A286541 -id:A286541 - cp's OEIS frontend

A286560 Compound filter (summands of A004001 & summands of A005185): a(n) = P(A286541(n), A286559(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

0, 0, 1, 2, 5, 41, 71, 71, 198, 313, 484, 922, 1153, 1201, 2105, 1565, 2588, 4046, 5001, 7443, 7443, 8851, 10671, 19589, 16570, 16935, 22254, 25313, 25313, 25313, 42891, 28793, 32768, 52795, 65504, 59178, 73355, 89033, 88632, 107660, 129045, 129045, 153471, 167646, 167646, 182446, 182446, 336130, 197244, 233297, 330472, 307358, 270167, 355325, 378466, 332156

Offset: 1

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A004001, A005185, A284019, A286541, A286559, A286569.

(define (A286560 n) (if (<= n 2) 0 (* (/ 1 2) (+ (expt (+ (A286541 n) (A286559 n)) 2) (- (A286541 n)) (- (* 3 (A286559 n))) 2))))

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A286541(n)+A286559(n))^2) - A286541(n) - 3*A286559(n)).

A286559 Compound filter (the left & right summand of Hofstadter Q-sequence): a(n) = P(Q(n-Q(n-1)), Q(n-Q(n-2))), 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, 2, 2, 5, 8, 8, 13, 13, 13, 25, 24, 25, 41, 32, 41, 50, 50, 61, 61, 61, 61, 113, 84, 86, 113, 113, 113, 113, 181, 128, 129, 181, 200, 163, 182, 221, 200, 221, 242, 242, 265, 265, 265, 265, 265, 481, 263, 290, 420, 363, 314, 422, 420, 365, 481, 420, 481, 481, 481, 481, 761, 512, 452, 687, 577, 513, 722, 761, 650, 687, 762, 723, 760, 722, 842, 760, 801

Offset: 1

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..67390
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A005185, A283467, A286541, A286560.

(define (A286559 n) (if (<= n 2) 0 (* (/ 1 2) (+ (expt (+ (A005185 (- n (A005185 (- n 1)))) (A005185 (- n (A005185 (- n 2))))) 2) (- (A005185 (- n (A005185 (- n 1))))) (- (* 3 (A005185 (- n (A005185 (- n 2)))))) 2))))

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A005185(n-A005185(n-1))+A005185(n-A005185(n-2)))^2) - A005185(n-A005185(n-1)) - 3*A005185(n-A005185(n-2))).

A302780 Restricted growth sequence transform of 4-tuple [H(H(n-1)), H(n-H(n-1)), Q(n-Q(n-1)), Q(n-Q(n-2))] where H = A004001 and Q = A005185.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 47, 50, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 80

Offset: 1

Antti Karttunen, Apr 27 2018

nonn

Restricted growth sequence transform of A286560: a filter sequence which includes both the summands of A004001 and the summands of A005185.
For all i, j: a(i) = a(j) => b(i) = b(j), where b is a sequence like A087740, A284019, A286569 or A302779.
For n > 1000 the duplicates get rare. In range [1000, 65536] there are only three cases: a(1353) = a(1354) = 1319, a(39361) = a(39362) = 39326, and a(46695) = a(46696) = 46659.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for Hofstadter-type sequences

Cf. A004001, A005185, A286560.

Cf. also A087740, A284019, A286541, A286569, A302779.

up_to = 65537;
first_n_of_A004001(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); (v); }; \\ Charles R Greathouse IV, Feb 26 2017
v004001 = first_n_of_A004001(up_to);
A004001(n) = v004001[n];
first_n_of_A005185(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[k-v[k-1]]+v[k-v[k-2]]); (v); }; \\
v005185 = first_n_of_A005185(up_to);
A005185(n) = v005185[n];
Aux302780(n) = if(n<3,0,[A004001(A004001(n-1)), A004001(n-A004001(n-1)), A005185(n-A005185(n-1)), A005185(n-A005185(n-2))]);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302780(n))),"b302780.txt");

cp's OEIS Frontend

A286560 Compound filter (summands of A004001 & summands of A005185): a(n) = P(A286541(n), A286559(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

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A286559 Compound filter (the left & right summand of Hofstadter Q-sequence): a(n) = P(Q(n-Q(n-1)), Q(n-Q(n-2))), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

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A302780 Restricted growth sequence transform of 4-tuple [H(H(n-1)), H(n-H(n-1)), Q(n-Q(n-1)), Q(n-Q(n-2))] where H = A004001 and Q = A005185.

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