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

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

0,3

Comments

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

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

Links

Table of n, a(n) for n=0..88.

Index entries for periodic sequences, period 32.

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).

Formula

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)

Mathematica

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

Prog

(Python)
def A105333(n): return (n*(n+1)>>1)&15 # Chai Wah Wu, Apr 17 2025
(PARI) a(n)=1613168760786662789121329248857645840\16^(n%32)%16 \\ Charles R Greathouse IV, Jun 03 2026
(PARI) a(n)=n*(n+1)/2%16 \\ Charles R Greathouse IV, Jun 03 2026

Crossrefs

Cf. A000217.

See A105198 for further information.

Sequence in context: A104619 A194037 A194101 * A126234 A387660 A259604

Adjacent sequences: A105330 A105331 A105332 * A105334 A105335 A105336

Keyword

nonn,easy,changed

Author

Oscar Takeshita, May 01 2005

Status

approved