Oscar Takeshita - cp's OEIS frontend

A105332 a(n) = n*(n+1)/2 mod 8.

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

Offset: 0

Oscar Takeshita, May 01 2005

Periodic with period length 16. - Ray Chandler, Apr 18 2025

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

Cf. A000217.

See A105198 for further information.

Mod[Accumulate[Range[0,110]],8] (* Harvey P. Dale, Nov 03 2013 *)

def A105332(n): return (n*(n+1)>>1)&7 # Chai Wah Wu, Apr 17 2025

A105333 a(n) = n*(n+1)/2 mod 16.

Original entry on oeis.org

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

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Triangular numbers mod 16. - Harvey P. Dale, Oct 12 2012

Periodic with period length 32. - Ray Chandler, Apr 18 2025

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

Cf. A000217.

See A105198 for further information.

Mod[#,16]&/@Accumulate[Range[0,90]] (* Harvey P. Dale, Oct 12 2012 *)

def A105333(n): return (n*(n+1)>>1)&15 # Chai Wah Wu, Apr 17 2025

From Chai Wah Wu, Apr 17 2025: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) - a(n-14) + a(n-15) - a(n-16) + a(n-17) - a(n-18) + a(n-19) - a(n-20) + a(n-21) - a(n-22) + a(n-23) - a(n-24) + a(n-25) - a(n-26) + a(n-27) - a(n-28) + a(n-29) - a(n-30) + a(n-31) for n > 30.
G.f.: x*(-x^28 - 2*x^27 - 4*x^26 - 6*x^25 - 9*x^24 + 4*x^23 - 16*x^22 + 12*x^21 - 25*x^20 + 18*x^19 - 20*x^18 + 6*x^17 - 17*x^16 + 8*x^15 - 16*x^14 + 8*x^13 - 17*x^12 + 6*x^11 - 20*x^10 + 18*x^9 - 25*x^8 + 12*x^7 - 16*x^6 + 4*x^5 - 9*x^4 - 6*x^3 - 4*x^2 - 2*x - 1)/((x - 1)*(x^2 + 1)*(x^4 + 1)*(x^8 + 1)*(x^16 + 1)). (End)

A105334 a(n) = n*(n+1)/2 mod 32.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 4, 13, 23, 2, 14, 27, 9, 24, 8, 25, 11, 30, 18, 7, 29, 20, 12, 5, 31, 26, 22, 19, 17, 16, 16, 17, 19, 22, 26, 31, 5, 12, 20, 29, 7, 18, 30, 11, 25, 8, 24, 9, 27, 14, 2, 23, 13, 4, 28, 21, 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, 21, 28, 4, 13, 23, 2, 14, 27, 9, 24

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 64. - Ray Chandler, Apr 18 2025

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

Cf. A000217.

See A105198 for further information.

Mod[Accumulate[Range[0,80]],32] (* Harvey P. Dale, Sep 01 2017 *)

def A105334(n): return (n*(n+1)>>1)&31 # Chai Wah Wu, Apr 17 2025

From Chai Wah Wu, Apr 17 2025: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) - a(n-14) + a(n-15) - a(n-16) + a(n-17) - a(n-18) + a(n-19) - a(n-20) + a(n-21) - a(n-22) + a(n-23) - a(n-24) + a(n-25) - a(n-26) + a(n-27) - a(n-28) + a(n-29) - a(n-30) + a(n-31) - a(n-32) + a(n-33) - a(n-34) + a(n-35) - a(n-36) + a(n-37) - a(n-38) + a(n-39) - a(n-40) + a(n-41) - a(n-42) + a(n-43) - a(n-44) + a(n-45) - a(n-46) + a(n-47) - a(n-48) + a(n-49) - a(n-50) + a(n-51) - a(n-52) + a(n-53) - a(n-54) + a(n-55) - a(n-56) + a(n-57) - a(n-58) + a(n-59) - a(n-60) + a(n-61) - a(n-62) + a(n-63) for n > 62.
G.f.: x*(-x^60 - 2*x^59 - 4*x^58 - 6*x^57 - 9*x^56 - 12*x^55 - 16*x^54 + 12*x^53 - 25*x^52 + 2*x^51 - 4*x^50 - 10*x^49 - 17*x^48 + 8*x^47 - 32*x^46 + 24*x^45 - 49*x^44 + 38*x^43 - 68*x^42 + 50*x^41 - 57*x^40 + 28*x^39 - 48*x^38 + 36*x^37 - 41*x^36 + 10*x^35 - 36*x^34 + 14*x^33 - 33*x^32 + 16*x^31 - 32*x^30 + 16*x^29 - 33*x^28 + 14*x^27 - 36*x^26 + 10*x^25 - 41*x^24 + 36*x^23 - 48*x^22 + 28*x^21 - 57*x^20 + 50*x^19 - 68*x^18 + 38*x^17 - 49*x^16 + 24*x^15 - 32*x^14 + 8*x^13 - 17*x^12 - 10*x^11 - 4*x^10 + 2*x^9 - 25*x^8 + 12*x^7 - 16*x^6 - 12*x^5 - 9*x^4 - 6*x^3 - 4*x^2 - 2*x - 1)/((x - 1)*(x^2 + 1)*(x^4 + 1)*(x^8 + 1)*(x^16 + 1)*(x^32 + 1)). (End)

