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%I A125127 #28 Jan 25 2019 08:30:51
%S A125127 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,
%T A125127 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,
%U A125127 7,15,31,63,113,191,241,123,1
%N A125127 Array L(k,n) read by antidiagonals: k-step Lucas numbers.
%H A125127 Freddy Barrera, <a href="/A125127/b125127.txt">Table of n, a(n) for n = 1..5050</a>
%H A125127 C. A. Charalambides, <a href="http://www.fq.math.ca/Scanned/29-4/charalambides.pdf">Lucas numbers and polynomials of order k and the length of the longest circular success run</a>, The Fibonacci Quarterly, 29 (1991), 290-297.
%H A125127 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>
%F A125127 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.
%F A125127 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
%e A125127 Table begins:
%e A125127 1 | 1  1  1   1   1   1    1    1    1    1
%e A125127 2 | 1  3  4   7  11  18   29   47   76  123
%e A125127 3 | 1  3  7  11  21  39   71  131  241  443
%e A125127 4 | 1  3  7  15  26  51   99  191  367  708
%e A125127 5 | 1  3  7  15  31  57  113  223  439  863
%e A125127 6 | 1  3  7  15  31  63  120  239  475  943
%e A125127 7 | 1  3  7  15  31  63  127  247  493  983
%e A125127 8 | 1  3  7  15  31  63  127  255  502 1003
%e A125127 9 | 1  3  7  15  31  63  127  255  511 1013
%o A125127 (Sage)
%o A125127 def L(k, n):
%o A125127     if n < 0:
%o A125127         return -1
%o A125127     a = [-1]*(k-1) + [k] # [-1, -1, ..., -1, k]
%o A125127     for i in range(1, n+1):
%o A125127         a[:] = a[1:] + [sum(a)]
%o A125127     return a[-1]
%o A125127 [L(k, n) for d in (1..12) for k, n in zip((d..1, step=-1), (1..d))] # _Freddy Barrera_, Jan 10 2019
%Y A125127 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.
%Y A125127 Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125129.
%K A125127 easy,nonn,tabl
%O A125127 1,5
%A A125127 _Jonathan Vos Post_, Nov 21 2006
%E A125127 Corrected by _Freddy Barrera_, Jan 10 2019

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