A283469 -id:A283469 - cp's OEIS frontend

A283470 a(n) = A004001(A004001(n-1)) XOR A004001(n-A004001(n-1)), a(1) = a(2) = 1.

1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 7, 7, 0, 0, 0, 0, 1, 0, 3, 2, 2, 1, 14, 14, 15, 15, 15, 0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 6, 7, 4, 4, 5, 4, 4, 3, 3, 3, 2, 29, 29, 30, 30, 30, 31, 31, 31, 31, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 4, 7, 4, 5, 10, 10, 11, 10, 9, 9, 14, 15, 15, 14, 14, 14, 13, 10, 11, 11, 10, 9, 9, 6, 6, 6, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A003987, A080677, A283468, A283469, A283472, A283471 (positions of zeros), A283473 (positions where coincides with A004001).

Cf. also A283677.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[BitXor[a[#], a[n - #]] &@ a[n - 1] + Boole[n <= 2], {n, 107}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283470 n) (if (<= n 2) 1 (A003987bi (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; A003987bi implements bitwise-XOR (see A003987). Code for A004001 given under that entry.

a(n) = A004001(A004001(n-1)) XOR A004001(A080677(n-1)), where XOR is bitwise-xor (A003987)
Other identities. For all n >= 1:
a(n) = A283469(n) - A283472(n).
A004001(n) = a(n) + 2*A283472(n).

Erroneous b-file replaced by a correct one - Antti Karttunen, Feb 24 2018

A283472 a(n) = A004001(A004001(n-1)) AND A004001(n-A004001(n-1)), a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 2, 2, 3, 0, 0, 4, 4, 4, 4, 4, 5, 4, 5, 5, 6, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 8, 8, 8, 10, 10, 10, 11, 11, 12, 12, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 16, 16, 17, 16, 17, 16, 17, 17, 16, 18, 18, 16, 16, 16, 17, 18, 18, 16, 16, 16, 17, 17, 17, 18, 20, 20, 20, 21, 22, 22, 24, 24, 24, 24

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004001, A004198, A283469, A283470, A283473 (positions of zeros).

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[BitAnd[a[#], a[n - #]] &@ a[n - 1] - Boole[n <= 2], {n, 97}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283472 n) (if (<= n 2) 0 (A004198bi (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; A004198bi implements bitwise-AND (see A004198). Code for A004001 given under that entry.

a(1) = a(2) = 0; for n > 2, a(n) = A004001(A004001(n-1)) AND A004001(A080677(n-1)), where AND is bitwise-and (A004198).
Other identities. For all n >= 1:
a(n) = A283469(n) - A283470(n).
A004001(n) = A283469(n) + a(n) = A283470(n) + 2*a(n).

A283473 Numbers n for which A004001(n) = A283470(n).

Original entry on oeis.org

1, 2, 5, 11, 12, 23, 24, 25, 26, 27, 50, 51, 52, 53, 54, 55, 56, 57, 58, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Equally, numbers n such that A004001(n) = A283469(n).

Antti Karttunen, Table of n, a(n) for n = 1..803

Positions of zeros in A283472.

Cf. A004001, A283469, A283470, A283471.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; With[{nn = 500}, Function[s, Select[Range@ nn, a@ # == s[[#]] &]]@ Table[BitXor[a[#], a[n - #]] &@ a[n - 1] + Boole[n <= 2], {n, nn}]] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

A283468 a(n) = A004001(A004001(n-1)) - A004001(n-A004001(n-1)), a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 1, 2, 2, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 2, 3, 3, 2, 3, 4, 4, 3, 4, 4, 3, 3, 3, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 3, 4, 4, 3, 4, 5, 6, 6, 5, 6, 7, 7, 6, 7, 7, 6, 6, 6, 5, 6, 7, 7, 6, 7, 7, 6, 6, 6, 5, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 18 2017

sign

The only negative terms seem to be -1's, occurring as a(1+(2^n)), for n >= 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004001, A080677, A283471 (positions of zeros), A283469, A283470, A283472.

Cf. also A283655.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[a[#] - a[n - #] &@ a[n - 1] + Boole[n <= 2], {n, 120}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283468 n) (if (<= n 2) 1 (- (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; Code for A004001 given under that entry.

a(1) = a(2) = 1; for n > 2, a(n) = A004001(A004001(n-1)) - A004001(A080677(n-1)).

A286541 Compound filter (the left & right summand of Hofstadter-Conway $10000 sequence): a(n) = P(A004001(A004001(n-1)), A004001(n-A004001(n-1))), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 5, 5, 8, 13, 19, 19, 25, 25, 25, 25, 32, 41, 51, 62, 62, 73, 86, 86, 99, 99, 99, 113, 113, 113, 113, 113, 128, 145, 163, 182, 202, 202, 222, 244, 267, 267, 290, 315, 315, 340, 340, 340, 366, 394, 394, 422, 422, 422, 451, 451, 451, 451, 481, 481, 481, 481, 481, 481, 512, 545, 579, 614, 650, 687, 687, 724, 763, 803, 844, 844, 885, 928, 972, 972

Offset: 1

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A004001, A286559, A286560.

Cf. A283468, A283469, A283470, A283472 and also A283471, A283473.

(define (A286541 n) (if (<= n 2) 0 (* (/ 1 2) (+ (expt (+ (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1))))) 2) (- (A004001 (A004001 (- n 1)))) (- (* 3 (A004001 (- n (A004001 (- n 1)))))) 2))))

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A004001(A004001(n-1))+A004001(n-A004001(n-1)))^2) - A004001(A004001(n-1)) - 3*A004001(n-A004001(n-1))).

cp's OEIS Frontend

A283470 a(n) = A004001(A004001(n-1)) XOR A004001(n-A004001(n-1)), a(1) = a(2) = 1.

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A283472 a(n) = A004001(A004001(n-1)) AND A004001(n-A004001(n-1)), a(1) = a(2) = 0.

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A283473 Numbers n for which A004001(n) = A283470(n).

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A283468 a(n) = A004001(A004001(n-1)) - A004001(n-A004001(n-1)), a(1) = a(2) = 1.

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A286541 Compound filter (the left & right summand of Hofstadter-Conway $10000 sequence): a(n) = P(A004001(A004001(n-1)), A004001(n-A004001(n-1))), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

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