A129756 -id:A129756 - cp's OEIS frontend

A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).

0, 1, 2, 3, 4, 7, 10, 13, 16, 21, 26, 31, 36, 43, 50, 57, 64, 73, 82, 91, 100, 111, 122, 133, 144, 157, 170, 183, 196, 211, 226, 241, 256, 273, 290, 307, 324, 343, 362, 381, 400, 421, 442, 463, 484, 507, 530, 553, 576, 601, 626, 651, 676, 703, 730, 757, 784, 813

Offset: 0

N. J. A. Sloane

Number of 0..n-1 arrays of 5 elements with zero 2nd differences. - R. H. Hardin, Nov 15 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Investigating the discrepancy property of de Bruijn sequences, University of Guelph (Canada, 2020).
János Pach and Pankaj K. Agarwal, Combinatorial Geometry, p. 220, 1995, Problem 13.10.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A129756 (first differences).

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), this sequence (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a:=[0,1,2,3,4,7];; for n in [7..60] do a[n]:=2*a[n-1]-a[n-2] +a[n-4]-2*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1+x^4)/((1-x)^2*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

f := n->n^2/4+3*n/2+g(n);
g := n->if n mod 2 = 0 then 3 elif n mod 4 = 1 then 9/4 else 13/4; fi;
seq(f(n), n=-3..50);

CoefficientList[Series[x*(1+x^4)/((1-x)^2*(1-x^4)), {x,0,60}], x] (* G. C. Greubel, Sep 12 2019 *)

concat([0], Vec(x*(1+x^4)/((1-x)^2*(1-x^4))+O(x^60))) \\ Charles R Greathouse IV, Sep 26 2012, modified by G. C. Greubel, Sep 12 2019

def A008811_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+x^4)/((1-x)^2*(1-x^4))).list()
A008811_list(60) # G. C. Greubel, Sep 12 2019

G.f.: x*(1+x^4)/((1-x)^2*(1-x^4)).
a(n) = 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6). - R. H. Hardin, Nov 15 2011
a(n) = (-2*(1+(-1)^n)*(-1)^floor(n/2) + 2*n^2 + 5 - (-1)^n)/8. - Tani Akinari, Jul 24 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x) - 2*cos(x))/4. - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/2)*Pi/4 + tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Aug 25 2022
a(n) = 2*floor((n^2 + 4)/8) + (n mod 2). - Ridouane Oudra, Sep 08 2023

A083219 a(n) = n - 2*floor(n/4).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Reinhard Zumkeller, Apr 22 2003

Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - Michel Lagneau, Apr 09 2013

a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - Kevin Ryde, Dec 13 2019

Altug Alkan, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A083220, A129756, A162751 (second highest bit removed).
Cf. A021913, A162330, A285869.
Essentially the same as A018837.

a:= n-> n - 2*floor(n/4):
seq(a(n), n=0..74);  # Alois P. Heinz, Jan 24 2021

Array[# - 2 Floor[#/4] &, 75, 0] (* Michael De Vlieger, Oct 02 2017 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* Harvey P. Dale, Oct 01 2018 *)

a(n) = n - n\4*2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A083220(n)/2.
a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.
G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Aug 28 2008
a(n) = n - A129756(n). - Michel Lagneau, Apr 09 2013
Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - Wolfdieter Lang, May 08 2017
a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - Kevin Ryde, Dec 13 2019
E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - Stefano Spezia, May 27 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - Amiram Eldar, Aug 21 2023

A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

Original entry on oeis.org

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1

Offset: 1

Omar E. Pol, Jul 02 2012

Column k lists the odd numbers repeated k times starting in row k^2.
1 together with the first differences of the row sums give the divisor function A000005.
T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;

Omar E. Pol, Diagram of divisors, figure 1, figure 2.

Row sums give A006218, n >= 1.
Columns (1-5): A005408, A109613, A130823, A129756, A130497.
Cf. A000005, A027750, A010766, A147861, A163100, A212119.

T(n,k) = Sum_{j=1..n} A212119(j,k).

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

A130497 Repetition of odd numbers five times.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 25, 25, 25, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 29, 29, 31, 31, 31

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 31 2007

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

Cf. A129756, A130496.

a:=[1,1,1,1,1,3];; for n in [7..80] do a[n]:=a[n-1]+a[n-5]-a[n-6]; od; a; # G. C. Greubel, Sep 12 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^5)/((1-x)*(1-x^5)) )); // G. C. Greubel, Sep 12 2019

P:=proc(q) local k,n; k:=[]; for n from 0 to q do k:=[op(k),2*floor(n/5)+1]; od; op(k); end: P(77);

Flatten[Table[#,{5}]&/@Range[1,31,2]] (* Harvey P. Dale, Mar 27 2013~ *)

my(x='x+O('x^80)); Vec((1+x^5)/((1-x)*(1-x^5))) \\ G. C. Greubel, Sep 12 2019

def A130497_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)*(1-x^5))).list()
A130497_list(80) # G. C. Greubel, Sep 12 2019

From R. J. Mathar, Mar 17 2010: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6).
G.f.: (1+x)*(1-x+x^2-x^3+x^4)/((1+x+x^2+x^3+x^4) * (1-x)^2 ). (End)
a(n) = 2*floor(n/5)+1 = A130496(n)+1. - Tani Akinari, Jul 24 2013

A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2n + 5, a(4n+2) = 2n + 3.

Original entry on oeis.org

1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, 17, 9, 19, 5, 21, 11, 23, 5, 25, 13, 27, 7, 29, 15, 31, 7, 33, 17, 35, 9, 37, 19, 39, 9, 41, 21, 43, 11, 45, 23, 47, 11, 49, 25, 51, 13, 53, 27, 55, 13, 57, 29, 59, 15, 61, 31, 63, 15, 65, 33, 67

Offset: 0

Paul Curtz, Sep 21 2014

Essentially a permutation of A129756 (odd numbers repeated four times).
a(-1) = 3, a(-2) = a(-3) = 1.
Distance between the first two (2*k+1)'s: 2*k+1 terms. Distance between the last two (2*n+1)'s: 4 terms. Essentially same distances as in -a(-n) = -1, -3, -1, -1, 1, 1, 1, 3, 1, 5, 3, 7, 3, 9, 5, 11, 3, 13, 7, 15, 5, 17, 9, 19, 5, 21, 11, 23, 7, 25, 13, 27, 7, ... .

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).

Cf. A010709, A061037, A121262, A129756, A176895, A246416.

I:=[1,5,3,7,1,9,5,11,3,13,7,15]; [n le 12 select I[n] else Self(n-4)+Self(n-8)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014

A247617:=n->(n+4)*(1-ceil((2-n)/4)-ceil((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq(A247617(n), n=0..50); # Wesley Ivan Hurt, Sep 21 2014

Table[(n + 4) (1 - Ceiling[(2 - n)/4] - Ceiling[(n - 2)/4])/2 + (n + 4) (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) - (n + 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 21 2014 *)

Vec(-(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1)/((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)) + O(x^100)) \\ Colin Barker, Sep 21 2014

a(n) = a(n-4) + a(n-8) - a(n-12).
a(n) * A246416(n) = A061037(n+2).
A246416(n+4) - a(n) = sequence of period 4: [1, 0, 0, 0].
a(n+4) - a(n) = sequence of period 8: [0, 4, 2, 4, 2, 4, 2, 4].
G.f.: -(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 21 2014
a(n) = a(n-8) + sequence of period 4: [2, 8, 4, 8] (= 2*A176895(n)).
a(-n) * A246416(-n) = A061037(n-2).
a(n) = (n+4)*(1-ceiling((2-n)/4)-ceiling((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 21 2014

A320661 a(n) = 2^(n+3) - 6*n - 7.

Original entry on oeis.org

1, 3, 13, 39, 97, 219, 469, 975, 1993, 4035, 8125, 16311, 32689, 65451, 130981, 262047, 524185, 1048467, 2097037, 4194183, 8388481, 16777083, 33554293, 67108719, 134217577, 268435299, 536870749, 1073741655, 2147483473, 4294967115, 8589934405, 17179868991

Offset: 0

Paul Curtz, Nov 14 2018

Companion to A247618 which has the same recurrence.

For this recurrence the main sequence is A000295.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000225, A000295, A010701, A016921, A100402, A129756, A247618.

List([0..40], n -> 2^(n+3) -6*n -7); # G. C. Greubel, Nov 15 2018

[2^(n+3) -6*n -7: n in [0..40]]; // G. C. Greubel, Nov 15 2018

a[n_]:=2^(n+3) - 6*n - 7; Array[a,32,0] (* Amiram Eldar, Nov 14 2018 *)

vector(40, n, n--; 2^(n+3) -6*n -7) \\ G. C. Greubel, Nov 15 2018

[2^(n+3) -6*n -7 for n in range(40)] # G. C. Greubel, Nov 15 2018

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n+1) = a(n-1) + 12*A000225(n). a(-1) = 3.
a(2*n) mod 9 = period 3: repeat [1, 4, 7].
a(2*n+1) mod 9 = 3.
a(n) mod 9 = period 6: repeat [1, 3, 4, 3, 7, 3].
a(n) mod 10 = period 20: repeat [1, 3, 3, 9, 7, 9, 9, 5, 3, 5, 5, 1, 9, 1, 1, 7, 5, 7, 7, 3] = Im(n). Im(n-1) = [3, 1, 3, 3,  9, 7, 9, 9,  5, 3, 5, 5,  1, 9, 1, 1,  7, 5, 7, 7].  Disordered [1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9].
a(n+1) - a(n) = 2^(n+3) - 6.
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: (1-x+6*x^2)/((1-2*x)*(1-x)^2).
E.g.f.: 8*exp(2*x) - (7 + 6*x)*exp(x). (End)

More terms from Amiram Eldar, Nov 14 2018

cp's OEIS Frontend

A008811 Expansion of x*(1+x^4)/((1-x)^2*(1-x^4)).

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A083219 a(n) = n - 2*floor(n/4).

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A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

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A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

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A130497 Repetition of odd numbers five times.

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A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2*n + 5, a(4n+2) = 2*n + 3.

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A320661 a(n) = 2^(n+3) - 6*n - 7.

A008811 Expansion of x(1+x^4)/((1-x)^2(1-x^4)).

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2n + 5, a(4n+2) = 2n + 3.