1,5

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

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

Table begins:

1.|.1..1..1...1...1...1....1....1....1....1.

2.|.1..3..7..11..18..29...47...76..123..199.

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.

n-step Lucas number analogue 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 The simplest sequence of positive numbers: the all 1's sequence. L(2, n) = 2-step Lucas numbers = A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3. L(3, n) = 3-step Lucas numbers = A001644 a(n)=a(n-1)+a(n-2)+a(n-3), a(0)=3, a(1)=1, a(2)=3. 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 A125128, A125129 for comments on partial sums of diagonals.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125128, A125129.

Sequence in context: A114588 A121745 A089312 this_sequence A058735 A107294 A161788

Adjacent sequences: A125124 A125125 A125126 this_sequence A125128 A125129 A125130

easy,nonn,tabl

Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 21 2006