A283025 -id:A283025 - cp's OEIS frontend

A283360 Absolute values of first differences of A283025.

0, 1, 2, 3, 2, 3, 5, 3, 3, 3, 7, 4, 4, 9, 4, 5, 6, 13, 6, 6, 6, 17, 9, 7, 19, 8, 8, 8, 22, 11, 8, 8, 23, 11, 9, 27, 11, 10, 11, 29, 11, 12, 12, 33, 11, 11, 19, 38, 11, 16, 14, 41, 15, 15, 13, 40, 14, 16, 16, 45, 15, 23, 48, 13, 20, 17, 15, 48, 21, 18, 53, 20, 19, 20

Offset: 1

Altug Alkan, Mar 05 2017

nonn

In order to compare this sequence and A005185 see the scatterplot in Links section. Note that A283025 is the sequence that focuses on the remainders when sum of first n terms of A005185 is divided by n. Absolute values of first differences of A283025 keeps the main characteristic of A005185 with deviations.

			a(4) = 3 because a(4) = abs(A283025(5) - A283025(4)) = abs(0 - 3) = 3.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Scatterplot of A005185 and A283360
Pavel Trojovský, On Some Properties of the Hofstadter-Mertens Function, Nonlinear Dynamics of Complex Systems, Vol. 2020, Article ID 1816756.

Cf. A005185, A283025.

Block[{a}, a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Abs@ Differences@ MapIndexed[Mod[#1, First[#2]] &, Accumulate@ Array[a[#] &, 75]]] (* Michael De Vlieger, Dec 18 2020 *)

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(#a, n, sum(k=1, n, a[k]) % n);
for (k=1, 1000, print1(abs(va[k+1] - va[k]) ", "));

a(n) = abs(A283025(n+1) - A283025(n)).

A282891 Remainder when sum of first n terms of A004001 is divided by n.

Original entry on oeis.org

0, 0, 1, 2, 4, 1, 3, 5, 8, 2, 6, 10, 2, 6, 10, 14, 2, 7, 13, 0, 6, 13, 21, 5, 13, 21, 2, 10, 18, 26, 3, 10, 18, 27, 2, 12, 23, 34, 7, 19, 32, 3, 16, 30, 44, 13, 27, 41, 7, 22, 37, 1, 16, 31, 47, 7, 22, 37, 53, 9, 24, 39, 54, 5, 20, 36, 53, 3, 21, 40, 59, 7, 27, 48, 70, 16, 38, 61, 6, 29, 53, 78, 20, 45, 70, 9, 34, 60, 87, 24

Offset: 1

Altug Alkan, Feb 24 2017

Numbers n such that a(n) = 0 are 1, 2, 20, 4743, 10936, ...

			a(5) = 4 since Sum_{k=1..5} A004001(k) = 1 + 1 + 2 + 2 + 3 = 9 and remainder when 9 is divided by 5 is 4.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A282891
Altug Alkan, Illustration Of Residue Classes Modulo 4

Cf. A004001, A282894, A283025.

A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
A004001(1):= 1: A004001(2):= 1:
L:= ListTools[PartialSums](map(A004001, [$1..1000])):
seq(L[i] mod i, i=1..1000); # Robert Israel, Feb 24 2017

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; MapIndexed[Last@ QuotientRemainder[#1, First@ #2] &, Accumulate@ Table[a@ n, {n, 90}]] (* Michael De Vlieger, Feb 24 2017, after Robert G. Wilson v at A004001 *)

first(n)=my(v=vector(n),s); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); for(k=1,n, s+=v[k]; v[k]=s%k); v \\ Charles R Greathouse IV, Feb 26 2017

a(n) = (Sum_{k=1..n} A004001(k)) mod 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

A283509 Remainder when sum of first n terms of the Hofstadter Q-sequence is divided by 2*n.

Original entry on oeis.org

1, 2, 4, 7, 0, 2, 5, 8, 12, 16, 20, 2, 6, 10, 16, 21, 27, 34, 3, 9, 15, 21, 27, 37, 45, 1, 9, 17, 25, 33, 45, 54, 63, 7, 18, 27, 37, 49, 60, 72, 3, 14, 26, 38, 50, 62, 74, 94, 8, 19, 35, 49, 61, 77, 93, 107, 11, 25, 41, 57, 73, 89, 113, 2, 15, 35, 52, 67, 88, 110, 129, 5, 25, 44, 64, 83, 105, 125, 146, 9, 31, 52, 73, 97

Offset: 1

Altug Alkan, Mar 09 2017

nonn

Sequence represents c(n, 2) where c(n, i) = (Sum_{k = 1..n} A005185(k)) mod (n*i). See also A283025 and corresponding illustration in Links section.

			a(4) = 7 since Sum_{k = 1..4} A005185(k) = 1 + 1 + 2 + 3 = 7 and remainder when 7 is divided by 8 is 7.

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

Cf. A005185, A076268, A283025.

A005185:= proc(n) option remember; procname(n-procname(n-1)) +procname(n-procname(n-2)) end proc:
A005185(1):= 1: A005185(2):= 1:
L:= ListTools[PartialSums](map(A005185, [$1..1000])):
seq(L[i] mod (2*i), i=1..1000); # after Robert Israel at A283025

a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Table[Mod[Total@ Array[a, n], 2 n], {n, 84}] (* Michael De Vlieger, Mar 13 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]) % (2*n))

A284214 Remainder when sum of first n terms of A006949 is divided by n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 0, 2, 4, 6, 9, 0, 3, 7, 12, 0, 4, 8, 12, 17, 1, 6, 12, 19, 0, 6, 13, 21, 29, 7, 16, 25, 0, 8, 16, 24, 33, 4, 13, 23, 34, 2, 12, 23, 35, 0, 12, 25, 38, 0, 12, 25, 39, 53, 12, 27, 42, 57, 13, 29, 45, 62, 16, 33, 50, 0, 16, 32, 48, 65, 11, 28, 46, 65, 8, 26, 45, 65, 5

Offset: 1

Altug Alkan, Mar 23 2017

nonn

Sequence represents e(n, 1) where e(n, i) = (Sum_{k=0..n-1} A006949(k)) mod (n*i).

See also alternative scatterplot and graph of this sequence in Links section.

			a(6) = 3 because Sum_{k=0..5} A006949(k) = 1 + 1 + 1 + 2 + 2 + 2 = 9 and remainder when 9 is divided by 6 is 3.

Altug Alkan, Table of n, a(n) for n = 1..20000
Altug Alkan, Alternative graph of A284214
Altug Alkan, Alternative scatterplot of A284214
Altug Alkan, Illustration of residue classes modulo 4

Cf. A006949, A282891, A283025.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1 - a[n - 1]] + a[n - 2 - a[n - 2]]; MapIndexed[Mod[#1, First@ #2] &, Accumulate@ Table[a@ n, {n, 0, 79}]] (* Michael De Vlieger, Mar 24 2017 *)

a(n) = (Sum_{k=0..n-1} A006949(k)) mod n.

cp's OEIS Frontend

A283360 Absolute values of first differences of A283025.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A282891 Remainder when sum of first n terms of A004001 is divided by n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

A283509 Remainder when sum of first n terms of the Hofstadter Q-sequence is divided by 2*n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A284214 Remainder when sum of first n terms of A006949 is divided by n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula