A130486 -id:A130486 - cp's OEIS frontend

A010879 Final digit of n.

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

Offset: 0

N. J. A. Sloane

Also decimal expansion of 137174210/1111111111 = 0.1234567890123456789012345678901234... - Jason Earls, Mar 19 2001
In general the base k expansion of A062808(k)/A048861(k) (k>=2) will produce the numbers 0,1,2,...,k-1 repeated with period k, equivalent to the sequence n mod k. The k-digit number in base k 123...(k-1)0 (base k) expressed in decimal is A062808(k), whereas A048861(k) = k^k-1. In particular, A062808(10)/A048861(10)=1234567890/9999999999=137174210/1111111111.
a(n) = n^5 mod 10. - Zerinvary Lajos, Nov 04 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).
Index entries for sequences related to final digits of numbers

Cf. A034948, A059988, A048861, A062808, A086457, A086458.
Cf. A008959, A008960, A070514. - Doug Bell, Jun 15 2015
Partial sums: A130488. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487.

a010879 = (`mod` 10)
a010879_list = cycle [0..9]  -- Reinhard Zumkeller, Mar 26 2012

[n mod(10): n in [0..90]]; // Vincenzo Librandi, Jun 17 2015

A010879 := proc(n)
    n mod 10 ;
end proc: # R. J. Mathar, Jul 12 2013

Table[10*FractionalPart[n/10], {n, 1, 300}] (* Enrique Pérez Herrero, Jul 30 2009 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 2, 3, 4, 5, 6, 7, 8, 9},81] (* Ray Chandler, Aug 26 2015 *)
PadRight[{},100,Range[0,9]] (* Harvey P. Dale, Oct 04 2021 *)

a(n)=n%10 \\ Charles R Greathouse IV, Jun 16 2011

def a(n): return n % 10 # Martin Gergov, Oct 17 2022

[power_mod(n,5,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 04 2009

a(n) = n mod 10.
Periodic with period 10.
From Hieronymus Fischer, May 31 and Jun 11 2007: (Start)
Complex representation: a(n) = 1/10*(1-r^n)*sum{1<=k<10, k*product{1<=m<10,m<>k, (1-r^(n-m))}} where r=exp(Pi/5*i) and i=sqrt(-1).
Trigonometric representation: a(n) = (256/5)^2*(sin(n*Pi/10))^2 * sum{1<=k<10, k*product{1<=m<10,m<>k, (sin((n-m)*Pi/10))^2}}.
G.f.: g(x) = (Sum_{k=1..9} k*x^k)/(1-x^10) = -x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8) / ( (x-1) *(1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) ).
Also: g(x) = x*(9*x^10-10*x^9+1)/((1-x^10)*(1-x)^2).
a(n) = n mod 2+2*(floor(n/2)mod 5) = A000035(n) + 2*A010874(A004526(n)).
Also: a(n) = n mod 5+5*(floor(n/5)mod 2) = A010874(n)+5*A000035(A002266(n)). (End)
a(n) = 10*{n/10}, where {x} means fractional part of x. - Enrique Pérez Herrero, Jul 30 2009
a(n) = n - 10*A059995(n). - Reinhard Zumkeller, Jul 26 2011
a(n) = n^k mod 10, for k > 0, where k mod 4 = 1. - Doug Bell, Jun 15 2015

Formula section edited for better readability by Hieronymus Fischer, Jun 13 2012

A010877 a(n) = n mod 8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The rightmost digit in the base-8 representation of n. Also, the equivalent value of the three rightmost digits in the base-2 representation of n. - Hieronymus Fischer, Jun 12 2007

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Partial sums: A130486. Other related sequences A130481, A130482, A130483, A130484, A130485.

Cf. A000035, A010887, A010873, A130909, A168181, A244413, A253513.

