cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Hieronymus Fischer, Jun 11 2007

Keywords

Comments

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.

Links

Crossrefs

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).

Programs

Formula

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)).

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

Views

Author

Hieronymus Fischer, May 31 2007

Keywords

Comments

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

Links

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488.

Programs

  • GAP
    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
  • Magma
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    a(n) = sum(k=0, n, k % 11); \\ Michel Marcus, Apr 28 2018
  • Sage
    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

Formula

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).

A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 66, 67, 69, 72, 76, 81, 87, 94, 102, 111, 121, 132, 132, 133, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 198, 199, 201, 204, 208, 213, 219, 226, 234, 243, 253, 264, 264, 265, 267, 270, 274, 279, 285, 292, 300
Offset: 0

Views

Author

Hieronymus Fischer, May 31 2007

Keywords

Comments

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 12, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

Links

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

Programs

  • GAP
    List([0..60], n-> Sum([0..n], k-> k mod 12 )); # G. C. Greubel, Sep 01 2019
  • Magma
    [&+[(k mod 12): k in [0..n]]: n in [0..60]]; // G. C. Greubel, Sep 01 2019
  • Maple
    seq(coeff(series(x*(1-12*x^11+11*x^12)/((1-x^12)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 01 2019
  • Mathematica
    Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,3,6,10,15,21,28,36,45,55,66,66},60] (* Harvey P. Dale, Jan 16 2024 *)
  • PARI
    a(n) = sum(k=0, n, k % 12); \\ Michel Marcus, Apr 29 2018
  • Sage
    [sum(k%12 for k in (0..n)) for n in (0..60)] # G. C. Greubel, Sep 01 2019

Formula

a(n) = 66*floor(n/12) + A010881(n)*(A010881(n) + 1)/2.
G.f.: (Sum_{k=1..11} k*x^k)/((1-x^12)*(1-x)).
G.f.: x*(1 - 12*x^11 + 11*x^12)/((1-x^12)*(1-x)^3).

A010884 Period 5: repeat [1,2,3,4,5].

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Partial sums are given by A130483(n)+n+1. - Hieronymus Fischer, Jun 08 2007
4115/33333 = 0.12345123451234512345... - Eric Desbiaux, Nov 03 2008

Links

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.
Cf. A177038 (decimal expansion of (195+sqrt(65029))/314).

Programs

Formula

a(n) = 1 + (n mod 5). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
G.f.: (5*x^4+4*x^3+3*x^2+2*x+1)/(1-x^5) = (5*x^6-6*x^5+1)/((1-x^5)*(1-x)^2).
a(n) = A010874(n)+1. (End)
a(n) = a(n-5). - Wesley Ivan Hurt, Jan 15 2022

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

Original entry on oeis.org

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

Views

Author

Hieronymus Fischer, Jun 11 2007

Keywords

Links

Crossrefs

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

Programs

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

Formula

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
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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