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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

%I

%S 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,

%T 48,24,11,128,255,248,126,106,54,26,13,256,511,503,254,227,116,57,29,

%U 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

%N Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

%C 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.

%C 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).

%C 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).

%H Antti Karttunen, <a href="/A265901/b265901.txt">Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array</a>

%H T. Kubo and R. Vakil, <a href="http://dx.doi.org/10.1016/0012-365X(94)00303-Z">On Conway's recursive sequence</a>, Discr. Math. 152 (1996), 225-252.

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

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

%e The top left corner of the array:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%e ...

%o (Scheme)

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

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

%Y Inverse permutation: A267102.

%Y Transpose: A265903.

%Y Cf. A265900 (main diagonal).

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

%Y Cf. A004001, A051135, A088359, A087686.

%Y Column 1: A188163.

%Y Column 2: A266109.

%Y Row 1: A000079 (2^n).

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

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

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

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

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

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

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

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

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

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

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

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

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

%Y Cf. also permutations A267111, A267112.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Dec 18 2015

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Last modified November 17 00:08 EST 2019. Contains 329209 sequences. (Running on oeis4.)