A347839 An array of the positive integers congruent to 2 modulo 3 (A016789), read by antidiagonals upwards, giving the present triangle.

2, 5, 8, 11, 20, 32, 14, 44, 80, 128, 17, 56, 176, 320, 512, 23, 68, 224, 704, 1280, 2048, 26, 92, 272, 896, 2816, 5120, 8192, 29, 104, 368, 1088, 3584, 11264, 20480, 32768, 35, 116, 416, 1472, 4352, 14336, 45056, 81920, 131072, 38, 140, 464, 1664, 5888, 17408, 57344, 180224, 327680, 524288

Offset: 1

Author

Wolfdieter Lang, Oct 21 2021

Keywords

• nonn
• easy
• tabl

Comments

This array a = (a(k, n))_{k >= 1,n >= 0} is underlying array A of A347834. See the first formula. It has a simple recurrence for the rows k, given the first column a(k, 0) = A347838(k), which lists the positive integers congruent to {2, 5, 11} modulo 12.

In the array one can add the negative of the powers of 4 as row for k = 0, i.e., -A000302(n), for n >= 0.

All positive numbers congruent to 2 modulo 3 (A017617) appear once in this array. Proof from the array A of A347834 of the positive integers congruent to {1,3,5,7} modulo 8, and the present first formula: The members of column n = 0 give all the positive integers congruent to {2, 5, 11} modulo 12 once, and the members of columns n >= 1 give all the positive integers congruent to 8 modulo 12 (A017617) once. These members combined lead to the positive integers congruent to 2 modulo 3.

Examples

			The array a(k, n) begins:
k \ n  0   1   2    3    4     5      6      7       8       9       10 ...
---------------------------------------------------------------------------
1:     2   8  32  128  512  2048   8192  32768  131072  524288  2097152 ...
2:     5  20  80  320 1280  5120  20480  81920  327680 1310720  5242880 ...
3:    11  44 176  704 2816 11264  45056 180224  720896 2883584 11534336 ...
4:    14  56 224  896 3584 14336  57344 229376  917504 3670016 14680064 ...
5:    17  68 272 1088 4352 17408  69632 278528 1114112 4456448 17825792 ...
6:    23  92 368 1472 5888 23552  94208 376832 1507328 6029312 24117248 ...
7:    26 104 416 1664 6656 26624 106496 425984 1703936 6815744 27262976 ...
8:    29 116 464 1856 7424 29696 118784 475136 1900544 7602176 30408704 ...
9:    35 140 560 2240 8960 35840 143360 573440 2293760 9175040 36700160 ...
10:   38 152 608 2432 9728 38912 155648 622592 2490368 9961472 39845888 ...
...
----------------------------------------------------------------------------
The triangle t(n,k) begins:
k \ n  0   1   2    3    4     5     6      7      8      9 ...
---------------------------------------------------------------
1:     2
2:     5   8
3:    11  20  32
4:    14  44  80  128
5:    17  56 176  320  512
6:    23  68 224  704 1280  2048
7:    26  92 272  896 2816  5120  8192
8:    29 104 368 1088 3584 11264 20480  32768
9:    35 116 416 1472 4352 14336 45056  81920 131072
10:   38 140 464 1664 5888 17408 57344 180224 327680 524288
...
-----------------------------------------------------------------
		

Crossrefs

Cf. A347834, A016789, A017617.

The rows k are given by -A000302 (for k=0), A004171, A003947(n+1), A002089, 2*A002042, ...

The columns n are given by 4^n*A347838 for n >= 0.

Programs

Maple

A := (n, k) -> 4^n*(3*(k + iquo(k, 3)) - 1):
for k from 1 to 10 do seq(A(n, k), n = 0..10) od;
# Alternatively:
gf  := n -> (4^n*((z*(z*(7*z + 3) + 3) - 1)))/((z - 1)^2*(1 + z + z^2)):
ser := n -> series(gf(n), z, 12):
col := (n, len) -> seq(coeff(ser(n), z, k), k = 1..len):
seq(print(col(n, 10)), n = 0..10); # Peter Luschny, Oct 26 2021

Mathematica

A[n_, k_] := 4^n (3(k + Quotient[k, 3]) - 1);
Table[A[n-k, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 07 2021, from Maple code *)

Formula

Array a:

a(k, n) = (3*A(k, n) + 1)/2, with the array A from A347834, for k >= 1, and n >= 0.

a(k, n) = 4^n*A347838(k) = 4^n*(2 + 3*k + 3*floor((k + 1)/3)).

Recurrence for rows k: a(k, n) = 4*a(k, n-1), for n >= 1, with a(k, 0) = A347838(k).

O.g.f.: expansion in z gives the o.g.f.s for rows k, also for k = 0: -A000302; expansion in x gives the o.g.f.s for columns n.

G(z, x) = (-1 + 3*z + 3*z^2 + 7*z^3)/((1 - z)*(1 - z^3)*(1 - 4*x)).

Triangle t:

t(k, n) = a(k-n, n), for k >= 1, and n = 0, 1, ..., k-1.