A190995 -id:A190995 - cp's OEIS frontend

A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.

0, 1, 1, 3, 2, 2, 4, 8, 3, 5, 3, 5, 7, 9, 11, 21, 5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55, 8, 12, 10, 18, 12, 16, 18, 34, 8, 12, 10, 18, 12, 16, 18, 34, 18, 26, 24, 44, 22, 30, 32, 60, 30, 46, 36, 64, 50, 66, 76, 144, 13, 19, 17, 31, 17, 23

Offset: 0

Philippe Deléham, Nov 06 2013

The rule for constructing the tree is the following:
.....x
.....|
.....y
..../ \
..y+x..3y-x
and the tree begins like this:
.........0......
.........|......
.........1......
......./   \....
......1.....3....
...../ \.../ \...
....2...2.4...8..
and so on.
Column 1 :  0,  1,  1,  2,  3,   5,   8, ... = A000045 (Fibonacci numbers).
Column 2 :  3,  2,  5,  7, 12,  19,  31, ... = A013655.
Column 3 :  4,  3,  7, 10, 17,  27,  44, ... = A022120.
Column 4 :  8,  5, 13, 18, 31,  49,  80, ... = A022138.
Column 5 :  7,  5, 12, 17, 29,  46,  75, ... = A022137.
Column 6 :  9,  7, 16, 23, 39,  62, 101, ... = A190995.
Column 7 : 11,  7, 18, 25, 43,  68, 111, ... = A206419.
Column 8 : 21, 13, 34, 47, 81, 128, 209, ... = ?
Column 9 : 11,  8, 19, 27, 46,  73, 119, ... = A206420.
The lengths of the rows are 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ... = A011782 .
The final numbers in the rows are 0, 1, 3, 8, 21, 55, 144, ... = A001906.
The middle numbers in the rows are 1, 2, 5, 13, 34, 89, ... =  A001519 .
Row sums for n>=1: 1, 4, 16, 64, 256, 1024, ... = 4^(n-1).

			The successive rows are:
  0
  1
  1, 3
  2, 2, 4, 8
  3, 5, 3, 5, 7, 9, 11, 21
  5, 7, 7, 13, 5, 7, 7, 13, 11, 17, 13, 23, 19, 25, 29, 55
  ...

Reinhard Zumkeller, Rows n = 0..13 of triangle, flattened

Cf. A230872, A230873.

Cf. A231330, A231331, A231335.

data Dtree = Dtree Dtree (Integer, Integer) Dtree
a230871 n k = a230871_tabf !! n !! k
a230871_row n = a230871_tabf !! n
a230871_tabf = [0] : map (map snd) (rows $ deleham (0, 1)) where
   rows (Dtree left (x, y) right) =
        [(x, y)] : zipWith (++) (rows left) (rows right)
   deleham (x, y) = Dtree
           (deleham (y, y + x)) (x, y) (deleham (y, 3 * y - x))
-- Reinhard Zumkeller, Nov 07 2013

T:= proc(n, k) T(n, k):= `if`(k=1 and n<2, n, (d->(1+2*d)*
      T(n-1, r)+(1-2*d)*T(n-2, iquo(r+1, 2)))(irem(k+1, 2, 'r')))
    end:
seq(seq(T(n, k), k=1..max(1, 2^(n-1))), n=0..7); # Alois P. Heinz, Nov 07 2013

T[n_, k_] := T[n, k] = If[k==1 && n<2, n, Function[d, r = Quotient[k+1, 2]; (1+2d) T[n-1, r] + (1-2d) T[n-2, Quotient[r+1, 2]]][Mod[k+1, 2]]];
Table[T[n, k], {n, 0, 7}, {k, 1, Max[1, 2^(n-1)]}] // Flatten (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

Incorrect formula removed by Michel Marcus, Sep 23 2023

A354265 Array read by ascending antidiagonals for n >= 0 and k >= 0. Generalized Lucas numbers, L(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)), where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2.

Original entry on oeis.org

2, 3, 1, 4, 4, 3, 5, 7, 7, 4, 6, 10, 11, 11, 7, 7, 13, 15, 18, 18, 11, 8, 16, 19, 25, 29, 29, 18, 9, 19, 23, 32, 40, 47, 47, 29, 10, 22, 27, 39, 51, 65, 76, 76, 47, 11, 25, 31, 46, 62, 83, 105, 123, 123, 76, 12, 28, 35, 53, 73, 101, 134, 170, 199, 199, 123

Offset: 0

Peter Luschny, May 29 2022

The definition declares the Lucas numbers for all integers n and k. It gives the classical Lucas numbers as L(0, n) = Lucas(n), where Lucas(n) = A000032(n) is extended in the usual way for negative n.

			Array starts:
[0]  2,  1,  3,  4,   7,  11,  18,  29,  47,   76, ... A000032
[1]  3,  4,  7, 11,  18,  29,  47,  76, 123,  199, ... A000032 (shifted)
[2]  4,  7, 11, 18,  29,  47,  76, 123, 199,  322, ... A000032 (shifted)
[3]  5, 10, 15, 25,  40,  65, 105, 170, 275,  445, ... A022088
[4]  6, 13, 19, 32,  51,  83, 134, 217, 351,  568, ... A022388
[5]  7, 16, 23, 39,  62, 101, 163, 264, 427,  691, ... A190995
[6]  8, 19, 27, 46,  73, 119, 192, 311, 503,  814, ... A206420
[7]  9, 22, 31, 53,  84, 137, 221, 358, 579,  937, ... A206609
[8] 10, 25, 35, 60,  95, 155, 250, 405, 655, 1060, ...
[9] 11, 28, 39, 67, 106, 173, 279, 452, 731, 1183, ...

Peter Luschny, The Fibonacci Function.

Cf. A000032, A000204, A022088, A022388, A190995, A206420, A206609.

Cf. A352744.

const FibMem = Dict{Int,Tuple{BigInt,BigInt}}()
function FibRec(n::Int)
    get!(FibMem, n) do
        n == 0 && return (BigInt(0), BigInt(1))
        a, b = FibRec(div(n, 2))
        c = a * (b * 2 - a)
        d = a * a + b * b
        iseven(n) ? (c, d) : (d, c + d)
    end
end
function Lucas(n, k)
    k ==  0 && return BigInt(n + 2)
    k == -1 && return BigInt(2 * n - 1)
    k <   0 && return (-1)^k * Lucas(1 - n, -k - 2)
    a, b = FibRec(k)
    c, d = FibRec(k - 1)
    n * (2 * a + b) + 2 * c + d
end
for n in -6:6
    println([Lucas(n, k) for k in -6:6])
end

phi := (1 + sqrt(5))/2: psi := (1 - sqrt(5))/2:
L := (n, k) -> phi^(k+1)*(n - psi) + psi^(k+1)*(n - phi):
seq(seq(simplify(L(n-k, k)), k = 0..n), n = 0..10);

L[n_, k_] := With[{c = Pi/2 + I*ArcCsch[2]},
             I^k Sec[c] (n Cos[c (k + 1)] - I Cos[c k]) ];
Table[Simplify[L[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm
(* Alternative: *)
L[n_, k_] := n*LucasL[k + 1] + LucasL[k];
Table[Simplify[L[n, k]], {n, 0, 6}, {k, 0, 6}] // TableForm

Functional equation extends Cassini's theorem:
L(n, k) = (-1)^k*L(1 - n, -k - 2).
L(n, k) = n*Lucas(k + 1) + Lucas(k).
L(n, k) = L(n, k-1) + L(n, k-2).
L(n, k) = i^k*sec(c)*(n*cos(c*(k + 1)) - i*cos(c*k)), where c = Pi/2 + i*arccsch(2), for all n, k in Z.
Using the generalized Fibonacci numbers F(n, k) = A352744(n, k),
L(n, k) = F(n, k+1) + F(n, k) + F(n, k-1) + F(n, k-2).

A190996 Fibonacci sequence beginning 10, 7.

Original entry on oeis.org

10, 7, 17, 24, 41, 65, 106, 171, 277, 448, 725, 1173, 1898, 3071, 4969, 8040, 13009, 21049, 34058, 55107, 89165, 144272, 233437, 377709, 611146, 988855, 1600001, 2588856, 4188857, 6777713, 10966570, 17744283, 28710853, 46455136, 75165989, 121621125

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

For n >= 5, the number a(n-3) is the dimension of a commutative Hecke algebra of affine type D_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019
From Greg Dresden and Yiming Wu, Sep 10 2023: (Start)
For n >= 3, a(n) is the number of ways to tile this shape of length n+2 with squares and dominos:
                        
 ||___________________||
|||_|||_|||_|||_|||
  ||                   ||. (End)
For n >= 3, a(n) is the number of edge covers of the kayak paddle graphs KP(3,3,n-3), where we interpret KP(3,3,0) as two C_3's with one common vertex. - Feryal Alayont, Sep 28 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551.
Eric Weisstein's World of Mathematics, Kayak Paddle Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A190994, A190995.

[n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

seq(coeff(series((10-3*x)/(1-x-x^2),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Jan 22 2019

Mathematica
```
LinearRecurrence[{1, 1}, {10, 7}, 100]
```

a(n)=7*fibonacci(n)+10*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011

[7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

a(n) = (5 + 2*sqrt(5)/5)*((1 + sqrt(5))/2)^n + (5 - 2*sqrt(5)/5)*((1 - sqrt(5))/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = 7*Fibonacci(n) + 10*Fibonacci(n-1). - Charles R Greathouse IV, Jun 08 2011
G.f.: (10-3*x)/(1-x-x^2). - Colin Barker, Jan 11 2012
a(n) = 4*Fibonacci(n+1) + 3*LucasL(n). - G. C. Greubel, Oct 26 2022
a(n) = A000285(n)+3*A000285(n-1). - Feryal Alayont, Sep 28 2024
a(2*n) = A000285(n)^2 + A000285(n-1)^2. - Greg Dresden, Feb 28 2025

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A230871 Construct a triangle as in the Comments, read nodes from left to right starting at the root and proceeding downwards.

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A354265 Array read by ascending antidiagonals for n >= 0 and k >= 0. Generalized Lucas numbers, L(n, k) = (psi^(k - 1)*(phi + n) - phi^(k - 1)*(psi + n)), where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2.

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A190996 Fibonacci sequence beginning 10, 7.

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A354265 Array read by ascending antidiagonals for n >= 0 and k >= 0. Generalized Lucas numbers, L(n, k) = (psi^(k - 1)(phi + n) - phi^(k - 1)(psi + n)), where phi = (1 + sqrt(5))/2 and psi = (1 - sqrt(5))/2.