A010874 -id:A010874 - cp's OEIS frontend

A076313 a(n) = floor(n/10) - (n mod 10).

0, -1, -2, -3, -4, -5, -6, -7, -8, -9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, 8, 7, 6, 5, 4

Offset: 0

Reinhard Zumkeller, Oct 06 2002

For n<100 equal to the negated alternating digital sum of n (see A055017). - Hieronymus Fischer, Jun 17 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A076314, A010879, A076309, A076310, A076311, A076312, A055017, A076314, A007953, A003132.

a076313 = uncurry (-) . flip divMod 10 -- Reinhard Zumkeller, Jun 01 2013

Table[Floor[n/10]-Mod[n,10],{n,0,100}] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,-1,-2,-3,-4,-5,-6,-7,-8,-9,1},100] (* Harvey P. Dale, Nov 02 2022 *)

a(n)=n\10-n%10 \\ Charles R Greathouse IV, Jan 30 2012

From Hieronymus Fischer, Jun 17 2007: (Start)
a(n) = 11*floor(n/10)-n.
a(n) = (n-11*(n mod 10))/10.
a(n) = 11*A002266(A004526(n))-n=11*A004526(A002266(n))-n.
a(n) = (n-11*A010879(n))/10.
a(n) = (n-11*A000035(n)-22*A010874(A004526(n)))/10.
a(n) = (n-11*A010874(n)-55*A000035(A002266(n)))/10.
G.f.: x*(-8*x^10+11*x^9-1)/((1-x^10)*(1-x)^2). (End)

A133875 n modulo 5 repeated 5 times.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 5^2 = 25.

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

Cf. A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133885, A133880, A133890, A133900, A133910.

[(1 + Floor(n/5)) mod 5 : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

A133875:=n->((1+floor(n/5)) mod 5); seq(A133875(n), n=0..100); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[1 + Floor[n/5], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 06 2014 *)
LinearRecurrence[{1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1},{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0},120] (* Harvey P. Dale, Dec 14 2017 *)

a(n) = (1 + floor(n/5)) mod 5.
a(n) = A010874(A002266(n+5)).
a(n) = 1 + floor(n/5) - 5*floor((n+5)/25).
a(n) = (((n+5) mod 25) - (n mod 5)) / 5.
a(n) = ((n + 5 - (n mod 5)) / 5) mod 5.
a(n) = A010874((n + 5 - A010874(n))/5).
a(n) = binomial(n+5, n) mod 5 = binomial(n+5, 5) mod 5.
a(n) = +a(n-1) -a(n-5) +a(n-6) -a(n-10) +a(n-11) -a(n-15) +a(n-16) -a(n-20) +a(n-21). - R. J. Mathar, Sep 03 2011
G.f.: ( 1+2*x^5+3*x^10+4*x^15 ) / ( (1-x)*(x^20+x^15+x^10+x^5+1) ). - R. J. Mathar, Sep 03 2011

A010883 Simple periodic sequence: repeat 1,2,3,4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130482(n) + n + 1. - Hieronymus Fischer, Jun 08 2007

1234/9999 = 0.123412341234... - Eric Desbiaux, Nov 03 2008

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

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

Cf. A177037 (decimal expansion of (9+2*sqrt(39))/15). - Klaus Brockhaus, May 01 2010

PadRight[{},120,{1,2,3,4}] (* Harvey P. Dale, Aug 02 2016 *)

a(n)=(n-1)%4+1 \\ Charles R Greathouse IV, Jun 11 2015

def A010883(n): return 1 + (n & 3) # Chai Wah Wu, May 25 2022

a(n) = 1 + (n mod 4). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010873(n) + 1.
Also a(n) = (1/2)*(5 - (-1)^n - 2*(-1)^((2*n - 1 + (-1)^n)/4)).
G.f.: g(x) = (4*x^3 + 3*x^2 + 2*x + 1)/(1 - x^4) = (4*x^5 - 5*x^4 + 1)/((1 - x^4)*(1-x)^2). (End)
a(n) = 5/2 - cos(Pi*n/2) - sin(Pi*n/2) - (-1)^n/2. - R. J. Mathar, Oct 08 2011

A099546 Odd part of n modulo 5.

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, Oct 23 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000265, A010874.

