A137176 Hyperlucas number array T(r,n) = L(n)^(r), read by ascending antidiagonals (r >= 0, n >= 0).

0, 0, 1, 0, 1, 3, 0, 1, 4, 4, 0, 1, 5, 8, 7, 0, 1, 6, 13, 15, 11, 0, 1, 7, 19, 28, 26, 18, 0, 1, 8, 26, 47, 54, 44, 29, 0, 1, 9, 34, 73, 101, 98, 73, 47, 0, 1, 10, 43, 107, 174, 199, 171, 120, 76, 0, 1, 11, 53, 150, 281, 373, 370, 291, 196, 123

Offset

0,6

Comments

In Theorem 17, Dil and Mezo (2008) connect the hyperlucas numbers (this array) with the incomplete Lucas numbers (A324242). - Petros Hadjicostas, Sep 03 2019

Links

Table of n, a(n) for n=0..65.

Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, arXiv:0803.4388 [math.NT], 2008.

Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, Applied Mathematics and Computation 206(2) (2008), 942-951.

Eric W. Weisstein, Steenrod Algebra.

Formula

T(r,n) = L(n)^(r) = Apply partial sum operator r times to Lucas numbers A000204.

From Petros Hadjicostas, Sep 03 2019: (Start)

T(r, n) = L(n)^(r) = Sum_{k = 0..n} L(k)^(r-1) for r >= 1, with T(0,n) = L(n)^(0) = L(n) = A000204(n), T(r,0) = L(0)^(r) = 0, and T(r,1) = L(1)^(r) = 1. (See Definition 13 in Dil and Mezo (2008).)

G.f. for row r: Sum_{n >= 0} L(n)^(r)*t^n = t * (1+2*t)/((1-t-t^2) * (1-t)^r). (Corrected from Proposition 14 in Dil and Mezo (2008).)

(End)

Example

The array T(r,n) = L(n)^(r) begins:
.....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS
r=0..|.0.|.1.|..3.|..4.|...7.|..11.|...18.|...29.|....47.|....76.|...123.|.A000204
r=1..|.0.|.1.|..4.|..8.|..15.|..26.|...44.|...73.|...120.|...196.|...319.|.A027961
r=2..|.0.|.1.|..5.|.13.|..28.|..54.|...98.|..171.|...291.|...487.|...806.|.A023537
r=3..|.0.|.1.|..6.|.19.|..47.|.101.|..199.|..370.|...661.|..1148.|..1954.|.A027963
r=4..|.0.|.1.|..7.|.26.|..73.|.174.|..373.|..743.|..1404.|..2552.|..4506.|.A027964
r=5..|.0.|.1.|..8.|.34.|.107.|.281.|..654.|.1397.|..2801.|..5353.|..9859.|.A053298
r=6..|.0.|.1.|..9.|.43.|.150.|.431.|.1085.|.2482.|..5283.|.10636.|.20495.|.new
r=7..|.0.|.1.|.10.|.53.|.203.|.634.|.1719.|.4201.|..9484.|.20120.|.40615.|.new
r=8..|.0.|.1.|.11.|.64.|.267.|.901.|.2620.|.6821.|.16305.|.36425.|.77040.|.new
r=9..|.0.|.1.|.12.|.76.|.343.|1244.|.3864.|10685.|.26990.|.63415.|140455.|.new
For example, T(4,5) = L(5)^(4) = L(0)^(3) + L(1)^(3) + L(2)^(3) + L(3)^(3) + L(4)^(3) + L(5)^(3) = 0 + 1 + 6 + 19 + 47 + 101 = 174. - Petros Hadjicostas, Sep 03 2019

Maple

L:= proc(r, n) option remember; `if`(n=0, 0, `if`(r=0,
      `if`(n<3, 2*n-1, L(0, n-2)+L(0, n-1)), L(r-1, n)+L(r, n-1)))
    end:
seq(seq(L(d-n, n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 03 2019

Mathematica

L[r_, n_] := L[r, n] = If[n == 0, 0, If[r == 0, If[n < 3, 2n-1, L[0, n-2] + L[0, n-1]], L[r-1, n] + L[r, n-1]]];
Table[L[d-n, n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

Crossrefs

Cf. A000204, A027961, A023537, A027963, A027964, A053298, A123736, A136431.

Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.

Sequence in context: A106356 A091613 A039727 * A143949 A124323 A250104

Adjacent sequences: A137173 A137174 A137175 * A137177 A137178 A137179

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Apr 04 2008

Status

approved