This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A016127 #46 Mar 31 2025 16:58:02
%S A016127 1,7,39,203,1031,5187,25999,130123,650871,3254867,16275359,81378843,
%T A016127 406898311,2034499747,10172515119,50862608363,254313107351,
%U A016127 1271565667827,6357828601279,31789143530683,158945718701991,794728595607107,3973642982229839,19868214919537803,99341074614466231
%N A016127 Expansion of g.f. 1/((1-2*x)*(1-5*x)).
%C A016127 With leading zero, binomial transform of A002450. - _Paul Barry_, Apr 11 2003
%C A016127 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
%C A016127 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
%H A016127 Vincenzo Librandi, <a href="/A016127/b016127.txt">Table of n, a(n) for n = 0..300</a>
%H A016127 Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, <a href="https://arxiv.org/abs/2307.08073">The sequence of higher order Mersenne numbers and associated binomial transforms</a>, arXiv:2307.08073 [math.NT], 2023.
%H A016127 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-10).
%F A016127 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
%F A016127 Binomial transform of A020989. - _Paul Barry_, May 18 2003
%F A016127 From _Paul Barry_, Aug 05 2005: (Start)
%F A016127 a(n) = Sum_{k=0..n} Sum_{j=0..n} 5^(n-j)*binomial(j,k).
%F A016127 a(n) = Sum_{k=0..n} 2^k*5^(n-k) = Sum_{k=0..n} 5^k*2^(n-k). (End)
%F A016127 For n > 2, a(n) = 9*a(n-1) - 24*a(n-2) + 20*a(n-3). - _Gary W. Adamson_, Dec 26 2007
%F A016127 From _Elmo R. Oliveira_, Mar 31 2025: (Start)
%F A016127 E.g.f.: exp(2*x)*(5*exp(3*x) - 2)/3.
%F A016127 a(n) = A005057(n+1)/3.
%F A016127 a(n) = 7*a(n-1) - 10*a(n-2). (End)
%t A016127 Join[{a=1,b=7},Table[c=7*b-10*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%t A016127 CoefficientList[Series[1/((1 - 2 x) (1 - 5 x)), {x, 0, 19}], x] (* _Michael De Vlieger_, Jan 31 2018 *)
%t A016127 LinearRecurrence[{7,-10},{1,7},30] (* _Harvey P. Dale_, Aug 18 2020 *)
%o A016127 (Sage) [lucas_number1(n,7,10) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A016127 (Sage) [(5^n - 2^n)/3 for n in range(1,21)] # _Zerinvary Lajos_, Jun 05 2009
%o A016127 (Magma) [(5^(n+1)-2^(n+1))/3: n in [0..30]]; // _Vincenzo Librandi_, Jun 08 2011
%o A016127 (PARI) Vec(1/((1-2*x)*(1-5*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%Y A016127 Cf. A001045, A002450, A005057, A020989.
%K A016127 nonn,easy
%O A016127 0,2
%A A016127 _N. J. A. Sloane_