A130909 -id:A130909 - cp's OEIS frontend

A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 175, 186, 198, 211, 225, 240, 240, 241, 243, 246, 250, 255, 261, 268, 276, 285, 295, 306, 318, 331, 345, 360, 360, 361, 363, 366, 370, 375, 381, 388

Offset: 0

Hieronymus Fischer, Jun 11 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Accumulate[Mod[Range[0,60],16]] (* Harvey P. Dale, May 30 2020 *)

a(n)=120*floor(n/16)+A130909(n)*(A130909(n)+1)/2. - G.f.: g(x)=(sum{1<=k<16, k*x^k})/((1-x^16)(1-x)). Also: g(x)=x(15x^16-16x^15+1)/((1-x^16)(1-x)^3).

a(n) = +a(n-1) +a(n-16) -a(n-17). G.f. ( 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 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +14*x^13 +15*x^14) ) / ( (1+x) *(1+x^2) *(1+x^4) *(1+x^8) *(x-1)^2 ). - R. J. Mathar, Nov 05 2011

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

A070370 a(n) = 5^n mod 16.

Original entry on oeis.org

1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1

Offset: 0

N. J. A. Sloane, May 12 2002

Period 4: repeat [1, 5, 9, 13].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1). [_R. J. Mathar_, Apr 20 2010]

Cf. A000351, A130909.

&cat [[1, 5, 9, 13]^^30]; // Wesley Ivan Hurt, Jul 06 2016

seq(op([1, 5, 9, 13]), n=0..50); # Wesley Ivan Hurt, Jul 06 2016

Table[Mod[5^n, 16], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)

a(n) = lift(Mod(5, 16)^n); \\ Michel Marcus, Mar 05 2016

[power_mod(5,n,16)for n in range(0,93)] # Zerinvary Lajos, Nov 26 2009

From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-4) for n>3.
G.f.: ( 1+5*x+9*x^2+13*x^3 ) / ( (1-x)*(1+x)*(1+x^2) ). (End)
a(n) = 7-2*((1+I)*(-I)^n+(1-I)*I^n+(-1)^n). - Bruno Berselli, Feb 07 2011
E.g.f.: 5*cosh(x) + 9*sinh(x) - 4*cos(x) - 4*sin(x). - G. C. Greubel, Mar 05 2016
a(n) = A130909(A000351(n)). - Michel Marcus, Jul 06 2016

A070439 a(n) = n^2 mod 16.

Original entry on oeis.org

0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0

Offset: 0

N. J. A. Sloane, May 12 2002

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

Cf. A054580, A130909.

[Modexp(n, 2, 16 ): n in [0..100]]; // Vincenzo Librandi, Mar 28 2016

Table[Mod[n^2,16],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2011 *)
CoefficientList[Series[(x + 4 x^2 + 9 x^3 + 9 x^5 + 4 x^6 + x^7) / (1 - x^8), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 25 2016 *)
PowerMod[Range[0,100],2,16] (* Harvey P. Dale, Jul 23 2023 *)

a(n)=n^2%16 \\ Charles R Greathouse IV, Apr 06 2016

G.f.: -x*(1+4*x+9*x^2+9*x^4+4*x^5+x^6) / ( (x-1)*(1+x)*(x^2+1)*(x^4+1) ). - R. J. Mathar, Jul 27 2015
a(n) = a(n-8). - G. C. Greubel, Mar 24 2016
a(n) = A130909(n^2). - Michel Marcus, Mar 24 2016

A126056 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 16.

Original entry on oeis.org

2, 3, 5, 7, 13, 1, 3, 15, 13, 9, 11, 15, 9, 15, 15, 11, 9, 1, 13, 7, 9, 5, 13, 1, 5, 9, 1, 3, 7, 1, 11, 7, 9, 11, 13, 13, 1, 1, 5, 11, 7, 7, 9, 1, 11, 9, 1, 9

Offset: 1

Artur Jasinski, Dec 17 2006

Cf. A000043, A130909, A126043-A126059.

Mod[MersennePrimeExponent[Range[48]], 16] (* Amiram Eldar, Oct 15 2024 *)

a(n) = A130909(A000043(n)). - Michel Marcus, Apr 07 2018

a(45)-a(47) from Ivan Panchenko, Apr 08 2018

a(48) from Amiram Eldar, Oct 15 2024

A167166 a(n) = n^7 mod 16.

Original entry on oeis.org

0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0

Offset: 0

Zerinvary Lajos, Oct 29 2009

Equivalently: n^(4*m+7) mod 16. - G. C. Greubel, Jun 04 2016

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).

Table[Mod[n^7, 16], {n, 0, 10}] (* G. C. Greubel, Jun 04 2016 *)
PowerMod[Range[0,100],7,16] (* or *) PadRight[{},100,{0,1,0,11,0,13,0,7,0,9,0,3,0,5,0,15}] (* Harvey P. Dale, Jul 29 2018 *)

a(n)=n^7%16 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,7,16)for n in range(0, 93)] #

From R. J. Mathar, Sep 30 2013: (Start)
a(n) = a(n-16).
G.f. -x*(1 +11*x^2 +13*x^4 +7*x^6 +9*x^8 +3*x^10 +5*x^12 +15*x^14) / ( (x-1)*(1+x)*(1+x^2)*(1+x^4)*(1+x^8) ). (End)
a(n) = A130909(A001015(n)). - Michel Marcus, Jun 04 2016

cp's OEIS Frontend

A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

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A010877 a(n) = n mod 8.

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A070370 a(n) = 5^n mod 16.

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A070439 a(n) = n^2 mod 16.

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A126056 Exponents p of the Mersenne primes 2^p - 1 (see A000043) read mod 16.

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A167166 a(n) = n^7 mod 16.

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