Cf. A099544, A099545, A099547, A099548, A099549, A099550, A099551.

a[n_] := Mod[n / 2^IntegerExponent[n, 2], 5]; Array[a, 100] (* Amiram Eldar, Aug 29 2024 *)

a(n) = (n>>valuation(n, 2))%5 \\Charles R Greathouse IV, May 14 2014

a(n) = A010874(A000265(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Aug 29 2024

A130488 a(n) = Sum_{k=0..n} (k mod 10) (Partial sums of A010879).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 90, 91, 93, 96, 100, 105, 111, 118, 126, 135, 135, 136, 138, 141, 145, 150, 156, 163, 171, 180, 180, 181, 183, 186, 190, 195, 201, 208, 216, 225, 225, 226, 228, 231, 235, 240, 246, 253

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 10, 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 (1,0,0,0,0,0,0,0,0,1,-1).

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

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

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

seq(coeff(series(x*(1-10*x^9+9*x^10)/((1-x^10)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Aug 31 2019

LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1}, {0,1,3,6,10,15,21,28,36,45, 45}, 60] (* G. C. Greubel, Aug 31 2019 *)

a(n) = sum(k=0, n, k % 10); \\ Michel Marcus, Apr 28 2018

def A130488(n):
    a, b = divmod(n,10)
    return 45*a+(b*(b+1)>>1) # Chai Wah Wu, Jul 27 2022

def A130488_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-10*x^9+9*x^10)/((1-x^10)*(1-x)^3)).list()
A130488_list(60) # G. C. Greubel, Aug 31 2019

a(n) = 45*floor(n/10) + A010879(n)*(A010879(n) + 1)/2.
G.f.: (Sum_{k=1..9} k*x^k)/((1-x^10)*(1-x)).
G.f.: x*(1 - 10*x^9 + 9*x^10)/((1-x^10)*(1-x)^3).

A156174 Period 5: repeat [1,-1,1,-1,0].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 06 2009

C(n) := a(n+4) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116(n+5). - Wolfdieter Lang, Feb 26 2014

With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 1, z = 1. - Michael Somos, Oct 17 2018

			G.f. = 1 - x + x^2 - x^3 + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Cf. A010874, A011558 (this read mod 2), A099443, A198517.

A156174:=n->[1,-1,1,-1,0][(n mod 5)+1]: seq(A156174(n), n=0..100); # Wesley Ivan Hurt, May 31 2015

CoefficientList[Series[(1 + x^2)/(1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* Wesley Ivan Hurt, May 31 2015 *)
a[ n_] := { -1, 1, -1, 0, 1}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := (-1)^Mod[n, 5] Sign @ Mod[n + 1, 5]; (* Michael Somos, Jun 17 2015 *)

a(n)=[1,-1,1,-1,0][n%5+1] \\ Charles R Greathouse IV, Oct 28 2011

{a(n) = (-1)^(n%5) * sign((n+1)%5)}; /* Michael Somos, Jun 17 2015 */

G.f.: (1+x^2)/(1 + x + x^2 + x^3 + x^4).
Sum_{i=0..n} a(i) = A198517(n). - Bruno Berselli, Nov 02 2011
From Wesley Ivan Hurt, May 31 2015: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 4.
a(n) = Sum_{i=0..3} A011558(n+2+i)*(-1)^i. (End)
Euler transform of length 5 sequence [-1, 1, 0, -1, 1]. - Michael Somos, Jun 17 2015
G.f.: (1-x)*(1-x^4)/((1-x^2)*(1-x^5)). - Michael Somos, Jun 17 2015
a(n) = -a(-2-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2015
a(n) = (2/5) * (cos(4*(n-2)*Pi/5) + cos(2*n*Pi/5) + cos(4*n*Pi/5) - cos(2*(n-3)*Pi/5) - cos(4*(n-3)*Pi/5) - cos(2*(n-1)*Pi/5) - cos(4*(n-1)*Pi/5) - cos((2*n+1)*Pi/5)). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (-1)^n * A099443(n). - Michael Somos, Oct 17 2018
a(5*n) = a(5*n + 2) = 1, a(5*n + 1) = a(5*n + 3) = -1, a(5*n + 4) = 0 for all n in Z. - Michael Somos, Nov 27 2019

A177154 Fractional part of the conversion from degrees Celsius to Fahrenheit.

