A030124 -id:A030124 - cp's OEIS frontend

A225376 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,5,11, Q starts with 4,6, R starts with 2; at each stage the smallest number not yet present in P,Q,R is appended to R; every number appears exactly once in the union of P,Q,R. Sequence gives P.

1, 5, 11, 20, 36, 60, 94, 140, 199, 272, 360, 465, 588, 730, 893, 1078, 1286, 1519, 1778, 2064, 2378, 2721, 3094, 3498, 3934, 4403, 4907, 5448, 6027, 6645, 7303, 8002, 8743, 9527, 10355, 11228

Offset: 1

N. J. A. Sloane, May 12 2013, based on email from Christopher Carl Heckman, May 06 2013

nonn

P can be extended for 10^6 terms, but it is not known if P,Q,R can be extended to infinity.
A probabilistic argument suggests that P, Q, R are infinite. - N. J. A. Sloane, May 19 2013
Martin Gardner (see reference) states that no such triple P,Q,R of sequences exists if it is required that P(1)

			The initial terms of P, Q, R are:
1     5    11    20    36    60    94   140   199   272   360
   4     6     9    16    24    34    46    59    73    88
      2     3     7     8    10    12    13    14    15

M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff.

Christopher Carl Heckman, Table of n, a(n) for n = 1..10002

Cf. A225377, A225378, A005228, A030124, A037257.

Hofstadter2 := proc (N) local h, dh, ddh, S, lbmex, i:
    h := 1, 5, 11: dh := 4, 6: ddh := 2:
    lbmex := 3: S := {h,dh,ddh}:
    for i from 4 to N do:
       while lbmex in S do: S := S minus {lbmex}: lbmex := lbmex + 1: od:
       ddh := ddh, lbmex:
       dh := dh, dh[-1] + lbmex:
       h := h, h[-1] + dh[-1]:
       S := S union {h[-1], dh[-1], ddh[-1]}:
       lbmex := lbmex + 1:
    od:
    if {h} intersect {dh} <> {} then: return NULL:
    elif {h} intersect {ddh} <> {} then: return NULL:
    elif {ddh} intersect {dh} <> {} then: return NULL:
    else: return [h]: fi:
end proc: # Christopher Carl Heckman, May 12 2013

Hofstadter2[N_] := Module[{P, Q, R, S, k, i}, P = {1, 5, 11}; Q = {4, 6}; R = {2}; k = 3; S = Join[P, Q, R]; For[i = 4, i <= N, i++, While[MemberQ[S, k], S = S~Complement~{k}; k++]; AppendTo[R, k]; AppendTo[Q, Q[[-1]] + k]; AppendTo[P, P[[-1]] + Q[[-1]]]; S = S~Union~{P[[-1]], Q[[-1]], R[[-1]]}; k++]; Which[P~Intersection~Q != {}, Return@Nothing, {P}~Intersection~R != {}, Return@Nothing, R~Intersection~Q != {}, Return@Nothing, True, Return@P]];
Hofstadter2[36] (* Jean-François Alcover, Mar 05 2023, after Christopher Carl Heckman's Maple code *)

Corrected and edited by Christopher Carl Heckman, May 12 2013

A225377 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,5,11, Q starts with 4,6, R starts with 2; at each stage the smallest number not yet present in P,Q,R is appended to R; every number appears exactly once in the union of P,Q,R. Sequence gives Q.

Original entry on oeis.org

4, 6, 9, 16, 24, 34, 46, 59, 73, 88, 105, 123, 142, 163, 185, 208, 233, 259, 286, 314, 343, 373, 404, 436, 469, 504, 541, 579, 618, 658, 699, 741, 784, 828, 873, 920, 968, 1017, 1067, 1118, 1170

Offset: 1

N. J. A. Sloane, May 12 2013, based on email from Christopher Carl Heckman, May 06 2013

nonn

P can be extended for 10^6 terms, but it is not known if P,Q,R can be extended to infinity.

A probabilistic argument suggests that P, Q, R are infinite. - N. J. A. Sloane, May 19 2013

			The initial terms of P, Q, R are:
1     5    11    20    36    60    94   140   199   272   360
   4     6     9    16    24    34    46    59    73    88
      2     3     7     8    10    12    13    14    15

Christopher Carl Heckman, Table of n, a(n) for n = 1..10001

Cf. A225376, A225378, A005228, A030124, A037257.

Maple
```
See A225376.
```

Corrected and edited by Christopher Carl Heckman, May 12 2013

A225378 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,5,11, Q starts with 4,6, R starts with 2; at each stage the smallest number not yet present in P,Q,R is appended to R; every number appears exactly once in the union of P,Q,R. Sequence gives R.

Original entry on oeis.org

2, 3, 7, 8, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64

Offset: 1

N. J. A. Sloane, May 12 2013, based on email from Christopher Carl Heckman, May 06 2013

nonn

P can be extended for 10^6 terms, but it is not known if P,Q,R can be extended to infinity.

A probabilistic argument suggests that P, Q, R are infinite. - N. J. A. Sloane, May 19 2013

			The initial terms of P, Q, R are:
1     5    11    20    36    60    94   140   199   272   360
   4     6     9    16    24    34    46    59    73    88
      2     3     7     8    10    12    13    14    15

Christopher Carl Heckman, Table of n, a(n) for n = 1..10000

Cf. A225376, A225377, A005228, A030124, A037257.

Maple
```
See A225376.
```

Corrected and edited by Christopher Carl Heckman, May 12 2013

A225850 Inverse of permutation in A167151.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 5, 10, 12, 14, 16, 7, 18, 20, 22, 24, 26, 9, 28, 30, 32, 34, 36, 38, 40, 11, 42, 44, 46, 48, 50, 52, 54, 56, 13, 58, 60, 62, 64, 66, 68, 70, 72, 74, 15, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 17, 96, 98, 100, 102, 104, 106, 108, 110, 112

Offset: 0

Reinhard Zumkeller, May 17 2013

nonn

For n > 0: a(A005228(n)) = 2*n-1 and a(A030124(n)) = 2*n.

For n > 0: A232739(n) = a(A232739(n+1))/2. - Antti Karttunen, Dec 04 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A167151.

Cf. also A005228, A030124, A232739, A232746, A232747, A232749, and also the permutation pair A232751/A232752.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a225850 = fromJust . (`elemIndex` a167151_list)

nmax = 100; A5228 = {1};
Module[{d = 2, k = 1}, Do[While[MemberQ[A5228, d], d++]; k += d; d++; AppendTo[A5228, k], {n, 1, nmax}]];
a46[n_] := For[k = 1, True, k++, If[A5228[[k]] > n, Return[k - 1]]];
a47[n_] := If[n == 1, 1, a46[n] (a46[n] - a46[n - 1])];
a48[n_] := a48[n] = If[n == 1, 0, a48[n-1] + (1 - (a46[n] - a46[n-1]))];
a49[n_] := If[n == 1, 0, a48[n] (a48[n] - a48[n - 1])];
a[n_] := If[n < 3, n, 2 (a47[n] + a49[n]) - (a46[n] - a46[n - 1])];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 09 2021 *)

(define (A225850 n) (if (< n 3) n (- (* 2 (+ (A232747 n) (A232749 n))) (- (A232746 n) (A232746 (- n 1))))))
;; Antti Karttunen, Dec 04 2013

If n < 3, a(n) = n, otherwise a(n) = (2*(A232747(n)+A232749(n))) - (A232746(n)-A232746(n-1)). - Antti Karttunen, Dec 04 2013

A121229 Beginning with a(1) = 1 and a(2) = 2, a(n) is not equal to the product of two consecutive (distinct) earlier terms.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Aug 21 2006

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A005228, A030124.

The complement is A286290, excluding the initial 1.

A121229 := proc(n)
    option remember;
    local a,ispr,i;
    if n <=2 then
        n;
    else
        for a from procname(n-1)+1 do
            ispr := false ;
            for i from 1 to n-2 do
                if procname(i)*procname(i+1) = a then
                    ispr := true ;
                    break;
                end if;
            end do:
            if not ispr then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 25 2017

a[n_] := a[n] = Module[{k, ispr, i}, If[n <= 2, n, For[k = a[n - 1] + 1, True, k++, ispr = False; For[i = 1, i <= n - 2, i++, If[a[i]*a[i + 1] == k, ispr = True; Break[]]]; If[!ispr, Return[k]]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 23 2022, after R. J. Mathar *)

from itertools import islice
def agen(): # generator of terms
    disallowed, prevk, k = {1, 2}, 2, 3; yield from [1, 2]
    while True:
        while k in disallowed: k += 1
        yield k; disallowed.update([k, k*prevk]); prevk = k
print(list(islice(agen(), 72))) # Michael S. Branicky, Sep 23 2022

A140778 a(n) is the smallest positive integer such that no number occurs twice in the union of the sequence and its absolute first differences.

Original entry on oeis.org

1, 3, 7, 12, 18, 8, 17, 28, 13, 27, 43, 19, 39, 60, 22, 45, 70, 26, 55, 85, 31, 63, 96, 34, 69, 105, 37, 77, 118, 42, 88, 135, 48, 97, 147, 52, 103, 156, 56, 113, 171, 59, 120, 184, 65, 131, 198, 71, 143, 216, 74, 149, 227, 79, 159, 240, 82, 165, 249, 86, 175, 265, 91, 183

Offset: 1

Franklin T. Adams-Watters, May 29 2008

This sequence and its first differences include every positive integer (exactly once).

			For a(5), the sequence to that point is [1,3,7,12], with absolute differences [2,4,5]. The next number cannot be 6, because then 6 would be in both the sequence and the first differences. Since all values smaller than 6 are taken, the difference must be positive and at least 6. A difference of 6 works, a(5) = 18.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A140779, A081145.

b:= proc() false end:
a:= proc(n) option remember; local k;
      if n=1 then b(1):= true; 1
    else for k while b(k) or (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
         b(k), b(abs(a(n-1)-k)):= true$2; k
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 14 2015

a[n_] := a[n] = Module[{}, If [n == 1, b[1] = True; 1, For[k = 1, b[k] || Function[t, b[t] || t == k][Abs[a[n-1] - k]], k++]; {b[k], b[Abs[a[n-1] - k]]} = {True, True}; k]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 22 2017, after Alois P. Heinz *)

IsInList(v, k) = for(i=1,#v,if(v[i]==k,return(1)));return(0) IsInDiff(v, k) = for(i=2,#v,if(abs(v[i]-v[i-1])==k,return(1)));return(0) NextA140778(v)={ local(i,d); if(#v==0,return(1)); i=2; while(1, d=abs(i-v[ #v]); if(!(i==d || IsInList(v,i) || IsInDiff(v,i) || IsInList(v,d) || IsInDiff(v,d)), return(i)); i++) } v=[];for(i=1,100,v=concat(v,NextA140778(v)));v

{u=0;a=1;for(n=1,99,u+=1<M. F. Hasler, May 13 2015

A232747 Inverse function to Hofstadter's A005228.

Original entry on oeis.org

1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 30 2013

nonn

This is an inverse function to Hofstadter's A005228 in the sense that for all n, n = a(A005228(n)). a(n) = 0 when n is not in A005228, but instead in its complement A030124.
Note that a(n)*A232749(n) = 0 for all n.
Used to compute the permutation A232751.

Index entries for Hofstadter-type sequences

A030124 gives the positions of zeros.

Cf. A005228, A232746, A232749, A232751.

nmax = 100; A5228 = {1}; Module[{d = 2, k = 1}, Do[While[MemberQ[A5228, d], d++]; k += d; d++; AppendTo[A5228, k], {n, 1, nmax}]];
a46[n_] := For[k = 1, True, k++, If[A5228[[k]] > n, Return[k - 1]]];
a[n_] := If[n == 1, 1, a46[n] (a46[n] - a46[n - 1])];
Array[a, nmax] (* Jean-François Alcover, Dec 09 2021 *)

a(1)=1, and for n>1, a(n) = A232746(n) * (A232746(n)-A232746(n-1)).

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A081688 0 followed by A030124 - 1.

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A037257 a() = 1,3,... [ A037257 ], differences = 2,... [ A037258 ] and 2nd differences [ A037259 ] are disjoint and monotonic; adjoin next free number to 2nd differences unless it would produce a duplicate in which case ignore.

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A167151 a(2n+1) = a(2n) + a(2n-1), a(2n)=least number not yet in the sequence, a(1)=1.

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A225850 Inverse of permutation in A167151.

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A121229 Beginning with a(1) = 1 and a(2) = 2, a(n) is not equal to the product of two consecutive (distinct) earlier terms.

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