A133888 -id:A133888 - cp's OEIS frontend

A133878 n modulo 8 repeated 8 times.

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 8^2=64.

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

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133888, A133880, A133890, A133900, A133910.

Flatten[Join[Table[PadRight[{},8,n],{n,7}],Table[PadRight[{},8,n],{n,0,7}]]] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=(1+floor(n/8)) mod 8.
a(n)=1+floor(n/8)-8*floor((n+8)/64).
a(n)=(((n+8) mod 64)-(n mod 8))/8.
a(n)=((n+8-(n mod 8))/8) mod 8.
G.f. g(x)=(1-x^8)(1+2x^8+3x^16+4x^24+5x^32+6x^40+7x^48)/((1-x)(1-x^64)).
G.f. g(x)=(1-x^8)*sum{0<=k<7, (k+1)*x^(8*k)}/((1-x)(1-x^64)).
G.f. g(x)=(7x^64-8x^56+1)/((1-x)(1-x^8)(1-x^64)).

A133891 a(n) = binomial(n+p,n) mod p, where p=12.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 16 2007

nonn

Periodic with length 6*12^2 = 864 = A133900(12).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636, A133910.
See A133872, A133873, A133875, A133877, A133884, A133886, A133888, A133889, A133890 for sequences with different values of p.
See A133900 for the respective periods regarding other values of p.

Table[Mod[Binomial[n+12,n],12],{n,0,110}] (* Harvey P. Dale, Oct 13 2017 *)

a(n) = binomial(n+12,12) mod 12.

A133898 Numbers m such that binomial(m+8,m) mod 8 = 0.

Original entry on oeis.org

56, 57, 58, 59, 60, 61, 62, 63, 120, 121, 122, 123, 124, 125, 126, 127, 184, 185, 186, 187, 188, 189, 190, 191, 248, 249, 250, 251, 252, 253, 254, 255, 312, 313, 314, 315, 316, 317, 318, 319, 376, 377, 378, 379, 380, 381, 382, 383, 440, 441, 442, 443, 444

Offset: 0

Hieronymus Fischer, Oct 20 2007

Partial sums of the sequence 56,1,1,1,1,1,1,1,57,1,1,1,1,1,1,1,57, ... which has period 8.

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

Cf. A000040, A133620, A133621, A133623, A133630, A133635.

Cf. A133878, A133888, A133890, A133900, A133910.

Select[Range[500],Mod[Binomial[#+8,#],8]==0&] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{56,57,58,59,60,61,62,63,120},60] (* Harvey P. Dale, Apr 07 2025 *)

a(n)=8*n+56-n%8*7 \\ Charles R Greathouse IV, Oct 13 2022

a(n)=8n+56-7*(n mod 8). [Corrected by Charles R Greathouse IV, Oct 13 2022]
G.f.: g(x)=(56+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8)/((1-x^8)(1-x)).
G.f.: g(x)=(56-55x-x^9) /((1-x^8)(1-x)^2).

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A133878 n modulo 8 repeated 8 times.

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A133891 a(n) = binomial(n+p,n) mod p, where p=12.

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A133898 Numbers m such that binomial(m+8,m) mod 8 = 0.

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