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

Author

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

Keywords

• nonn
• easy

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.

Examples

			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.
		

References

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

Links

• E. Namlı, Tribonacci ve lucas sayılarının üreteç fonksiyonları yardımıyla oluşan bazı özdeşlikler, Master's thesis, Bursa Uludag University (Turkey), 2020, 2-16.
• Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1,-1).

Crossrefs

Cf. A000045, A000073, A001333, A000129.

Programs

Mathematica

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

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)

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.

Extensions

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