Table[Mod[n,8],{n,0,120}]   (* Harvey P. Dale, Apr 21 2011 *)

vector(100,i,i)%8 \\ Charles R Greathouse IV, Jul 16 2011

def A010877(n): return n&7 # Chai Wah Wu, Jul 09 2022

Complex representation: a(n) = (1/8)*(1-r^n)*Sum_{k=1..7} k*Product_{m=1..7, m<>k} (1 - r^(n-m)) where r = exp(Pi/4*i) = (1+i)*sqrt(2)/2 and i=sqrt(-1).
Trigonometric representation: a(n) = 256*(sin(n*Pi/8))^2*Sum_{k=1..7} k*Product_{m=1..7, m<>k} (sin((n-m)*Pi/8))^2.
G.f.: g(x) = (Sum_{k=1..7}, k*x^k)/(1-x^8).
Also: g(x) = x(7x^8-8x^7+1)/((1-x^8)(1-x)^2). - Hieronymus Fischer, May 31 2007
a(n) = n mod 2 + 2*(floor(n/2) mod 4) = A000035(n) + 2*A010873(A004526(n)).
a(n) = n mod 4 + 4*(floor(n/4) mod 2) = A010873(n) + 4*A000035(A002265(n)).
a(n) = n mod 2 + 2*(floor(n/2) mod 2) + 4*(floor(n/4) mod 2) = A000035(n) + 2*A000035(A004526(n)) + 4*A000035(A002265(n)). - Hieronymus Fischer, Jun 12 2007
a(n) = (1/2)*(7 - (-1)^n - 2*(-1)^(b/4) - 4*(-1)^((b - 2 + 2*(-1)^(b/4))/8)) where b = 2n - 1 + (-1)^n. - Hieronymus Fischer, Jun 12 2007
General formula for period 2^k: a(n) = (1/2)*(2^k - 1 - Sum_{j=0..k-1} 2^j*(-1)^p(j,n)) where p(j,n) is defined recursively by p(0,n)=n, p(j,n) = (1/4)*(2*p(j-1,n) - 1 + (-1)^p(j-1,n)). - Hieronymus Fischer, Jun 14 2007
a(n) = floor(1234567/99999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor(48913/2396745*8^(n+1)) mod 8. - Hieronymus Fischer, Jan 04 2013

Formula section re-edited for better readability by Hieronymus Fischer

A010878 a(n) = n mod 9.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Periodic with period of length 9. The digital root of n (A010888) is a very similar sequence.

The rightmost digit in the base-9 representation of n. Also, the equivalent value of the two rightmost digits in the base-3 representation of n. - Hieronymus Fischer, Jun 11 2007

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

Partial sums: A130487. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486.

a010878 = (`mod` 9)
a010878_list = cycle [0..8]  -- Reinhard Zumkeller, Jan 09 2013

A010878 := proc(n)
    modp(n,9) ;
end proc:
seq(A010878(n),n=0..100) ; # R. J. Mathar, Sep 09 2015

Array[Mod[#, 9]&, 105, 0] (* Jean-François Alcover, Jan 30 2018 *)
PadRight[{},120,Range[0,8]] (* Harvey P. Dale, Dec 19 2018 *)

a(n)=n%9 \\ Charles R Greathouse IV, Sep 24 2015

Complex representation: a(n)=(1/9)*(1-r^n)*sum{1<=k<9, k*product{1<=m<9,m<>k, (1-r^(n-m))}} where r=exp(2*pi/9*i) and i=sqrt(-1). Trigonometric representation: a(n)=(256/9)^2*(sin(n*pi/9))^2*sum{1<=k<9, k*product{1<=m<9,m<>k, (sin((n-m)*pi/9))^2}}. G.f.: g(x)=(sum{1<=k<9, k*x^k})/(1-x^9). Also: g(x)=x(8x^9-9x^8+1)/((1-x^9)(1-x)^2). - Hieronymus Fischer, May 31 2007
a(n) = n mod 3 + 3*(floor(n/3)mod 3) = A010872(n) + 3*A010872(A002264(n)). - Hieronymus Fischer, Jun 11 2007
a(n) = floor(12345678/999999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor(1513361/96855122*9^(n+1)) mod 9. - Hieronymus Fischer, Jan 04 2013

A010880 a(n) = n mod 11.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0

Offset: 0

N. J. A. Sloane

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

Partial sums: A130489. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488.

