A016127 Expansion of g.f. 1/((1-2*x)*(1-5*x)).

1, 7, 39, 203, 1031, 5187, 25999, 130123, 650871, 3254867, 16275359, 81378843, 406898311, 2034499747, 10172515119, 50862608363, 254313107351, 1271565667827, 6357828601279, 31789143530683, 158945718701991, 794728595607107, 3973642982229839, 19868214919537803, 99341074614466231

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

With leading zero, binomial transform of A002450. - Paul Barry, 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, Aug 05 2005

Equals term (1,2) in M^n, M = the 3 X 3 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 - 5*x^2 - 4*x + 20. - Gary W. Adamson, Mar 12 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.
• Index entries for linear recurrences with constant coefficients, signature (7,-10).

Crossrefs

Cf. A001045, A002450, A005057, A020989.

Programs

Magma

[(5^(n+1)-2^(n+1))/3: n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

Mathematica

Join[{a=1,b=7},Table[c=7*b-10*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
CoefficientList[Series[1/((1 - 2 x) (1 - 5 x)), {x, 0, 19}], x] (* Michael De Vlieger, Jan 31 2018 *)
LinearRecurrence[{7,-10},{1,7},30] (* Harvey P. Dale, Aug 18 2020 *)

PARI

Vec(1/((1-2*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

[lucas_number1(n,7,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Sage

[(5^n - 2^n)/3 for n in range(1,21)] # Zerinvary Lajos, Jun 05 2009

Formula

a(n) = (5^(n+1) - 2^(n+1))/3 = Sum_{i=0..n} 5^i*2^(n-1) = 5*a(n-1) + 2^n = 2*a(n-1) + 5^n. - Henry Bottomley, Apr 07 2003

Binomial transform of A020989. - Paul Barry, May 18 2003

From Paul Barry, Aug 05 2005: (Start)

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). (End)

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

From Elmo R. Oliveira, Mar 31 2025: (Start)

E.g.f.: exp(2*x)*(5*exp(3*x) - 2)/3.

a(n) = A005057(n+1)/3.

a(n) = 7*a(n-1) - 10*a(n-2). (End)