A378943 Numbers obtained from the tribonacci triangle formed by the number of connection points in the paths obtained by Pell walk on the square grid.

2, 3, 7, 13, 25, 46, 86, 158, 292, 537, 989, 1819, 3347, 6156, 11324, 20828, 38310, 70463, 129603, 238377, 438445, 806426, 1483250, 2728122, 5017800, 9229173, 16975097, 31222071, 57426343, 105623512, 194271928, 357321784, 657217226, 1208810939, 2223349951, 4089378117

Offset

1,1

Comments

The study was inspired by the stepping system discussed in the study titled Pell Walks by Thomas Koshy (2014). In the study, the points formed in this walk are called connected points. There is a direct connection between the Pell-Lucas number sequence and the tribonacci number sequence.

The connection points obtained from Thomas Koshy (2014) were examined on the Pascal triangle. The intermediate sequences of this A number sequence were examined. The counting numbers were extracted from the sequence. The resulting new number sequence is 1,1,4,9,20,40,79,150,283,523... This is called the D number sequence. It is similar to A023607, which is a convolution of Fibonacci and Lucas numbers.

References

Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer Science-Business Media New York, 2014, pp. 227-253.

Links

Table of n, a(n) for n=1..36.

Esra Namlı, Tribonacci ve lucas sayılarının üreteç fonksiyonları yardımıyla oluşan bazı özdeşlikler, Master's thesis, Bursa Uludag University (in Turkey), 2020, pp. 2-16.

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1,-1).

Formula

a(1)=2, a(2)=3, a(3)=7, and a(n+3) = a(n)+a(n+1)+a(n+2)+2 if n is odd, a(n+3) = a(n)+a(n+1)+ a(n+2)+1 if n is even.

From Elmo R. Oliveira, Apr 12 2026: (Start)

a(n) = a(n-1) + 2*a(n-2) - a(n-4) - a(n-5).

G.f.: x*(2 + x)/((-1 + x^2)*(-1 + x + x^2 + x^3)). (End)

Example

For n = 4 a(4) = a(3) + a(2) + a(1) + 1 = 2 + 3 + 7 + 1 = 13.
For n = 5 a(5) = a(4) + a(3) + a(2) + 2 = 3 + 7 + 13 + 2 = 25.

Mathematica

LinearRecurrence[{1, 2, 0, -1, -1}, {2, 3, 7, 13, 25}, 36] (* Hugo Pfoertner, Jan 09 2025 *)

Prog

(Python)
def a(n, memo={}):
    if n == 1:
        return 2
    elif n == 2:
        return 3
    elif n == 3:
        return 7
    if n in memo:
        return memo[n]
    if n % 2 == 1:
        memo[n] = a(n - 3, memo) + a(n - 2, memo) + a(n - 1, memo) + 2
    else:
        memo[n] = a(n - 3, memo) + a(n - 2, memo) + a(n - 1, memo) + 1
    return memo[n]
sequence = [a(n) for n in range(1, n)]
print(sequence)

Crossrefs

Cf. A000045, A000073, A000129, A001333, A023607.

Sequence in context: A213968 A213967 A128695 * A024504 A256494 A344432

Adjacent sequences: A378940 A378941 A378942 * A378944 A378945 A378946

Keyword

nonn,easy

Author

Irem Eski, Arzu Caglar, Cansu Aytar, and Emre Kaya, Dec 11 2024

Extensions

Edited - N. J. A. Sloane, Jan 23 2025

Status

approved