PadRight[{},80,Range[0,10]] (* Harvey P. Dale, Nov 13 2012 *)

a(n)=n%11 \\ Charles R Greathouse IV, Oct 07 2015

[power_mod(n,11,11) for n in range(0, 78)] # Zerinvary Lajos, Nov 07 2009

From Hieronymus Fischer, Sep 30 2007: (Start)
a(n) = (1/11)*(1-r^n)*sum{1<=k<11, k*product{1<=m<11,m<>k, (1-r^(n-m))}} where r=exp(2*Pi/11*i) and i=sqrt(-1).
a(n) = (1024/11)^2*(sin(n*Pi/11))^2*sum{1<=k<11, k*product{1<=m<11,m<>k, (sin((n-m)*Pi/11))^2}}.
G.f.: (sum{1<=k<11, k*x^k})/(1-x^11).
G.f.: x*(10*x^11-11*x^10+1)/((1-x^11)*(1-x)^2). (End)

More terms from Correction. Typo at the sum formula for the g.f.: the summation index should not read "1<=k<10" but "1<=k<11" (see corrected formula).

A010881 Simple periodic sequence: n mod 12.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 0

N. J. A. Sloane

The value of the rightmost digit in the base-12 representation of n. - Hieronymus Fischer, Jun 11 2007

			a(27) = 3 since 27 = 12*2+3.

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

Partial sums: A130490. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

Mod[Range[0, 100], 12] (* Paolo Xausa, Feb 02 2024 *)

A010881(n)=n%12 \\ M. F. Hasler, Sep 25 2014

From Hieronymus Fischer, May 31 2007: (Start)
a(n) = n mod 12.
Complex representation: a(n) = (1/12)*(1-r^n)*Sum_{k=1..11} k*Product_{m=1..11, m<>k} (1-r^(n-m)) where r = exp(Pi/6*i) = (sqrt(3)+i)/2 and i = sqrt(-1).
Trigonometric representation: a(n) = (512/3)^2*(sin(n*Pi/12))^2*Sum_{k=1..11} k*Product_{m=1..11, m<>k} (sin((n-m)*Pi/12))^2.
G.f.: (Sum_{k=1..11} k*x^k)/(1-x^12).
G.f.: x*(11*x^12-12*x^11+1)/((1-x^12)*(1-x)^2). (End)
From Hieronymus Fischer, Jun 11 2007: (Start)
a(n) = (n mod 2)+2*(floor(n/2) mod 6) = A000035(n)+2*A010875(A004526(n)).
a(n) = (n mod 3)+3*(floor(n/3) mod 4) = A010872(n)+3*A010873(A002264(n)).
a(n) = (n mod 4)+4*(floor(n/4) mod 3) = A010873(n)+4*A010872(A002265(n)).
a(n) = (n mod 6)+6*(floor(n/6) mod 2) = A010875(n)+6*A000035(A152467(n)).
a(n) = (n mod 2)+2*(floor(n/2) mod 2)+4*(floor(n/4) mod 3) = A000035(n)+2*A000035(A004526(n))+4*A010872(A002265(n)). (End)
a(A001248(k) + 17) = 6 for k>2. - Reinhard Zumkeller, May 12 2010
a(n) = A034326(n+1)-1. - M. F. Hasler, Sep 25 2014

A130487 a(n) = Sum_{k=0..n} (k mod 9) (Partial sums of A010878).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 36, 37, 39, 42, 46, 51, 57, 64, 72, 72, 73, 75, 78, 82, 87, 93, 100, 108, 108, 109, 111, 114, 118, 123, 129, 136, 144, 144, 145, 147, 150, 154, 159, 165, 172, 180, 180, 181, 183, 186, 190, 195, 201, 208, 216, 216, 217, 219, 222, 226

Offset: 0

Hieronymus Fischer, May 31 2007

Let A be the Hessenberg n X n matrix defined by A[1,j]=j mod 9, 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 (1,0,0,0,0,0,0,0,1,-1).

