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A125127
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Array L(k,n) read by antidiagonals: k-step Lucas numbers.
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7
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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
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OFFSET
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1,5
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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