A005185 -id:A005185 - cp's OEIS frontend

A239913 a(n) = n - Q(n), where Q(n) is Hofstadter's Q-sequence A005185.

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 7, 8, 8, 9, 10, 11, 8, 11, 12, 11, 12, 13, 14, 11, 15, 16, 14, 14, 17, 17, 16, 18, 18, 18, 19, 19, 20, 21, 22, 23, 16, 25, 25, 21, 24, 27, 24, 25, 28, 25, 28, 27, 28, 29, 30, 23, 31, 34, 28, 32, 35, 30, 30, 34, 34, 33, 35

Offset: 1

N. J. A. Sloane, Apr 09 2014

nonn

Just as for A005185, it is not known if this sequence exists for all n.

Cf. A005185.

from functools import lru_cache
@lru_cache(maxsize=None)
def A239913(n): return n-1 if n<=2 else A239913(a:=1+A239913(n-1))+A239913(b:=2+A239913(n-2))-a-b+n # Chai Wah Wu, Apr 13 2023

For n >= 3, a(n) = a(a(n-1)+1) + a(a(n-2)+2) - a(n-1) - a(n-2) + n - 3. - Chai Wah Wu, Apr 13 2023

A280706 a(n) = Sum_{k=1..n} q(k+1-q(k)), where q(k) = A005185(k); partial sums of A283467.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 13, 16, 19, 23, 26, 30, 35, 39, 44, 49, 54, 60, 66, 72, 78, 86, 92, 100, 108, 116, 124, 132, 142, 150, 159, 169, 179, 189, 200, 211, 221, 232, 243, 254, 266, 278, 290, 302, 314, 330, 340, 354, 368, 380, 394, 410, 424, 438, 454, 468, 484, 500, 516, 532, 552, 568, 585, 606, 622, 639, 658, 678, 698, 719, 740

Offset: 1

Antti Karttunen after Altug Alkan's A284173, Mar 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for Hofstadter-type sequences

Cf. A005185, A284173.

Partial sums of A283467.

a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Accumulate@ Table[a[n + 1 - a[n]], {n, 72}] (* Michael De Vlieger, Mar 22 2017 *)

a(n) = if(n<3, 1, a(n - a(n - 1)) + a(n - a(n - 2)));
for(n=1, 72, print1(sum(k=1, n, a(k + 1 - a(k))),", ")) \\ Indranil Ghosh, Mar 22 2017

;; Code for A005185 given under that entry.
;; With memoization-macro definec:
(definec (A280706 n) (if (= 1 n) 1 (+ (A280706 (- n 1)) (A283467 n))))
;; As an explicit sum (slower):
(define (A280706 n) (add (lambda (k) (A005185 (- (+ k 1) (A005185 k)))) 1 n))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(1) = 1, for > 1, a(n) = A283467(n) + a(n-1).

A284173(n) = a(n) modulo n.

A283673 a(n) = lcm(q(n - q(n+1) + 2), q(n - q(n) + 2)) where q(n) = A005185(n).

Original entry on oeis.org

1, 2, 2, 2, 6, 6, 3, 3, 3, 4, 15, 4, 5, 20, 5, 30, 30, 6, 6, 6, 6, 8, 24, 24, 8, 8, 8, 8, 10, 72, 72, 10, 110, 90, 99, 11, 110, 11, 132, 132, 12, 12, 12, 12, 12, 16, 70, 154, 112, 48, 84, 112, 112, 14, 16, 112, 16, 16, 16, 16, 20, 272, 238, 357, 304, 272, 380, 20, 340, 357, 399

Offset: 1

Altug Alkan, Mar 14 2017

See the order of chaotic subsequences in scatterplot link.

			a(4) = lcm(A005185(4 - A005185(5) + 2), A005185(4 - A005185(4) + 2)) = lcm(A005185(3), A005185(3)) = lcm(2, 2) = 2.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A283673

Cf. A005185, A283672, A283677.

q[1] = q[2] = 1; q[n_] := q[n] = q[n - q[n - 1]] + q[n - q[n - 2]]; Table[LCM[q[n - q[n + 1] + 2], q[n - q[n] + 2]], {n, 71}] (* Indranil Ghosh, Mar 14 2017 *)

a=vector(1001); a[1]=a[2]=1; for(n=3, #a, a[n]=a[n-a[n-1]]+a[n-a[n-2]]); va = vector(1000, n, lcm(a[n+2-a[n+1]],a[n+2-a[n]]))

A284173 a(n) = (Sum_{k=1..n} q(k+1-q(k))) mod n where q(k) = A005185(k).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 5, 7, 9, 1, 2, 4, 7, 9, 12, 15, 0, 3, 6, 9, 12, 17, 20, 0, 4, 8, 12, 16, 22, 26, 31, 4, 9, 14, 20, 26, 31, 37, 3, 8, 14, 20, 26, 32, 38, 1, 4, 11, 18, 23, 30, 39, 46, 53, 6, 12, 20, 28, 36, 44, 56, 1, 9, 21, 28, 36, 46, 57, 68, 9, 20, 30, 39, 48, 60, 69, 2, 12

Offset: 1

Altug Alkan, Mar 21 2017

nonn

Sequence represents d(n, 1, 1) where d(n, i, j) = (Sum_{k=1..n} q(k+j-q(k))) mod (n*i) where q(k) = A005185(k).

