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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.

%I A378943 #39 Apr 17 2025 16:50:57
%S A378943 2,3,7,13,25,46,86,158,292,537,989,1819,3347,6156,11324,20828,38310,
%T A378943 70463,129603,238377,438445,806426,1483250,2728122,5017800,9229173,
%U A378943 16975097,31222071,57426343,105623512,194271928,357321784,657217226,1208810939,2223349951,4089378117
%N 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.
%C A378943 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.
%C A378943 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.
%D A378943 T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer Science-Business Media New York, 2014, 227-253.
%H A378943 E. Namlı, <a href="http://hdl.handle.net/11452/11494">Tribonacci ve lucas sayılarının üreteç fonksiyonları yardımıyla oluşan bazı özdeşlikler</a>, Master's thesis, Bursa Uludag University (Turkey), 2020, 2-16.
%H A378943 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,0,-1,-1).
%F A378943 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.
%e A378943 For n = 4 a(4) = a(3) + a(2) + a(1) + 1 = 2 + 3 + 7 + 1 = 13.
%e A378943 For n = 5 a(5) = a(4) + a(3) + a(2) + 2 = 3 + 7 + 13 + 2 = 25.
%t A378943 LinearRecurrence[{1, 2, 0, -1, -1}, {2, 3, 7, 13, 25}, 36] (* _Hugo Pfoertner_, Jan 09 2025 *)
%o A378943 (Python)
%o A378943 def a(n, memo={}):
%o A378943     if n == 1:
%o A378943         return 2
%o A378943     elif n == 2:
%o A378943         return 3
%o A378943     elif n == 3:
%o A378943         return 7
%o A378943     if n in memo:
%o A378943         return memo[n]
%o A378943     if n % 2 == 1:
%o A378943         memo[n] = a(n - 3, memo) + a(n - 2, memo) + a(n - 1, memo) + 2
%o A378943     else:
%o A378943         memo[n] = a(n - 3, memo) + a(n - 2, memo) + a(n - 1, memo) + 1
%o A378943     return memo[n]
%o A378943 sequence = [a(n) for n in range(1, n)]
%o A378943 print(sequence)
%Y A378943 Cf. A000045, A000073, A001333, A000129.
%K A378943 nonn,easy
%O A378943 1,1
%A A378943 _Irem Eski_, _Arzu Caglar_, _Cansu Aytar_, and _Emre Kaya_, Dec 11 2024
%E A378943 Edited - _N. J. A. Sloane_, Jan 23 2025