|
%I #63 Feb 27 2024 19:38:22
%S 1,1,8,57,407,2906,20749,148149,1057792,7552693,53926643,385039194,
%T 2749201001,19629446201,140155324408,1000716717057,7145172343807,
%U 51016923123706,364263634209749,2600862362591949,18570300172353392
%N Generalized Fibonacci numbers.
%C Row m=7 of A135597.
%C For n >= 1, row sums of triangle for numbers 7^k*C(m,k) with duplicated diagonals. - _Vladimir Shevelev_, Apr 13 2012
%C For n >= 1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6,7} containing no subwords ii, (i=0,1,...,6). - _Milan Janjic_, Jan 31 2015
%H Vincenzo Librandi, <a href="/A015453/b015453.txt">Table of n, a(n) for n = 0..1000</a>
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 18.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,1)
%F a(n) = 7*a(n-1) + a(n-2).
%F a(n) = Sum_{k=0..n} 6^k*A055830(n,k). - _Philippe Deléham_, Oct 18 2006
%F G.f.: (1-6*x)/(1-7*x-x^2). - _Philippe Deléham_, Nov 20 2008
%F For n >= 2, a(n) = F_(n)(7) + F_(n+1)(7), where F_(n)(x) is Fibonacci polynomial (cf. A049310): F_(n)(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - _Vladimir Shevelev_, Apr 13 2012
%F a(n) = A054413(n) - 6*A054413(n-1). - _R. J. Mathar_, Jul 06 2012
%F a(n) = Sum_{k=0..n} A046854(n-1,k)*7^k. - _R. J. Mathar_, Feb 14 2024
%t LinearRecurrence[{7, 1}, {1, 1}, 30] (* _Vincenzo Librandi_, Nov 08 2012 *)
%t CoefficientList[Series[(1-6*x)/(1-7*x-x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017 *)
%o (Magma) [n le 2 select 1 else 7*Self(n-1) + Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 08 2012
%o (PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-7*x-x^2)) \\ _G. C. Greubel_, Dec 19 2017
%o (Sage) [lucas_number1(n+1, 7, -1) - 6*lucas_number1(n, 7, -1) for n in (0..30)] # _G. C. Greubel_, Dec 24 2021
%Y Cf. A054413, A055830.
%Y Row m=7 of A135597.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
|
