A381534 A084849 interleaved with positive even numbers.

1, 2, 4, 4, 11, 6, 22, 8, 37, 10, 56, 12, 79, 14, 106, 16, 137, 18, 172, 20, 211, 22, 254, 24, 301, 26, 352, 28, 407, 30, 466, 32, 529, 34, 596, 36, 667, 38, 742, 40, 821, 42, 904, 44, 991, 46, 1082, 48, 1177, 50, 1276, 52

Offset: 1

Author

Ali Sada, Feb 26 2025

Keywords

• nonn
• easy

Comments

To construct the sequence, we start with two 1’s on separate lines:

1,

1,

Next, we zigzag natural numbers between the lines, leaving spaces:

1,,3,,5,,7,,9,,11,...

1,2,,4,,6,,8,,10,_...

To fill the spaces, we insert the sum of the numbers in the previous column:

1, 2, 3, 7, 5, 16, 7, 29, 9, 46, 11, 67...

1, 2, 4, 4, 11, 6, 22, 8, 37, 10, 56,...

a(n) is the second sequence. The first sequence is A354008(k), for k > 2.

The first sequence is odd numbers interleaved with A130883. (From M. F. Hasler via Seqfan.)

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.)

Examples

			A084849(0) = 1, so a(1) = 1.
a(2) is the first positive even number, 2.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A354008, A005843, A000124.

Programs

Mathematica

LinearRecurrence[{0,3,0,-3,0,1},{1,2,4,4,11,6},60] (* Harvey P. Dale, May 09 2025 *)

Formula

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