BBC Russian

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)).

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: A373853 A357578 A125150 * A257801 A257726 A183089` `Adjacent sequences: A265898 A265899 A265900 * A265902 A265903 A265904`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Antti Karttunen, Dec 18 2015`
	STATUS	`approved`