VOOZH about

URL: https://oeis.org/A015453

⇱ A015453 - OEIS

login
A015453
Generalized Fibonacci numbers.
5
1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408, 1000716717057, 7145172343807, 51016923123706, 364263634209749, 2600862362591949, 18570300172353392
OFFSET
0,3
COMMENTS
Row m=7 of A135597.
For n >= 1, row sums of triangle for numbers 7^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012
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
LINKS
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 18.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = 7*a(n-1) + a(n-2).
a(n) = Sum_{k=0..n} 6^k*A055830(n,k). - Philippe Deléham, Oct 18 2006
G.f.: (1-6*x)/(1-7*x-x^2). - Philippe Deléham, Nov 20 2008
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
a(n) = A054413(n) - 6*A054413(n-1). - R. J. Mathar, Jul 06 2012
a(n) = Sum_{k=0..n} A046854(n-1,k)*7^k. - R. J. Mathar, Feb 14 2024
MATHEMATICA
LinearRecurrence[{7, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
CoefficientList[Series[(1-6*x)/(1-7*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)
PROG
(Magma) [n le 2 select 1 else 7*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012
(PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-7*x-x^2)) \\ G. C. Greubel, Dec 19 2017
(SageMath) [lucas_number1(n+1, 7, -1) - 6*lucas_number1(n, 7, -1) for n in (0..30)] # G. C. Greubel, Dec 24 2021
CROSSREFS
Row m=7 of A135597.
Sequence in context: A097114 A022038 A277671 * A181246 A281355 A281912
KEYWORD
nonn,easy
STATUS
approved