A391114 Array read by antidiagonals upwards: A(n, k) = if(n mod 2 = 0, floor((n+2^k-2) / 2^k), n * (k*2^k+1) + 1).

4, 1, 10, 10, 1, 26, 2, 28, 1, 66, 16, 1, 76, 1, 162, 3, 46, 1, 196, 1, 386, 22, 2, 126, 1, 484, 1, 898, 4, 64, 1, 326, 1, 1156, 1, 2050, 28, 2, 176, 1, 806, 1, 2692, 1, 4610, 5, 82, 1, 456, 1, 1926, 1, 6148, 1, 10242, 34, 3, 226, 1, 1128, 1, 4486, 1, 13828, 1, 22530

Offset

1,1

Comments

Does for every column k iterating n -> A(n, k) always reach 1 (one cycle 1 -> k*2^k+2 -> k+1 -> ... -> 1)? For k=1 we have the Collatz or 3x+1 problem (see A006370).

Answer: no, because for the column k = 10 the sequence starts with 10242, 11, 112652, 111, 1136752, 1111, 11377752, 11112, 11, ... showing a cycle and doesn't reach 1. - Werner Schulte, Jan 05 2026

Links

Table of n, a(n) for n=1..66.

Formula

A(2*n+1, k) = A(2*n-1, k) + 2 * A002064(k).

A(n, k) = A(n-2, k) + A(n-2^k, k) - A(n-2-2^k, k).

G.f. of column k: ((k*2^k+2) * x + x^2 + (2*k*2^k+2) * (x^3 - x^(2^k+1)) / (1 - x^2) + k*2^k * x^(2^k+1)) / ((1 - x^2) * (1 - x^(2^k))).

Example

The array A(n, k) for k > 0 and n > 0 starts:
n \ k :  1   2   3   4    5    6     7     8     9     10     11     12      13
===============================================================================
   1  :  4  10  26  66  162  386   898  2050  4610  10242  22530  49154  106498
   2  :  1   1   1   1    1    1     1     1     1      1      1      1       1
   3  : 10  28  76 196  484 1156  2692  6148 13828  30724  67588 147460  319492
   4  :  2   1   1   1    1    1     1     1     1      1      1      1       1
   5  : 16  46 126 326  806 1926  4486 10246 23046  51206 112646 245766  532486
   6  :  3   2   1   1    1    1     1     1     1      1      1      1       1
   7  : 22  64 176 456 1128 2696  6280 14344 32246  71688 157704 344072  745480
   8  :  4   2   1   1    1    1     1     1     1      1      1      1       1
   9  : 28  82 226 586 1450 3466  8074 18442 41482  92170 202762 442378  958474
  10  :  5   3   2   1    1    1     1     1     1      1      1      1       1
  11  : 34 100 276 716 1772 4236  9868 22540 50700 112652 247820 540684 1171468
  12  :  6   3   2   1    1    1     1     1     1      1      1      1       1
  13  : 40 118 326 846 2094 5006 11662 26638 59918 133134 292878 638990 1384462
  etc.

Mathematica

A[n_, k_] := If[EvenQ[n], Floor[(n + 2^k - 2)/2^k], n*(k*2^k + 1) + 1]; Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 29 2025 *)

Prog

(PARI) A(n, k) = if(n%2==0, floor((n+2^k-2)/2^k), n*(k*2^k+1)+1)

Crossrefs

Cf. A000012 (row 2), A002064, A006370 (column 1), A391054 (column 2), A391075 (column 3).

Sequence in context: A349809 A182971 A062145 * A385108 A178216 A307529

Adjacent sequences: A391111 A391112 A391113 * A391115 A391116 A391117

Keyword

nonn,easy,tabl

Author

Werner Schulte, Nov 29 2025

Status

approved