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A125127 Array L(k,n) read by antidiagonals: k-step Lucas numbers. 7
1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 1, 3, 7, 7, 1, 1, 3, 7, 11, 11, 1, 1, 3, 7, 15, 21, 18, 1, 1, 3, 7, 15, 26, 39, 29, 1, 1, 3, 7, 15, 31, 51, 71, 47, 1, 1, 3, 7, 15, 31, 57, 99, 131, 76, 1, 1, 3, 7, 15, 31, 63, 113, 191, 241, 123, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Freddy Barrera, Table of n, a(n) for n = 1..5050

C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.

Eric Weisstein's World of Mathematics, Lucas n-Step Number

FORMULA

L(k,n) = L(k,n-1) + L(k,n-2) + ... + L(k,n-k); L(k,n) = -1 for n < 0, and L(k,0) = k.

G.f. for row k: x*(dB(k,x)/dx)/(1-B(k,x)), where B(k,x) = x + x^2 + ... + x^k. - Petros Hadjicostas, Jan 24 2019

EXAMPLE

Table begins:

1 | 1  1  1   1   1   1    1    1    1    1

2 | 1  3  4   7  11  18   29   47   76  123

3 | 1  3  7  11  21  39   71  131  241  443

4 | 1  3  7  15  26  51   99  191  367  708

5 | 1  3  7  15  31  57  113  223  439  863

6 | 1  3  7  15  31  63  120  239  475  943

7 | 1  3  7  15  31  63  127  247  493  983

8 | 1  3  7  15  31  63  127  255  502 1003

9 | 1  3  7  15  31  63  127  255  511 1013

PROG

(Sage)

def L(k, n):

    if n < 0:

        return -1

    a = [-1]*(k-1) + [k] # [-1, -1, ..., -1, k]

    for i in range(1, n+1):

        a[:] = a[1:] + [sum(a)]

    return a[-1]

[L(k, n) for d in (1..12) for k, n in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jan 10 2019

CROSSREFS

n-step Lucas number analog of A092921 Array F(k, n) read by antidiagonals: k-generalized Fibonacci numbers (and see related A048887, A048888). L(1, n) = "1-step Lucas numbers" = A000012. L(2, n) = 2-step Lucas numbers = A000204. L(3, n) = 3-step Lucas numbers = A001644. L(4, n) = 4-step Lucas numbers = A001648 Tetranacci numbers A073817 without the leading term 4. L(5, n) = 5-step Lucas numbers = A074048 Pentanacci numbers with initial conditions a(0)=5, a(1)=1, a(2)=3, a(3)=7, a(4)=15. L(6, n) = 6-step Lucas numbers = A074584 Esanacci ("6-anacci") numbers. L(7, n) = 7-step Lucas numbers = A104621 Heptanacci-Lucas numbers. L(8, n) = 8-step Lucas numbers = A105754. L(9, n) = 9-step Lucas numbers = A105755. See A000295, A125129 for comments on partial sums of diagonals.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125129.

Sequence in context: A133116 A059959 A192812 * A051120 A114476 A260419

Adjacent sequences:  A125124 A125125 A125126 * A125128 A125129 A125130

KEYWORD

easy,nonn,tabl

AUTHOR

Jonathan Vos Post, Nov 21 2006

EXTENSIONS

Corrected by Freddy Barrera, Jan 10 2019

STATUS

approved

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Last modified April 4 05:15 EDT 2020. Contains 333212 sequences. (Running on oeis4.)