A010874 -id:A010874 - cp's OEIS frontend

A068908 Number of partitions of n modulo 5.

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

Offset: 0

Henry Bottomley, Mar 05 2002

nonn

Of the partitions of numbers from 1 to 100000: 36256 are 0, 15758 are 1, 16133 are 2, 16028 are 3 and 15825 are 4 modulo 5, largely because the number of partitions of 5m+4 is always a multiple of 5.

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A000041, A010874, A068906.

Cf. A040051, A068907, A068909, A020919.

Mod[PartitionsP[Range[0,110]],5] (* Harvey P. Dale, Dec 20 2023 *)

a(n) = numbpart(n) % 5; \\ Michel Marcus, Jul 14 2022

a(n) = A010874(A000041(n)) = A068906(5, n).

a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009]

A130489 a(n) = Sum_{k=0..n} (k mod 11) (Partial sums of A010880).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110, 110, 111, 113, 116, 120, 125, 131, 138, 146, 155, 165, 165, 166, 168, 171, 175, 180, 186, 193, 201, 210, 220, 220, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 275, 276

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 11, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488.

a:=[0,1,3,6,10,15,21,28,36,45, 55,55];; for n in [13..61] do a[n]:=a[n-1]+a[n-11]-a[n-12]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,36,45,55,55]; [n le 12 select I[n] else Self(n-1) + Self(n-11) - Self(n-12): n in [1..61]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-11*x^10+10*x^11)/((1-x^11)*(1-x)^3), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Aug 31 2019

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1}, {0,1,3,6,10,15,21,28,36,45, 55,55}, 60] (* G. C. Greubel, Aug 31 2019 *)
Accumulate[PadRight[{},80,Range[0,10]]] (* Harvey P. Dale, Jul 21 2021 *)

a(n) = sum(k=0, n, k % 11); \\ Michel Marcus, Apr 28 2018

def A130489_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-11*x^10+10*x^11)/((1-x^11)*(1-x)^3)).list()
A130489_list(60) # G. C. Greubel, Aug 31 2019

a(n) = 55*floor(n/11) + A010880(n)*(A010880(n) + 1)/2.
G.f.: (Sum_{k=1..10} k*x^k)/((1-x^11)*(1-x)).
G.f.: x*(1 - 11*x^10 + 10*x^11)/((1-x^11)*(1-x)^3).

A379115 a(n) = A328845(n) mod 5, where A328845 is the first Fibonacci based variant of arithmetic derivative.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 15 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..81839

Cf. A010874, A328845, A374125, A374205, A379116 (positions of 0's), A379117 (their characteristic function).

A379115[n_] := If[n <= 1, 0, Mod[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 5]];
Array[A379115, 100, 0] (* Paolo Xausa, Dec 16 2024 *)

A379115(n) = if(n<=1, 0, my(f=factor(n)); (n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]))%5);

a(n) = A010874(A328845(n)) = A010874(A374125(n)).

A010885 Period 6: repeat [1, 2, 3, 4, 5, 6].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3

Offset: 0

N. J. A. Sloane

Partial sums are given by A130484(n)+n+1. - Hieronymus Fischer, Jun 08 2007

41152/333333 = 0.123456123456123456... [Eric Desbiaux, Nov 03 2008]

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.

Cf. A177158 (decimal expansion of (103+2*sqrt(4171))/162). [From Klaus Brockhaus, May 03 2010]

&cat[[1..6]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A010885:=n->(21-3*cos(n*Pi)-4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/6: seq(A010885(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[Range[6],{n,15}]] (* Harvey P. Dale, Aug 01 2011 *)
PadRight[{},90,Range[6]] (* Harvey P. Dale, Sep 23 2019 *)

a(n) = 1 + (n mod 6). - Paolo P. Lava, Nov 21 2006
a(n) = A010875(n)+1. G.f.: g(x)=(Sum_{0<=k<6} (k+1)*x^k)/(1-x^6). Also g(x)=(6*x^7-7*x^6+1)/((1-x^6)*(1-x)^2). - Hieronymus Fischer, Jun 08 2007
From Wesley Ivan Hurt, Jun 17 2016: (Start)
G.f.: (1+2*x+3*x^2+4*x^3+5*x^4+6*x^5)/(1-x^6).
a(n) = (21-3*cos(n*Pi)-4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/6.
a(n) = a(n-6) for n>5. (End)

A010889 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8,9,10.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130488(n)+n+1. - Hieronymus Fischer, Jun 08 2007

Continued fraction expansion of (232405+sqrt(71216963807))/348378. [From Klaus Brockhaus, May 15 2010]

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A004526, A002264, A002265, A002266.

Cf. A177933 (decimal expansion of (232405+sqrt(71216963807))/348378). [From Klaus Brockhaus, May 15 2010]

PadRight[{},120,Range[10]] (* Harvey P. Dale, Feb 22 2015 *)

def a(n): return n % 10 + 1 # Paul Muljadi, Aug 06 2024

a(n) = 1 + (n mod 10) - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010879(n)+1.
G.f.: (Sum_{k=0..9} (k+1)*x^k)/(1-x^10).
G.f.: (10x^11-11x^10+1)/((1-x^10)(1-x)^2). (End)

More terms from Klaus Brockhaus, May 15 2010

A053840 (Sum of digits of n written in base 5) modulo 5.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the fourth row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010874, A053824, A141803.

