - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005

a(n)=sum{k=0..n, C(n, k)*A001045(n-k)*2^k}=sum{k=0..n, C(n, k)*A001045(k)*2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Apr 22 2005

Coefficients of series expansion of (1+4x)/(1-x-16x^2+16x^3) at point x=0. - Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006

a(n) = A125118(n,3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006

a(n)=Sum_{k, 0<=k<=n}2^(n-k)*A128908(n,k), n>=1 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2008]

a(n)=Sum_{k, 0<=k<=n} A106566(n,k)*A100335(k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

If we define f(m,j,x)=sum(binomial(m,k)*stirling2(k,j)*x^(m-k),k=j..m) then a(n-1)=f(2*n,4,-2), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Apr 26 2009]

a(n)=A014551(n)*A001045(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 08 2009]

[seq((4^n-1)/3, n=0..40)];

a:=n->sum(4^(n-j), j=1..n): seq(a(n), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007

A002450:=1/(4*z-1)/(z-1); [S. Plouffe in his 1992 dissertation, dropping the initial zero.]

with(finance):seq(add(futurevalue( 2, 3, k), k=0..n)/2, n=-1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

lst={}; Do[p=(4^n-1)/3; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]

b = {}; a = {1}; Do[s = FromDigits[a, 4]; AppendTo[b, s]; AppendTo[a, 1], {n, 1, 50}]; b [From Artur Jasinski (grafix(AT)csl.pl), Sep 30 2008]

(MAGMA) [ (4^n-1)/3: n in [0..25] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 28 2008]

(Other) sage: [lucas_number1(n, 5, 4) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

(Other) sage: [gaussian_binomial(n, 1, 4) for n in xrange(0, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

a(n) = (A007583(n)-1)/2.

Partial sums of powers of 4, A000302.

a(n)=A000975(2n)/2.

A084160(n) = 2*a(n).

Cf. A002446, A024036, A020988, A080674, A047849, A007583.

Cf. A080355, A112627, A113860, A129735.

Cf. A018215.

A160967, A139391. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 31 2009]

Sequence in context: A026027 A002054 A028948 this_sequence A084241 A026855 A097113

Adjacent sequences: A002447 A002448 A002449 this_sequence A002451 A002452 A002453

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Artur Jasinski (grafix(AT)csl.pl), Jan 27 2006

More terms from Artur Jasinski (grafix(AT)csl.pl), Nov 12 2007

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009