0,2

With leading zero, binomial transform of A002450. - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003

The sequence of fractions a(n)/(n+1) is the 3rd binomial transform of the sequence of fractions J(n+1)/(n+1) where J(n) is A001045(n). - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005

Equals term (1,2) in M^n, M = the 3x3 matrix [1, 1, 3; 1, 3, 1; 3, 1, 1]. a(n)/ a(n-1) tends to 5, a root to the charpoly x^3 - 5x^2 -4x + 20. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 12 2009]

a(n) =(5^(n+1)-2^(n+1))/3 =sum_i{0<=i<=n}5^i*2^(n-1) =5a(n-1)+2^n =2a(n-1)+5^n. - Henry Bottomley (se16(AT)btinternet.com), Apr 07 2003

Binomial transform of A020989. - Paul Barry (pbarry(AT)wit.ie), May 18 2003

a(n)=sum{k=0..n, sum{j=0..n, 5^(n-j)*binomial(j, k)}}; a(n)=sum{k=0..n, 2^k*5^(n-k)}=sum{k=0..n, 5^k*2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Aug 05 2005

For n>2, a(n) = 9*a(n-1) - 24*a(n-2) + 20*a(n-3). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2007

(Other) sage: [lucas_number1(n, 7, 10) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]

(Other) sage: [(5^n - 2^n)/3 for n in xrange(1, 21)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]

Sequence in context: A026752 A026379 A026708 this_sequence A099460 A092923 A164550

Adjacent sequences: A016124 A016125 A016126 this_sequence A016128 A016129 A016130

nonn

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