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A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)). 15
1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).

Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array

T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Index entries for Hofstadter-type sequences

Index entries for sequences that are permutations of the natural numbers

FORMULA

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

EXAMPLE

The top left corner of the array:

   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...

   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...

   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...

   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...

   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...

  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...

  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...

  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...

  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...

  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...

  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...

  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...

  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...

  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...

  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...

  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...

  ...

PROG

(Scheme)

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))

(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

CROSSREFS

Inverse permutation: A267102.

Transpose: A265903.

Cf. A265900 (main diagonal).

Cf. A162598 (row index of n in array), A265332 (column index of n in array).

Cf. A004001, A051135, A088359, A087686.

Column 1: A188163.

Column 2: A266109.

Row 1: A000079 (2^n).

Row 2: A000225 (2^n - 1, from 3 onward).

Row 3: A000325 (2^n - n, from 5 onward).

Row 4: A000918 (2^n - 2, from 6 onward).

Row 5: A084634 (?, from 9 onward).

Row 6: A132732 (2^n - 2n + 2, from 10 onward).

Row 7: A000295 (2^n - n - 1, from 11 onward).

Row 8: A036563 (2^n - 3).

Row 9: A084635 (?, from 17 onward).

Row 12: A048492 (?, from 20 onward).

Row 13: A249453 (?, from 22 onward).

Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).

Row 15: A000247 (2^n - n - 2, from 25 onward).

Row 16: A028399 (2^n - 4).

Cf. also permutations A267111, A267112.

Sequence in context: A026237 A308301 A125150 * A257801 A257726 A183089

Adjacent sequences:  A265898 A265899 A265900 * A265902 A265903 A265904

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Dec 18 2015

STATUS

approved

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Last modified October 21 14:20 EDT 2019. Contains 328301 sequences. (Running on oeis4.)