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Array L(k,n) read by antidiagonals: k-step Lucas numbers.
7

%I #28 Jan 25 2019 08:30:51

%S 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 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 7,15,31,63,113,191,241,123,1

%N Array L(k,n) read by antidiagonals: k-step Lucas numbers.

%H Freddy Barrera, <a href="/A125127/b125127.txt">Table of n, a(n) for n = 1..5050</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>

%F 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 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 Table begins:

%e 1 | 1 1 1 1 1 1 1 1 1 1

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

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

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

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

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

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

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

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

%o (Sage)

%o def L(k, n):

%o if n < 0:

%o return -1

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

%o for i in range(1, n+1):

%o a[:] = a[1:] + [sum(a)]

%o return a[-1]

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

%K easy,nonn,tabl

%O 1,5

%A _Jonathan Vos Post_, Nov 21 2006

%E Corrected by _Freddy Barrera_, Jan 10 2019