A105335 a(n) = n*(n+1)/2 mod 64.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 2, 14, 27, 41, 56, 8, 25, 43, 62, 18, 39, 61, 20, 44, 5, 31, 58, 22, 51, 17, 48, 16, 49, 19, 54, 26, 63, 37, 12, 52, 29, 7, 50, 30, 11, 57, 40, 24, 9, 59, 46, 34, 23, 13, 4, 60, 53, 47, 42, 38, 35, 33, 32, 32, 33, 35, 38, 42, 47, 53, 60, 4, 13

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 128. - Ray Chandler, Apr 18 2025

Index entries for linear recurrences with constant coefficients, order 127.

See A105198 for further information.

Cf. A018819.

Mod[Accumulate[Range[0,80]],64] (* Harvey P. Dale, Jul 17 2020 *)

G.f. = p(x)/q(x) where p(x) has degree 124, and q(x) = (x-1)*(x^2+1)*(x^4+1)*(x^8+1)*(x^16+1)*(x^32+1)*(x^64+1), which has coefficients which are alternately +1 and -1 (when expanded). Compare the g.f. for binary partitions, A018819. - Harvey P. Dale and N. J. A. Sloane, Jul 17 2020

A105336 a(n) = n*(n+1)/2 mod 128.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 8, 25, 43, 62, 82, 103, 125, 20, 44, 69, 95, 122, 22, 51, 81, 112, 16, 49, 83, 118, 26, 63, 101, 12, 52, 93, 7, 50, 94, 11, 57, 104, 24, 73, 123, 46, 98, 23, 77, 4, 60, 117, 47, 106, 38, 99, 33, 96, 32, 97, 35, 102, 42

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 256. - Ray Chandler, Apr 18 2025

Index entries for linear recurrences with constant coefficients, order 255.

See A105198 for further information.

Mod[Accumulate[Range[0,70]],128] (* Harvey P. Dale, Oct 16 2013 *)

A105337 a(n) = n*(n+1)/2 mod 256.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 20, 44, 69, 95, 122, 150, 179, 209, 240, 16, 49, 83, 118, 154, 191, 229, 12, 52, 93, 135, 178, 222, 11, 57, 104, 152, 201, 251, 46, 98, 151, 205, 4, 60, 117, 175, 234, 38, 99

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 512. - Ray Chandler, Apr 18 2025

Index entries for linear recurrences with constant coefficients, order 511.

See A105198 for further information.

Mod[Accumulate[Range[0,70]],256] (* Harvey P. Dale, Nov 20 2020 *)

A105338 a(n) = n*(n+1)/2 mod 512.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 16, 49, 83, 118, 154, 191, 229, 268, 308, 349, 391, 434, 478, 11, 57, 104, 152, 201, 251, 302, 354, 407, 461, 4, 60, 117, 175, 234

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 1024. - Ray Chandler, Apr 18 2025

Index entries for linear recurrences with constant coefficients, order 1023.

See A105198 for further information.

Mod[#,512]&/@Accumulate[Range[0,60]] (* Harvey P. Dale, Dec 01 2018 *)

A105339 a(n) = n*(n+1)/2 mod 1024.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 11, 57, 104, 152, 201, 251, 302, 354, 407, 461, 516, 572, 629

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Periodic with period length 2048. - Ray Chandler, Apr 18 2025

Index entries for linear recurrences with constant coefficients, order 2047.