Original entry on oeis.org

0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2

Offset: 0

Reinhard Zumkeller, May 03 2010

nonn

From Klaus Brockhaus, May 06 2010: (Start)
Periodic sequence: Repeat 0, 8, 6, 4, 2.
Decimal expansion of 8642/99999. (End)

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A029925, A029926, A085259, A085268.

PadRight[{}, 100, {0, 8, 6, 4, 2}] (* Paolo Xausa, Jan 29 2024 *)

(A085259(n) + a(n)/10 - 32) * 5 / 9 = n;
A029925(n) - A085259(n) = floor(a(n)/5);
a(n) = A010879(A022966(A005843(A010874(n)))).
G.f.: 2*x*(4+3*x+2*x^2+x^3)/(1-x^5). - Klaus Brockhaus, May 06 2010

A277543 a(n) = n/5^m mod 5, where 5^m is the greatest power of 5 that divides n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 19 2016

a(n) is the rightmost nonzero digit in the base 5 expansion of n (A007091).

			a(20) = (20/5 mod 5) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A007091, A010874, A132739.

Cf. A277550, A277551, A277555, A277548 (positions of 1, 2, 3 and 4 in this sequence).

Table[Mod[n/5^IntegerExponent[n, 5], 5], {n, 1, 160}]

a(n) = n/5^valuation(n, 5) % 5; \\ Michel Marcus, Oct 20 2016

a(n) = A132739(n) mod 5 = A010874(A132739(n)). - Michel Marcus, Oct 20 2016

A010887 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130486(n)+n+1. - Hieronymus Fischer, Jun 08 2007

1371742/11111111 = 0.123456781234567812345678... - Eric Desbiaux, Nov 03 2008

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

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

Cf. A177034 (decimal expansion of (9280+3*sqrt(13493990))/14165). - Klaus Brockhaus, May 01 2010

a010887 = (+ 1) . flip mod 8
a010887_list = cycle [1..8]
-- Reinhard Zumkeller, Nov 09 2014, Mar 04 2014

PadRight[{},90,Range[8]] (* Harvey P. Dale, May 10 2022 *)

def A010887(n): return 1 + (n & 7) # Chai Wah Wu, May 25 2022

a(n) = 1 + (n mod 8) - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = (1/2)*(9 - (-1)^n - 2*(-1)^(b/4) - 4*(-1)^((b - 2 + 2*(-1)^(b/4))/8)) where b = 2n - 1 + (-1)^n.
Also a(n) = A010877(n) + 1.
G.f.: g(x) = (1/(1-x^8))*Sum_{k=0..7} (k+1)*x^k.
Also: g(x) = (8x^9 - 9x^8 + 1)/((1-x^8)*(1-x)^2). (End)

A060154 Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 4, 3, 0, 2, 0, 0, 1, 2, 3, 4, 1, 0, 1, 0, 1, 0, 3, 4, 0, 0, 1, 0, 0, 1, 8, 1, 4, 1, 1, 1, 2, 1, 0, 1, 4, 0, 0, 5, 0, 2, 0, 0, 0, 0, 1, 2, 9, 1, 1, 6, 1, 3, 1, 1, 1, 0, 1, 4, 3, 6, 8, 0, 0, 4, 4, 0, 2, 0, 0, 1, 2, 9, 4, 5, 0, 1, 1, 3, 0, 1, 0, 1, 0

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3) = 5^3 mod 3 = 125 mod 3 = 2.
Rows start:
  0, 1, 1, 1, 1, ...
  0, 0, 2, 0, 2, ...
  0, 1, 0, 1, 3, ...
  0, 0, 1, 0, 4, ...
  0, 1, 2, 1, 0, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A057427, A015910, A056969.
Columns include A000004, A000035 (several times), A010872, A010874, A010876, A021559 and other periodic sequences.
Diagonals include A000004 and A057427.
Cf. A114448.

T(n, k) = A051129(n, k)-n*A060155(n, k).

cp's OEIS Frontend

A076313 a(n) = floor(n/10) - (n mod 10).

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A133875 n modulo 5 repeated 5 times.

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A010883 Simple periodic sequence: repeat 1,2,3,4.

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A099546 Odd part of n modulo 5.

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A130488 a(n) = Sum_{k=0..n} (k mod 10) (Partial sums of A010879).

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A156174 Period 5: repeat [1,-1,1,-1,0].

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A177154 Fractional part of the conversion from degrees Celsius to Fahrenheit.

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