Altug Alkan, Table of n, a(n) for n = 1..10000
Robert Israel, Illustration Of Residue Classes Modulo 8

Cf. A005185, A280706, A283025, A283467, A283509.

N:= 1000: # to get a(1) to a(N)
B[1]:= 1:
B[2]:= 1:
for n from 3 to N do
  B[n]:= B[n-B[n-1]] + B[n-B[n-2]];
od:
seq(add(B[k+1-B[k]], k=1..n) mod n, n=1..N); # Robert Israel, Mar 22 2017

q[n_]:=If[n<3, 1, q[n - q[n - 1]] + q[n - q[n - 2]]]; a[n_]:=Mod[Sum[q[k + 1 - q[k]],{k, n}], n]; Table[a[n], {n, 100}] (* Indranil Ghosh, Mar 21 2017 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[n-a[n-1]]+a[n-a[n-2]]); vector(#a, n, sum(k=1, n, a[k+1-a[k]]) % n)

(define (A284173 n) (modulo (A280706 n) n)) ;; Other code as in A280706, A283467 and A005185 - Antti Karttunen, Mar 22 2017

a(n) = A280706(n) mod n. - Antti Karttunen, Mar 22 2017

A108583 Primes of the form 1 + Product_{j=1..k} b(j), where b(n) = b(n-1) + 2*A005185(n) and b(1) = 2.

Original entry on oeis.org

3, 13, 2593, 2426113

Offset: 0

Roger L. Bagula, Jul 05 2005

Next term contains 886 digits. - G. C. Greubel, Dec 19 2022

G. C. Greubel, Table of n, a(n) for n = 0..4

Cf. A005185, A108585.

Hofstadter[n_]:= Hofstadter[n]= If[n<2, 1, Hofstadter[n-Hofstadter[n- 1]] +Hofstadter[n-Hofstadter[n-2]]]; (* A005185 *)
b[n_]:= b[n]= If[n==1, 2, b[n-1] +2*Hofstadter[n]]; (* A108585 *)
p[n_]:= p[n]= Product[b[j], {j,n}];
Select[Table[p[n] +1, {n, 500}], PrimeQ]

Edited by G. C. Greubel, Dec 19 2022

A108585 a(n) = a(n-1) + 2*A005185(n+1), with a(1) = 2.

Original entry on oeis.org

2, 6, 12, 18, 26, 36, 46, 58, 70, 82, 98, 114, 130, 150, 168, 188, 210, 232, 256, 280, 304, 328, 360, 388, 416, 448, 480, 512, 544, 584, 618, 652, 692, 734, 772, 812, 856, 898, 942, 988, 1034, 1082, 1130, 1178, 1226, 1274, 1338, 1386, 1436, 1496, 1552, 1604

Offset: 1

Roger L. Bagula, Jul 05 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A005185, A108583.

Hofstadter[n_]:= Hofstadter[n]= If[n<3, 1, Hofstadter[n-Hofstadter[n- 1]] + Hofstadter[n-Hofstadter[n-2]]];
a[n_]:= a[n]= If[n==1, 2, a[n-1] +2*Hofstadter[n+1]];
Table[a[n], {n,60}]

@CachedFunction
def H(n): return 1 if (n<3) else H(n-H(n-1)) + H(n-H(n-2))
def a(n): return 2 if (n==1) else a(n-1) + 2*H(n+1)
[a(n) for n in range(1,61)] # G. C. Greubel, Dec 19 2022

A169637 The number of permutations of the first n elements of the Hofstaedter Q-sequence (A005185), augmented by Q(0)=1.

Original entry on oeis.org

1, 1, 1, 4, 20, 60, 420, 3360, 15120, 151200, 831600, 3326400, 43243200, 302702400, 1513512000, 24216192000, 411675264000, 3705077376000, 70396470144000, 703964701440000, 14783258730240000, 162615846032640000, 1246721486250240000, 7480328917501440000

Offset: 0

Roger L. Bagula, Apr 04 2010

nonn

An augmented Hofstadter sequence 1,1,1,2,3,3,... is defined by adding a single 1 in front of A005185. a(n) is the number of permutations using the first n+1 elements, 1 up to A005185(n), of this augmented sequence.