Cf. A010060, A053837-A053844.

Mod[Total@ IntegerDigits[#, 5], 5] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)

a(n) = vecsum(digits(n,5)) % 5; \\ Michel Marcus, May 16 2016

a(n) = A010874(A053824(n)). - Andrey Zabolotskiy, May 18 2016

A070471 a(n) = n^3 mod 5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 12 2002

Decimal expansion of 1324/99999. - Vincenzo Librandi, Dec 09 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A010874.

CoefficientList[Series[-x (1 + 3 x + 2 x^2 + 4 x^3)/((x - 1) (1 + x + x^2 + x^3 + x^4)), {x, 0, 100}], x] (* Vincenzo Librandi, May 21 2014 *)
PowerMod[Range[0, 100], 3, 5] (* G. C. Greubel, Mar 26 2016 *)
Table[If[Mod[n, 5] == 0, 0, ModularInverse[n, 5]], {n, 0, 100}] (* Jean-François Alcover, May 03 2017 *)

my(x='x+O('x^99)); concat(0, Vec(-x*(1+3*x+2*x^2+4*x^3)/((x-1)*(1+x+x^2+x^3+x^4)))) \\ Altug Alkan, Mar 27 2016

a(n) = lift(Mod(n, 5)^3); \\ Michel Marcus, Jun 03 2025

[power_mod(n,3,5) for n in (0..101)] # Zerinvary Lajos, Oct 29 2009

a(n) = n^7 mod 5 since 7 == 3 (mod 5-1).
G.f.: -x*(1+3*x+2*x^2+4*x^3) / ( (x-1)*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Dec 10 2010
a(n) = a(n-5). - G. C. Greubel, Mar 26 2016
a(n) = 10 - Sum_{k=1..4} a(n-k) for n > 3. - Nicolas Bělohoubek, Jun 03 2025

A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 66, 67, 69, 72, 76, 81, 87, 94, 102, 111, 121, 132, 132, 133, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 198, 199, 201, 204, 208, 213, 219, 226, 234, 243, 253, 264, 264, 265, 267, 270, 274, 279, 285, 292, 300

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 12, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

List([0..60], n-> Sum([0..n], k-> k mod 12 )); # G. C. Greubel, Sep 01 2019

[&+[(k mod 12): k in [0..n]]: n in [0..60]]; // G. C. Greubel, Sep 01 2019

seq(coeff(series(x*(1-12*x^11+11*x^12)/((1-x^12)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 01 2019

Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,3,6,10,15,21,28,36,45,55,66,66},60] (* Harvey P. Dale, Jan 16 2024 *)

a(n) = sum(k=0, n, k % 12); \\ Michel Marcus, Apr 29 2018

[sum(k%12 for k in (0..n)) for n in (0..60)] # G. C. Greubel, Sep 01 2019

a(n) = 66*floor(n/12) + A010881(n)*(A010881(n) + 1)/2.
G.f.: (Sum_{k=1..11} k*x^k)/((1-x^12)*(1-x)).
G.f.: x*(1 - 12*x^11 + 11*x^12)/((1-x^12)*(1-x)^3).

A379225 a(n) = A113177(n) mod 5, where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 20 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..81839

Cf. A000045, A030426, A010874, A082116, A113177, A374208, A374124, A379226 (positions of 0's), A379227.

Cf. also A379115.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i, 1])));
A379225(n) = (A113177(n)%5);

a(n) = A010874(A113177(n)) = A010874(A374124(n)).

A010884 Period 5: repeat [1,2,3,4,5].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130483(n)+n+1. - Hieronymus Fischer, Jun 08 2007

4115/33333 = 0.12345123451234512345... - Eric Desbiaux, Nov 03 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.

Cf. A177038 (decimal expansion of (195+sqrt(65029))/314).

PadRight[{},120,Range[5]] (* Harvey P. Dale, Dec 08 2018 *)

a(n)=n%5+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 5). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
G.f.: (5*x^4+4*x^3+3*x^2+2*x+1)/(1-x^5) = (5*x^6-6*x^5+1)/((1-x^5)*(1-x)^2).
a(n) = A010874(n)+1. (End)
a(n) = a(n-5). - Wesley Ivan Hurt, Jan 15 2022

cp's OEIS Frontend

A068908 Number of partitions of n modulo 5.

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A130489 a(n) = Sum_{k=0..n} (k mod 11) (Partial sums of A010880).

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A379115 a(n) = A328845(n) mod 5, where A328845 is the first Fibonacci based variant of arithmetic derivative.

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A010885 Period 6: repeat [1, 2, 3, 4, 5, 6].

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A010889 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8,9,10.

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A053840 (Sum of digits of n written in base 5) modulo 5.

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A070471 a(n) = n^3 mod 5.

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