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A381534 A084849 interleaved with positive even numbers.

%I A381534 #29 Jul 07 2025 16:13:05
%S A381534 1,2,4,4,11,6,22,8,37,10,56,12,79,14,106,16,137,18,172,20,211,22,254,
%T A381534 24,301,26,352,28,407,30,466,32,529,34,596,36,667,38,742,40,821,42,
%U A381534 904,44,991,46,1082,48,1177,50,1276,52
%N A381534 A084849 interleaved with positive even numbers.
%C A381534 To construct the sequence, we start with two 1’s on separate lines:
%C A381534   1,
%C A381534   1,
%C A381534 Next, we zigzag natural numbers between the lines, leaving spaces:
%C A381534   1,_,3,_,5,_,7,_,9,_,11,_...
%C A381534   1,2,_,4,_,6,_,8,_,10,_...
%C A381534 To fill the spaces, we insert the sum of the numbers in the previous column:
%C A381534   1, 2, 3, 7, 5, 16, 7, 29, 9, 46, 11, 67...
%C A381534   1, 2, 4, 4, 11, 6, 22, 8, 37, 10, 56,...
%C A381534 a(n) is the second sequence. The first sequence is A354008(k), for k > 2.
%C A381534 The first sequence is odd numbers interleaved with A130883. (From M. F. Hasler via Seqfan.)
%C A381534 The numbers we find by adding the columns are: 2,4,7,11,16,22,29,37,46,56,67,…. which is A000124 (n >= 1). The sequence is constructed by alternating the even indexed terms of this sequence (1,4,11,22,37,56…) with the numbers (added by “zigzag” to the second row before we add the columns to get the missing numbers); namely the even numbers 2*n (n >= 1). Therefore, the sequence seems to be A000124(2n)  (n>=0), interleaved with A005843(n); (n>=1). (From _David James Sycamore_ via Seqfan.)
%H A381534 Harvey P. Dale, <a href="/A381534/b381534.txt">Table of n, a(n) for n = 1..1000</a>
%H A381534 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A381534 G.f.: -x*(-2*x^4+2*x^3-x^2-2*x-1)/(-x^6+3*x^4-3*x^2+1). - _Michel Marcus_ Feb 27 2025
%e A381534 A084849(0) = 1, so a(1) = 1.
%e A381534 a(2) is the first positive even number, 2.
%t A381534 LinearRecurrence[{0,3,0,-3,0,1},{1,2,4,4,11,6},60] (* _Harvey P. Dale_, May 09 2025 *)
%Y A381534 Cf. A354008, A005843, A000124.
%K A381534 nonn,easy
%O A381534 1,2
%A A381534 _Ali Sada_, Feb 26 2025