			For n=3, the first 4 elements of the augmented sequence are (1,1,1,2), with a(3)=4 permutations, namely (1,1,1,2), (1,1,2,1), (1,2,1,1) and (2,1,1,1).

Rémy Sigrist, PARI program for A169637

Cf. A005185.

f[0] = 1; f[1] = 1; f[2] = 1;
f[n_] := f[n] = f[n - f[n - 1]] + f[n - f[n - 2]];
a[m_] := Length[Permutations[Table[f[i], {i, 0, m}]]];
(* b = Table[a[m], {m, 0, 10}]  *)
(* A much better way to compute the terms is to use the multinomials of the multiplicities of the terms of A005229! - Joerg Arndt, Dec 23 2014 *)

PARI
```
See Links section.
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Definition clarified, comment and example added - R. J. Mathar, Dec 08 2010

More terms from Rémy Sigrist, Jun 29 2021

A283672 a(n) = gcd(q(n - q(n+1) + 2), q(n - q(n) + 2)) where q(n) = A005185(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 3, 4, 1, 4, 5, 1, 5, 1, 1, 6, 6, 6, 6, 8, 2, 2, 8, 8, 8, 8, 10, 1, 1, 10, 1, 1, 1, 11, 1, 11, 1, 1, 12, 12, 12, 12, 12, 16, 2, 1, 2, 4, 2, 2, 2, 14, 16, 2, 16, 16, 16, 16, 20, 1, 1, 1, 1, 1, 1, 20, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 22, 1, 1, 23, 1, 1, 1, 1, 23, 24

Offset: 1

Altug Alkan, Mar 14 2017

nonn

			a(4) = gcd(A005185(4 - A005185(5) + 2), A005185(4 - A005185(4) + 2)) = gcd(A005185(3), A005185(3)) = gcd(2, 2) = 2.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative scatterplot of A283672

Cf. A005185, A283673.

q[1] = q[2] = 1; q[n_] := q[n] = q[n - q[n - 1]] + q[n - q[n - 2]]; Table[GCD[q[n - q[n + 1] + 2], q[n - q[n] + 2]], {n, 88}] (* Indranil Ghosh, Mar 14 2017 *)

a=vector(1001); a[1]=a[2]=1; for(n=3, #a, a[n]=a[n-a[n-1]]+a[n-a[n-2]]); va = vector(1000, n, gcd(a[n+2-a[n+1]], a[n+2-a[n]]))

A302779 Restricted growth sequence transform of ordered pair [Q(n-Q(n-1)), Q(n-Q(n-2))], the left & right summand of Hofstadter Q-sequence A005185.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 9, 10, 9, 11, 11, 12, 12, 12, 12, 13, 14, 15, 13, 13, 13, 13, 16, 17, 18, 16, 19, 20, 21, 22, 19, 22, 23, 23, 24, 24, 24, 24, 24, 25, 26, 27, 28, 29, 30, 31, 28, 32, 25, 28, 25, 25, 25, 25, 33, 34, 35, 36, 37, 38, 39, 33, 40, 36, 41, 42, 43, 39, 44, 43, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 49, 53, 53, 53, 53, 53, 53

Offset: 1

Antti Karttunen, Apr 27 2018

nonn

Restricted growth sequence transform of A286559.

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

Cf. A005185, A286559, A302780.

up_to = 65537;
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];
Aux302779(n) = if(n<3,0,[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,Aux302779(n))),"b302779.txt");

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

A239913 a(n) = n - Q(n), where Q(n) is Hofstadter's Q-sequence A005185.

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A280706 a(n) = Sum_{k=1..n} q(k+1-q(k)), where q(k) = A005185(k); partial sums of A283467.

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A283673 a(n) = lcm(q(n - q(n+1) + 2), q(n - q(n) + 2)) where q(n) = A005185(n).

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A284173 a(n) = (Sum_{k=1..n} q(k+1-q(k))) mod n where q(k) = A005185(k).

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A108583 Primes of the form 1 + Product_{j=1..k} b(j), where b(n) = b(n-1) + 2*A005185(n) and b(1) = 2.

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A108585 a(n) = a(n-1) + 2*A005185(n+1), with a(1) = 2.

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A169637 The number of permutations of the first n elements of the Hofstaedter Q-sequence (A005185), augmented by Q(0)=1.

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A283672 a(n) = gcd(q(n - q(n+1) + 2), q(n - q(n) + 2)) where q(n) = A005185(n).

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