A005185 -id:A005185 - cp's OEIS frontend

A027619 Numbers k such that Hofstadter Q-sequence Q(k) (A005185) satisfies Q(k) = k/2.

2, 50, 56, 128, 156, 166, 208, 238, 272, 308, 336, 392, 474, 476, 512, 618, 658, 666, 710, 734, 836, 868, 1016, 1064, 1376, 1386, 1424, 1432, 1832, 2216, 2280, 2334, 2606, 2638, 2676, 2700, 2740, 2782, 2786, 2912, 2922, 2948, 2954, 3758

Offset: 1

G. R. Bower (fsgrb(AT)aurora.alaska.edu)

nonn

q[n_] := q[n] = If[n <= 2, 1, q[n - q[n-1]] + q[n - q[n-2]]];
Reap[For[n = 1, n <= 4000, n++, If[q[n] == n/2, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 02 2019 *)

Conjecture: a(n)/n^2 is bounded. - Benoit Cloitre, Oct 26 2002

A072231 a(n) = floor(n^2/A005185(n-1)), where A005185 is Hofstadter's Q-sequence.

Original entry on oeis.org

1, 4, 4, 5, 8, 9, 9, 12, 13, 16, 20, 18, 21, 24, 22, 28, 28, 29, 32, 33, 36, 40, 44, 36, 44, 48, 45, 49, 52, 56, 48, 60, 64, 57, 58, 68, 68, 65, 72, 72, 73, 76, 77, 80, 84, 88, 92, 72, 100, 100, 86, 96, 108, 97, 100, 112, 101, 112, 108, 112, 116, 120, 99, 124, 136, 114, 128

Offset: 1

Roger L. Bagula, Jul 05 2002

nonn

Cf. A005185.

Corrected and extended by Robert G. Wilson v, Jun 05 2004

Edited by N. J. A. Sloane, Jun 07 2004

A086267 a(n) = 3 + (H(n) mod 6) + floor(r) where H()=A005185() and r = (H(n) - 2*H(n+1) + H(n+2) - 4) / H(n).

Original entry on oeis.org

1, 0, 2, 5, 4, 5, 7, 7, 2, 2, 2, 4, 4, 4, 6, 5, 6, 7, 7, 2, 2, 3, 2, 6, 4, 4, 6, 6, 7, 6, 4, 7, 7, 4, 5, 3, 4, 6, 5, 6, 7, 7, 2, 2, 2, 3, 2, 5, 3, 3, 2, 7, 4, 2, 3, 6, 5, 2, 4, 4, 5, 4, 7, 6, 3, 4, 8, 5, 5, 7, 3, 4, 6, 5, 7, 5, 2, 6, 7, 3, 4, 3, 3, 6, 4, 5, 7, 7, 6, 2, 2, 2, 2, 3, 2, 7, 7, 6, 2, 5, 2, 2, 3, 4, 3

Offset: 1

Roger L. Bagula, Aug 28 2003

A005185 := proc(n)
        option remember;
        if n<=2 then
                1
        elif n > procname(n-1) and n > procname(n-2) then
                procname(n-procname(n-1))+procname(n-procname(n-2));
        end if;
end proc:
A086267 := proc(n)
        local H ;
        H := A005185(n) ;
        H-2*A005185(n+1)+A005185(n+2)-4;
        %/H ;
        3+ floor(%)+ (H mod 6) ;
end proc:
seq(A086267(n),n=1..50) ; # R. J. Mathar, Oct 10 2011

Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]] Hofstadter[1] = Hofstadter[2] = 1 Digits=502 a=Table[Hofstadter[n], {n, 1, Digits}]; b=Table[Floor[(a[[n]]-2*a[[n+1]]+a[[n+2]]-4)/a[[n]]]+Mod[a[[n]], 6]+3, {n, 1, Digits-2}] ListPlot[b]

A087722 Strictly decreasing domain set of A005185.

Original entry on oeis.org

9, 14, 17, 19, 21, 24, 26, 28, 30, 31, 33, 35, 37, 39, 40, 41, 43, 44, 46, 48, 50, 52, 53, 55, 57, 58, 61, 62, 63, 64, 65, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 82, 83, 84, 86, 87, 88, 90, 92, 93, 94, 100, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 116, 118, 119

Offset: 1

Roger L. Bagula, Sep 29 2003

nonn

Cf. A005185.

digits=750;
Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]];
Hofstadter[1] = Hofstadter[2] = 1;
a1=Table[Hofstadter[n], {n, 1, digits}];
f[x_, y_] := Abs[x-y]/; x-y<0;
f[x_, y_] := 0/; x-y>=0;
b=Table[If[f[a1[[n]], a1[[n-1]]]>0, a1[[n]], 0], {n, 2, digits}];
c=Delete[Union[b], 1]

A116590 a(0)=1; a(n)=b(n+2)+b(n), where b(n)=A005185(n) is the Hofstadter Q-sequence: b(1)=b(2)=1; b(n)=b(n-b(n-1))+b(n-b(n-2)) for n > 2.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 11, 12, 14, 14, 16, 18, 17, 20, 20, 21, 23, 23, 24, 24, 28, 26, 30, 30, 30, 32, 32, 36, 33, 37, 37, 38, 39, 41, 41, 41, 44, 44, 45, 47, 47, 48, 48, 48, 56, 48, 57, 54, 53, 56, 58, 56, 58, 62, 58, 64, 62, 64, 64, 72, 65, 71, 71, 66, 71, 74, 73, 76, 78, 77

Offset: 0

Roger L. Bagula, Mar 27 2006

A similar definition applied to the Fibonacci sequence (A000045) leads to the Lucas sequence (A000032).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000032, A005185.

a116590 n = a116590_list !! n
a116590_list = 1 : zipWith (+) a005185_list (drop 2 a005185_list)
-- Reinhard Zumkeller, Apr 25 2012

b:=proc(n) option remember; if n<=2 then 1 else b(n-b(n-1))+b(n-b(n-2)): fi: end: a[0]:=1: for n from 1 to 71 do a[n]:=b(n)+b(n+2) od: seq(a[n],n=0..71);

F[0] = 0; F[1] = 1; F[2] = 1; F[n_] := F[n] = F[n - F[n - 1]] + F[n - F[n - 2]] L[0] = 1; L[n_] := L[n] = F[n - 1] + F[n + 1]
Table[L[n], {n, 1, 200}]

a(n) = A005185(n+2) + A005185(n) for n>=1.

Edited by N. J. A. Sloane, Apr 15 2006

A121459 Let f(n) = A004001(n)^2 - A005185(n)^2. Then a(n) = f(abs(f(n-1))) + f(abs(n - f(n-1))).

Original entry on oeis.org

0, -3, -14, -18, -7, -9, -47, -51, 0, -15, -15, -48, -17, -36, 57, -151, 0, -63, 0, -11, 0, 25, 26, 368, 29, -5, -96, -33, 0, -144, 2275, -466, -180, 433, 472, 0, -43, 316, 0, 0, 47, -302, 49, 152, 1122, 945, 1273, 10170, 589, 1310, 121, 54, 3117, 0, 177, 2141, -1280, -5, 310, 0

Offset: 1

Roger L. Bagula, Sep 06 2006

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A004001, A005185.

HConway[n_]:= HConway[n]= If[n<3, 1, HConway[HConway[n-1]] + HConway[n-HConway[n -1]]];
Hofstadter[n_]:= Hofstadter[n]= If[n<3, 1, Hofstadter[n -Hofstadter[n-1]] + Hofstadter[n -Hofstadter[n-2]]];
f[n_]:= f[n]= HConway[n]^2 - Hofstadter[n]^2;
Table[f[Abs[f[n-1]]] + f[Abs[n -f[n-1]]], {n,60}] (* corrected by G. C. Greubel, Feb 12 2020 *)

Edited by N. J. A. Sloane, Oct 01 2006

Terms a(31) and beyond corrected by G. C. Greubel, Feb 12 2020

A176048 Triangle t(n,m) = A005185(m+1)+A005185(n-m+1)-A005185(n+1), read by rows 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, -1, 0, 1, 0, 1, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0

Offset: 0

Roger L. Bagula and Gary W. Adamson, Apr 07 2010

			  1;
 1, 1;
 1, 0, 1;
 1, 0, 0, 1;
 1, 1, 1, 1, 1;
 1, 0, 1, 1, 0, 1;
 1, 0, 0, 1, 0, 0, 1;
 1, 1, 1, 1, 1, 1, 1, 1;
 1, 0, 1, 1, 0, 1, 1, 0, 1;

A176048 := proc(n,m)
        A005185(m+1)+A005185(n-m+1)-A005185(n+1) ;
end proc:
seq(seq(A176048(n,m),m=0..n),n=0..12) ; # R. J. Mathar, Jul 11 2012

Installed a sequence where definition and terms match, removed erroneous row sums. - R. J. Mathar, Jul 11 2012

A283451 a(n) = Sum_{k=1..n} (-1)^k * A005185(k).

Original entry on oeis.org

-1, 0, -2, 1, -2, 2, -3, 2, -4, 2, -4, 4, -4, 4, -6, 3, -7, 4, -7, 5, -7, 5, -7, 9, -5, 9, -7, 9, -7, 9, -11, 6, -11, 9, -12, 7, -13, 9, -12, 10, -13, 10, -14, 10, -14, 10, -14, 18, -6, 19, -11, 17, -9, 21, -9, 19, -13, 17, -15, 17, -15, 17, -23, 10, -21, 17, -18, 15, -24, 16, -21

Offset: 1

Altug Alkan, Mar 07 2017

			a(7) = -3 because -1 + 1 - 2 + 3 - 3 + 4 - 5 = -3.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A283451

Cf. A005185, A076268.

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

A283669 a(n) = q(n-q(n+1)+2) - q(n-q(n)+2) where q(n) = A005185(n).

Original entry on oeis.org

0, -1, -1, 0, -1, -1, 0, 0, 0, 0, -2, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, -1, 1, 0, -1, 1, 2, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -4, 3, -2, -4, 2, 2, -2, 0, 0, -2, 0, 0, 0, 0, 0, -1, 3, 4, -3, 1, -1, 0, 3, 4, 2, 1, -2, -1, 2, -2, 1, -1, 0, 1, 1, 0, -2, 1, -1, -1, 0, 0, 0, 0

Offset: 1

Altug Alkan, Mar 13 2017

sign

Altug Alkan, Scatterplot of A283669 for 10^6 terms

Cf. A005185.

q[1] = q[2] = 1; q[n_] := q[n] = q[n - q[n - 1]] + q[n - q[n - 2]]; a[n_]:= q[n - q[n + 1] + 2] - q[n - q[n] + 2] ; Table[a[n], {n, 90}] (* Indranil Ghosh, Mar 13 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, a[n+2-a[n+1]]-a[n+2-a[n]])

A284124 Remainder when 4*n is divided by A005185(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 3, 2, 0, 4, 2, 0, 4, 0, 0, 1, 8, 6, 10, 8, 0, 4, 8, 0, 2, 6, 12, 0, 4, 8, 4, 9, 13, 16, 14, 11, 8, 20, 9, 6, 3, 7, 4, 8, 12, 16, 20, 0, 4, 0, 24, 12, 4, 6, 10, 0, 4, 22, 12, 16, 20, 24, 12, 25, 12, 36, 23, 8, 3, 0, 25, 22, 12, 23, 20, 31, 14, 32, 29, 19, 16

Offset: 1

Altug Alkan, Mar 20 2017

			a(5) = 2 because remainder when 4*5 = 20 is divided by A005185(5) = 3 is 2.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A284124

Cf. A005185, A008586.

a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Table[Mod[4 n, a@ n], {n, 81}] (* Michael De Vlieger, Mar 20 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(1000, n, (4*n)%a[n])

a(n) = A008586(n) mod A005185(n) for n > 0.

cp's OEIS Frontend

A027619 Numbers k such that Hofstadter Q-sequence Q(k) (A005185) satisfies Q(k) = k/2.

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Mathematica

Formula

A072231 a(n) = floor(n^2/A005185(n-1)), where A005185 is Hofstadter's Q-sequence.

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Extensions

A086267 a(n) = 3 + (H(n) mod 6) + floor(r) where H()=A005185() and r = (H(n) - 2*H(n+1) + H(n+2) - 4) / H(n).

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Maple

Mathematica

A087722 Strictly decreasing domain set of A005185.

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Mathematica

A116590 a(0)=1; a(n)=b(n+2)+b(n), where b(n)=A005185(n) is the Hofstadter Q-sequence: b(1)=b(2)=1; b(n)=b(n-b(n-1))+b(n-b(n-2)) for n > 2.

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Haskell

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A121459 Let f(n) = A004001(n)^2 - A005185(n)^2. Then a(n) = f(abs(f(n-1))) + f(abs(n - f(n-1))).

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A176048 Triangle t(n,m) = A005185(m+1)+A005185(n-m+1)-A005185(n+1), read by rows 0<=m<=n.

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Maple

Extensions

A283451 a(n) = Sum_{k=1..n} (-1)^k * A005185(k).

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PARI

A283669 a(n) = q(n-q(n+1)+2) - q(n-q(n)+2) where q(n) = A005185(n).

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A284124 Remainder when 4*n is divided by A005185(n).

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