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

a:=[0,1,3,6,10,15,21,28,36,36];; for n in [11..71] do a[n]:=a[n-1]+a[n-9]-a[n-10]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,36,36]; [n le 10 select I[n] else Self(n-1) + Self(n-9) - Self(n-10): n in [1..71]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-9*x^8+8*x^9)/((1-x^9)*(1-x)^3), x, n+1), x, n), n = 0 .. 70); # G. C. Greubel, Aug 31 2019

Accumulate[PadRight[{},120,Range[0,8]]] (* Harvey P. Dale, Dec 19 2018 *)
Accumulate[Mod[Range[0,100],9]] (* Harvey P. Dale, Oct 16 2021 *)

a(n) = sum(k=0, n, k % 9); \\ Michel Marcus, Apr 28 2018

def A130487_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-9*x^8+8*x^9)/((1-x^9)*(1-x)^3)).list()
A130487_list(70) # G. C. Greubel, Aug 31 2019

a(n) = 36*floor(n/9) + A010878(n)*(A010878(n) + 1)/2.
G.f.: (Sum_{k=1..8} k*x^k)/((1-x^9)*(1-x)).
G.f.: x*(1 - 9*x^8 + 8*x^9)/((1-x^9)*(1-x)^3).

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).

A130909 Simple periodic sequence (n mod 16).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Hieronymus Fischer, Jun 11 2007

The value of the rightmost digit in the base-16 representation of n. Also, the equivalent value of the two rightmost digits in the base-4 representation of n. Also, the equivalent value of the four rightmost digits in the base-2 representation of n.

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

Cf. partial sums A130910. Other related sequences A010872, A010873, A130481, A130482, A130483, A130486.

See A010877 for a general formula in terms of powers of -1 (for period 2^k).

a(n)=n%16 \\ Charles R Greathouse IV, Jul 13 2016

def A130909(n): return n&15 # Chai Wah Wu, Jan 18 2023

a(n) = n mod 16 = n-16*floor(n/16).
G.f.: g(x) = (Sum_{k=1..15} k*x^k)/(1-x^16).
G.f.: g(x) = x(15x^16-16x^15+1)/((1-x^16)(1-x)^2).
a(n) = A000035(n) + 2*A010877(A004526(n)).
a(n) = A010873(n) + 4*A010873(A002265(n)).
a(n) = A010877(n) + 8*A000035(floor(n/8)).
a(n) = (1/2)*(15 - ( - 1)^n - 2*( - 1)^(b/4) - 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8) - 8*( - 1)^((b - 6 + ( - 1)^n + 2*( - 1)^(b/4) + 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8))/16)) where b = 2n - 1 + ( - 1)^n.
a(n) = n mod 2+2*(floor(n/2)mod 2)+4*(floor(n/4)mod 2)+8*(floor(n/8)mod 2).
a(n) = (1/2)*(15-(-1)^n-2*(-1)^floor(n/2)-4*(-1)^floor(n/4)-8*(-1)^floor(n/= 8)).
Complex representation: a(n) = (1/16)*(1-r^n)*sum{1<=k<16, k*product{1<=m<16,m<>k, (1-r^(n-m))}} where r=exp(Pi/8*i)=(sqrt(2+sqrt(2))+i*sqrt(2-sqrt(2)))/2 and i=sqrt(-1).
Trigonometric representation: a(n) = 2^22*(sin(n*Pi/16))^2*sum{1<=k<16, k*product{1<=m<16,m<>k, (sin((n-m)*Pi/16))^2}}.
a(n) = (1/2)*(15-(-1)^p(0,n)-2*(-1)^p(1,n)-4*(-1)^p(2,n)-8*(-1)^p(3,n)) where p(k,n) is defined recursively by p(0,n)=n, p(k,n)=1/4*(2*p(k-1,n)-1+(-1)^p(k-1,n)).

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)

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).

cp's OEIS Frontend

A010879 Final digit of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A010877 a(n) = n mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A010878 a(n) = n mod 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A010880 a(n) = n mod 11.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

A010881 Simple periodic sequence: n mod 12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A130487 a(n) = Sum_{k=0..n} (k mod 9) (Partial sums of A010878).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica