A096535 -id:A096535 - cp's OEIS frontend

A122276 If b(n-1) + b(n-2) < n then a(n) = 0, otherwise a(n) = 1, where b(i) = A096535(i).

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 2

Klaus Brockhaus, Aug 29 2006

nonn

Conjecture: lim {n -> infinity} x_n / y_n = 1, where x_n is the number of j <= n such that A096535(j) = A096535(j-1) + A096535(j-2) and y_n is the number of j <= n such that A096535(j) = A096535(j-1) + A096535(j-2) - j. Computational support: x_n / y_n = 0.9999917 for n = 10^9.

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A096535, A122277.

f[s_] := f[s] = Append[s, Mod[s[[ -2]] + s[[ -1]], Length[s]]]; t = Nest[f, {1, 1}, 106]; s = {}; Do[AppendTo[s, If[t[[n]] + t[[n + 1]] < n + 1, 0, 1]], {n, 105}]; s (* Robert G. Wilson v Sep 02 2006 *)

{m=107;a=1;b=1;for(n=2,m,d=divrem(a+b,n);print1(d[1],",");a=b;b=d[2])}

a(n) = floor((A096535(n-1)+A096535(n-2))/n)

A096274 Indices of zeros in A096535.

Original entry on oeis.org

2, 8, 13, 20, 25, 595, 1044, 7932, 74247, 14693476, 16766626, 24072338, 72643740, 1881945888, 3304284638, 5163731431, 5669949197, 16209038688, 23714508403, 56796564073, 181057353263, 323874989643, 406930606305, 539293061152, 1751203649485, 2136659012156

Offset: 1

Jim Nastos, Jun 24 2004

nonn

Suggested by Leroy Quet.

Cf. A096535: a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

Cf. A132678.

/* C program from Peter Pein */
#include 
main(int argc, char *argv[])
{ long long a0=1, a1=1, n=1, tmp, nmax;
if (argc != 2) { fprintf(stderr,"%s n\ncalculates the indices of the first n zeros in A096535\n", argv[0]);
return(1); }
nmax=atol(argv[1]);
while (nmax-- > 0) {
while(a1 != 0) {
tmp = (a0 + a1) % ++n; a0 = a1; a1 = tmp; }
printf("%lld\n",n++); a1 = a0; a0 = 0; }
return 0; }

import Data.List (elemIndices)
a096274 n = a096274_list !! (n-1)
a096274_list = elemIndices 0 a096535_list
-- Reinhard Zumkeller, Oct 19 2011

a = b = 1; lst = {}; Do[c = Mod[a + b, n]; If[c == 0, AppendTo[lst, n]; Print@n]; a = b; b = c, {n, 2, 10^9}] (* Robert G. Wilson v, Dec 17 2007 *)

a(13) from Robert G. Wilson v, Jun 23 2004
a(14) - a(16) from Robert G. Wilson v, Aug 30 2006
Extended to a(26) by Zak Seidov, Peter Pein (petsie(AT)dordos.net) and Martin Fuller, Nov 22 2007

A132678 Indices of 1's in A096535.

Original entry on oeis.org

0, 1, 3, 4, 56, 67, 670, 7740, 41842, 47345, 89440, 93196, 189277, 247372, 321327, 474346, 826237, 1988987, 2364721, 2886736, 2937246, 5426145, 12969551, 34658342, 109686031, 373121462, 681070488, 1000410504, 4064275165

Offset: 1

Zak Seidov, Nov 15 2007

nonn

			No more terms <=14059654470.
Last calculated terms in A096535 are: n=14059654470, a(n-2)=12346157556, a(n-1)=1713496920, a(n)=6.

Cf. A096274 = indices of zeros in A096535.

Cf. A096274, A096535.

import Data.List (elemIndices)
a132678 n = a132678_list !! (n-1)
a132678_list = elemIndices 1 a096535_list
-- Reinhard Zumkeller, Oct 19 2011

A197877 Smallest number m such that A096535(m) = n.

Original entry on oeis.org

2, 0, 5, 6, 465, 7, 16, 208, 37, 17, 11, 58, 40, 35, 84, 18, 45, 29, 395, 30, 68, 59, 41, 62, 32, 191, 325, 109, 369, 33, 89, 72, 36, 85, 42, 47, 64, 51, 101, 88, 77, 49, 125, 394, 1124, 249, 76, 50, 54, 65, 119, 74, 2193, 61, 483, 133, 80, 186, 95, 990, 468

Offset: 0

Reinhard Zumkeller, Oct 19 2011

nonn

A096535(a(n)) = n and A096535(m) <> n for m < a(n), concerning first conjecture in A096535.

			a(0) = A096274(1) = 2; a(1) = A132678(1) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a197877 n = a197877_list !! n
a197877_list = map (fromJust . (`elemIndex` a096535_list)) [0..]

A079777 a(0) = 0, a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) (mod n).

Original entry on oeis.org

0, 1, 1, 2, 3, 0, 3, 3, 6, 0, 6, 6, 0, 6, 6, 12, 2, 14, 16, 11, 7, 18, 3, 21, 0, 21, 21, 15, 8, 23, 1, 24, 25, 16, 7, 23, 30, 16, 8, 24, 32, 15, 5, 20, 25, 0, 25, 25, 2, 27, 29, 5, 34, 39, 19, 3, 22, 25, 47, 13, 0, 13, 13, 26, 39, 0, 39, 39, 10, 49, 59, 37, 24, 61, 11, 72, 7, 2, 9, 11, 20

Offset: 0

Joseph L. Pe, Mar 08 2003

nonn

Robert G. Wilson v, Table of n, a(n) for n = 0..10001 [a(9917) corrected by _Georg Fischer_, Dec 03 2023]

Cf. A000045, A058981, A096534, A096535, Zeros in A073853.

l = {1, 1}; For[i = 3, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l
f[s_] := f[s] = Append[s, Mod[s[[ -2]] + s[[ -1]], Length[s]]]; Nest[f, {0, 1}, 80] (* Robert G. Wilson v, Dec 16 2007 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==Mod[a[n-1]+a[n-2],n]},a,{n,80}] (* Harvey P. Dale, Nov 29 2019 *)

Edited by Robert G. Wilson v, Dec 16 2007

A122277 Length of n-th run of zeros in A122276.

Original entry on oeis.org

5, 3, 5, 4, 2, 2, 2, 1, 4, 3, 2, 1, 5, 2, 4, 2, 2, 1, 3, 1, 2, 1, 3, 2, 1, 2, 4, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 1, 2, 2, 3, 2, 1, 5, 2, 2, 2, 2, 1, 4, 4, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 2, 3, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 5, 1, 2, 3, 3, 3, 2, 1, 2

Offset: 1

Klaus Brockhaus, Aug 29 2006

nonn

A run of zeros in A122276 corresponds to a section of A096535 where a(j) = a(j-1) + a(j-2) holds.

Cf. A096535, A122276, A122278 (records), A122279 (where records occur).

f[s_] := f[s] = Append[s, Mod[s[[ -2]] + s[[ -1]], Length[s]]]; k = 435; t = Nest[f, {1, 1}, k]; s = {}; Do[ AppendTo[s, If[t[[n]] + t[[n + 1]] < n + 1, 0, 1]], {n, k}]; Length /@ Select[Split@s, Union@# == {0} &] (* Robert G. Wilson v Sep 02 2006 *)

{m=1000;a=1;b=1;c=0;for(n=2,m,d=divrem(a+b,n);if(d[1]==0,c++,if(c>0,print1(c,",");c=0));a=b;b=d[2])}

A096534 a(1) = 0; a(2) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 1, 6, 7, 3, 10, 1, 11, 12, 8, 4, 12, 16, 9, 5, 14, 19, 10, 5, 15, 20, 8, 0, 8, 8, 16, 24, 7, 31, 3, 34, 0, 34, 34, 28, 21, 7, 28, 35, 18, 7, 25, 32, 8, 40, 48, 36, 31, 13, 44, 1, 45, 46, 32, 18, 50, 6, 56, 62, 53, 49, 35, 16, 51, 67, 47, 42, 16, 58, 74, 56, 53, 31, 5, 36

Offset: 1

Franklin T. Adams-Watters, Jun 23 2004

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A079777, A096535.

A072987 is a closely related sequence.

N:= 100: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 0: A[2]:= 1:
for n from 3 to N do
  A[n]:= A[n-1] + A[n-2] mod n
od:
convert(A,list); # Robert Israel, Nov 13 2017

l = {0, 1}; For[i = 3, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l

A121343 a(n) = Fibonacci(n) mod n(n+1)/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 13, 21, 34, 0, 23, 66, 51, 62, 10, 35, 67, 19, 1, 45, 89, 1, 229, 168, 275, 298, 236, 319, 59, 155, 125, 309, 376, 407, 485, 630, 628, 419, 466, 615, 370, 517, 343, 663, 830, 988, 1033, 168, 624, 700, 746, 1167, 158, 872, 1105, 609, 610, 59, 1181, 0, 1, 125

Offset: 0

N. J. A. Sloane, Aug 29 2006

			a(11)=23 since Fib(11)=89==23(mod (11*12/2)).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A023173, A000217, A096535.

a:= proc(n) local r, M, p, m; r, M, p, m:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1)/2;
      do if irem(p, 2, 'p')=1 then r:= r.M mod m fi;
         if p=0 then break fi; M:= M.M mod m
      od; r[1, 2]
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 26 2016

f[n_] := If[n == 0, 0, Mod[Fibonacci@n, n(n + 1)/2]]; f /@ Range[0, 62] (* Robert G. Wilson v, Aug 31 2006 *)
Join[{0},Mod[First[#],Last[#]]&/@With[{nn=70},Thread[{Fibonacci[ Range[ nn]], Accumulate[Range[nn]]}]]] (* Harvey P. Dale, May 21 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
a(n)=lift(fibmod(n,n*(n+1)/2)) \\ Charles R Greathouse IV, Jun 20 2017

A000045(n) modulo A000217(n).

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A122278 Records in A122277.

Original entry on oeis.org

5, 7, 11, 12, 13, 19, 20, 21, 25, 27, 28, 29

Offset: 1

Klaus Brockhaus, Aug 30 2006

nonn

Cf. A096535, A122276, A122277, A122279 (where records occur).

More terms from Martin Fuller, Nov 22 2007

A122279 Where records occur in A122277.

Original entry on oeis.org

1, 114, 472, 86520, 397603, 514911, 5123504, 382611481, 1166422075, 24846586495, 62401902289, 344065155571

Offset: 1

Klaus Brockhaus, Aug 30 2006

nonn

Cf. A096535, A122276, A122277, A122278 (records).

More terms from Martin Fuller, Nov 22 2007

cp's OEIS Frontend

A122276 If b(n-1) + b(n-2) < n then a(n) = 0, otherwise a(n) = 1, where b(i) = A096535(i).

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A096274 Indices of zeros in A096535.

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A132678 Indices of 1's in A096535.

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A197877 Smallest number m such that A096535(m) = n.

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A079777 a(0) = 0, a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) (mod n).

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A122277 Length of n-th run of zeros in A122276.

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A096534 a(1) = 0; a(2) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

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A121343 a(n) = Fibonacci(n) mod n(n+1)/2.

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