See A105198 for further information.

Mod[Accumulate[Range[0,100]],1024] (* Harvey P. Dale, Jun 30 2017 *)

A105340 a(n) = n*(n+1)/2 mod 2048.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 32

Offset: 0

Oscar Takeshita, May 01 2005

Period 4096. - Charles R Greathouse IV, Oct 16 2012

Index entries for linear recurrences with constant coefficients, order 4095.

See A105198 for further information.

Cf. A000217.

Mod[Accumulate[Range[0,70]],2048] (* Harvey P. Dale, Sep 18 2019 *)

a(n)=n*(n+1)/2%2048 \\ Charles R Greathouse IV, Oct 16 2012

def A105340(n): return (n*(n+1)>>1)&2047 # Chai Wah Wu, Apr 17 2025

Removed formulas that were wrong at indices >= 65 because the "mod 2048" part of the definition had been ignored R. J. Mathar, Jan 26 2010

A105198 a(n) = n(n+1)/2 mod 4.

Original entry on oeis.org

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

Offset: 0

Oscar Takeshita, Apr 11 2005

0,1,3,2,2,3,1,0 repeated indefinitely.

If N is any power of 2 then n(n+1)/2 mod N is a repeating pattern of length 2N. Moreover, the first N digits form a permutation P of A={0,1,...,N-1}. The subsequent N digits are P in the reversed order. The technique is useful for the generation of arbitrarily large pseudo-random permutations.

Antti Karttunen, Table of n, a(n) for n = 0..8191
O. Y. Takeshita and D. J. Costello, Jr., New Deterministic Interleaver Designs for Turbo-Codes, IEEE Trans. Inform. Theory, vol. 46, no. 6, pp. 1988-2006, Sept. 2000.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Cf. triangular numbers A000217, A105332-A105340.

One less than A110549, A133882 shifted once right, with zero inserted to front.

[ -Ceiling(n/2)*(-1)^n mod 4 : n in [0..100]]; // Wesley Ivan Hurt, Jul 13 2014

for n from 0 to 300 do printf(`%d,`, n*(n+1)/2 mod 4) od: # James Sellers, Apr 21 2005
A105198:=n->-ceil(n/2)*(-1)^n mod 4: seq(A105198(n), n=0..100); # Wesley Ivan Hurt, Jul 13 2014

Table[Mod[-Ceiling[n/2] (-1)^n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 13 2014 *)

Vec((x+2*x^2+2*x^4+x^5)/(1-x+x^2-x^3+x^4-x^5+x^6-x^7) + O(x^90)) \\ Michel Marcus, Jul 13 2014

def A105198(n): return (n*(n+1)>>1)&3 # Chai Wah Wu, Apr 17 2025

(define (A105198 n) (modulo (* 1/2 n (+ 1 n)) 4)) ;; Antti Karttunen, Aug 10 2017

From Paul Barry, Jul 26 2005: (Start)
G.f.: (x + 2x^2 + 2x^4 + x^5)/(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7).
a(n) = cos(3*Pi*n/4 + Pi/4)/2 + (1/2 - sqrt(2)/2)*sin(3*Pi*n/4 + Pi/4) - (1/2 + sqrt(2)/2)*cos(Pi*n/4 + Pi/4) - sin(Pi*n/4 + Pi/4)/2 - cos(Pi*n/2)/2 + sin(Pi*n/2)/2 + 3/2. (End)
a(n) = (((n+1)^5 - n^5 - 1) mod 120)/30. - Gary Detlefs, Mar 25 2012
a(n) = -ceiling(n/2)*(-1)^n mod 4. - Wesley Ivan Hurt, Jul 13 2014

More terms from James Sellers, Apr 21 2005

cp's OEIS Frontend

User: Oscar Takeshita

A105332 a(n) = n*(n+1)/2 mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A105333 a(n) = n*(n+1)/2 mod 16.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A105334 a(n) = n*(n+1)/2 mod 32.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A105335 a(n) = n*(n+1)/2 mod 64.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A105336 a(n) = n*(n+1)/2 mod 128.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A105337 a(n) = n*(n+1)/2 mod 256.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A105338 a(n) = n*(n+1)/2 mod 512.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A105339 a(n) = n*(n+1)/2 mod 1024.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica