A088911 -id:A088911 - cp's OEIS frontend

A131030 Period 6: repeat [16, 7, 7, 16, 25, 25].

16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16, 7, 7, 16, 25, 25, 16

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Sixth column of triangular array T defined in A131022.

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. A131022. Other columns of T are in A088911, A131026, A131027, A131028, A131029.

m:=79; [ [16, 7, 7, 16, 25, 25][(n-1) mod 6 + 1]: n in [1..m] ];

seq(op([16, 7, 7, 16, 25, 25]), n=0..30); # Wesley Ivan Hurt, Oct 02 2018

{m=79; for(n=1, m, r=(n-1)%6; print1(if(r==0||r==3, 16, if(r==1||r==2, 7, 25)), ","))}

a(1) = a(4) = 16, a(2) = a(3) = 7, a(5) = a(6) = 25; for n > 6, a(n) = a(n-6).
G.f.: x*(16 - 25*x + 25*x^2)/((1-x)*(1 - x + x^2)).
a(n) = 16 + 9*cos(n*Pi/3) - 3*sqrt(3)*sin(n*Pi/3). - Wesley Ivan Hurt, Sep 26 2018

A291962 Decimal repunits written in base 2.

Original entry on oeis.org

0, 1, 1011, 1101111, 10001010111, 10101101100111, 11011001000000111, 100001111010001000111, 101010011000101011000111, 110100111110110101111000111, 1000010001110100011010111000111, 1010010110010001100001100111000111, 1100111011110101111010000000111000111

Offset: 0

Felix Fröhlich, Sep 06 2017

Interpreting A002275 as binary numbers and converting to decimal gives A000225. This sequence gives the resulting terms of the "reverse" operation.

The n least significant bits of a(n) seem to converge to A088911 as n increases.

Seiichi Manyama, Table of n, a(n) for n = 0..301

Cf. A000225, A002275, A007088, A088911.

Table[FromDigits@ IntegerDigits[Floor[10^n/9], 2], {n, 0, 12}] (* Michael De Vlieger, Sep 06 2017 *)
FromDigits[IntegerDigits[#,2]]&/@Table[FromDigits[PadRight[{},n,1]],{n,0,20}] (* Harvey P. Dale, Apr 01 2023 *)

a(n) = subst(Pol(binary((10^n-1)/9)), x, 10)

def a(n): return 0 if n == 0 else int(bin(int("1"*n))[2:])
print([a(n) for n in range(13)]) # Michael S. Branicky, Apr 26 2022

a(n) = A007088(A002275(n)).

A153234 a(n) = floor(2^n/9).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 14, 28, 56, 113, 227, 455, 910, 1820, 3640, 7281, 14563, 29127, 58254, 116508, 233016, 466033, 932067, 1864135, 3728270, 7456540, 14913080, 29826161, 59652323, 119304647, 238609294, 477218588, 954437176, 1908874353, 3817748707, 7635497415, 15270994830

Offset: 0

Paul Curtz, Dec 21 2008

Partial sums of A113405. - Mircea Merca, Dec 28 2010
Dubickas proves that infinitely many terms of this sequence are composite. - Charles R Greathouse IV, Feb 04 2016
Parity from a(4) onward gives A088911 (Period 6: repeat [1, 1, 1, 0, 0, 0]). - Jeremy Gardiner, Nov 04 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,3,-2).

Cf. A113405.

[Round((2*2^n-9)/18): n in [0..40]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(2^n/9),n=0..25); # Mircea Merca, Dec 28 2010

Table[Floor[2^n/9], {n, 0, 37}] (* Michael De Vlieger, Mar 26 2016 *)

a(n)=2^n\9 \\ Charles R Greathouse IV, Oct 07 2015

x='x+O('x^44); concat(vector(4), Vec(x^4/((x-1)*(2*x-1)*(1+x)*(x^2-x+1)))) \\ Altug Alkan, Mar 25 2016

[floor(2^n/9) for n in (0..40)] # G. C. Greubel, Jun 05 2019

a(n+1) - 2*a(n) = A088911(n+3).
a(n) + a(n+3) = 2^n - 1 = A000225(n), n > 0.
From Mircea Merca, Dec 28 2010: (Start)
a(n) = round((2*2^n-9)/18) = floor((2^n-1)/9) = ceiling((2^n-8)/9).
a(n) = a(n-6) + 7*2^(n-6), n > 5. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + 3*a(n-4) - 2*a(n-5).
G.f.: x^4 / ( (1-2*x)*(1-x^2)*(1-x+x^2) ).
a(n) + a(n+1) = A111927(n). - R. J. Mathar, Apr 08 2013

More terms from Vincenzo Librandi, Jun 25 2011

A131025 Antidiagonal sums of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 9, 6, 16, 11, 27, 22, 50, 50, 101, 114, 215, 255, 471, 552, 1024, 1145, 2169, 2290, 4460, 4460, 8921, 8556, 17477, 16383, 33861, 31674, 65536, 62255, 127791, 124510, 252302, 252302, 504605, 514446, 1019051, 1048575, 2067627

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

nonn

			For first seven rows of T see A131022 or A129339.

Michael De Vlieger, Table of n, a(n) for n = 1..6644
Scott M. Bailey and Donald M. Larson, The A(1)-module structure of the homology of Brown-Gitler spectra, arXiv:2107.01316 [math.AT], 2021.

Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131023 (first subdiagonal of T), A131024 (row sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

m:=43; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];

CoefficientList[Series[(1 - 3 x^2 + 2 x^4 + 2 x^6 - 2 x^8 + x^9)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - 2 x^2)*(1 - 3 x^2 + 3 x^4)), {x, 0, 42}], x] (* Michael De Vlieger, Oct 26 2021 *)

{m=43; M=matrix(m, m); for(j=1, m, M[j, 1]=if((j-1)%6<3, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, print1(sum(k=1, (j+1)\2, M[j-k+1, k]), ","))}

G.f.: (1-3*x^2+2*x^4+2*x^6-2*x^8+x^9)/((1-x)*(1+x)*(1-x+x^2)*(1-2*x^2)*(1-3*x^2+3*x^4)).

A112713 Expansion of x/(1 - x + x^5 - x^6).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0

Offset: 0

Paul Barry, Sep 15 2005

Decimal expansion of 10000/900009. - Elmo R. Oliveira, May 08 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,-1,1).

Cf. A000035, A133872, A088911, A131078, A112712.

CoefficientList[Series[x/(1 - x + x^5 - x^6), {x, 0, 100}], x] (* or *) LinearRecurrence[{1, 0, 0, 0, -1, 1}, {0, 1, 1, 1, 1, 1}, 100] (* Harvey P. Dale, Feb 16 2014 *)

G.f.: x/(1 - x + x^5 - x^6) = x/((1-x)*(1+x)*(1-x+x^2-x^3+x^4)).
a(n) = a(n-1) - a(n-5) + a(n-6).
a(n) = a(n-10).
a(n) = (Sum_{k=0..floor((n+2)/2)} (-1)^(k+1)*C(n-k+2, k-1)*F(n-2*k+2)) mod 2.
a(n) = A112712(n) mod 2.

Incorrect g.f. removed by Georg Fischer, May 15 2019

A131023 First subdiagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 37, 101, 256, 593, 1267, 2534, 4825, 8921, 16384, 30581, 58975, 117950, 242461, 504605, 1048576, 2156201, 4371451, 8742902, 17308657, 34085873, 67108864, 132623405, 263652487, 527304974, 1059392917, 2133134741

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Also first differences of main diagonal A129339.

			For first seven rows of T see A131022 or A129339.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6).

Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

m:=34; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n+1, n]: n in [1..m-1] ];

{m=33; v=concat([1, 2, 3, 4],vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v}

a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).
G.f.: x*(1-3*x+2*x^2+x^3)/((1-2*x)*(1-3*x+3*x^2)).
a(n) = A057681(n-1) + 2^(n-2), a(1) = 1. - Bruno Berselli, Feb 17 2011

A131024 Row sums of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

Original entry on oeis.org

1, 3, 7, 11, 16, 22, 36, 73, 175, 431, 1024, 2290, 4824, 9649, 18571, 34955, 65536, 124510, 242460, 484921, 989527, 2038103, 4194304, 8565754, 17308656, 34617313, 68703187, 135812051, 268435456, 532087942, 1059392916, 2118785833, 4251920575, 8546887871

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Sum of n-th row equals (n+1)-th term of main diagonal minus (n+1)-th term of first column. A088911 has offset 0, so a(n) = A129339(n+1) - A088911(n).

			For first seven rows of T see A131022 or A129339.

Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,15,-6).

Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131023 (first subdiagonal of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

m:=32; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ &+[ M[j, k]: k in [1..j] ]: j in [1..m] ];

lista(m) = my(M=matrix(m, m)); for(j=1, m, M[j, 1]=if((j-1)%6<3, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, print1(sum(k=1, j, M[j, k]), ", "))

G.f.: x*(1-3*x+3*x^2-3*x^3+6*x^4-4*x^5+x^6)/((1-x)*(1+x)*(1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).

A350589 Sum over all partitions of [n] of the number of blocks containing their own index.

Original entry on oeis.org

0, 1, 3, 9, 30, 112, 464, 2109, 10411, 55351, 314772, 1903878, 12189432, 82274309, 583389847, 4332513061, 33607736990, 271657081128, 2283282938288, 19916981288017, 179994994948647, 1682624910161483, 16247280435775188, 161833756265886822, 1660836884761337248

Offset: 0

Alois P. Heinz, Jan 07 2022

nonn

Also the number of partitions of [n] where the first k elements are marked (1 <= k <= n) and at least k blocks contain their own index: a(3) = 9 = 5 + 3 + 1: 1'23, 1'2|3, 1'3|2, 1'|23, 1'|2|3, 1'3|2', 1'|2'3, 1'|2'|3, 1'|2'|3'.

			a(3) = 9 = 1 + 1 + 2 + 2 + 3: 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..575
Wikipedia, Partition of a set

Cf. A000110, A005490, A088911, A108087, A259691, A347420, A350648.

b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> add(b(n-i, i), i=1..n):
seq(a(n), n=0..25);

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];
a[n_] := Sum[b[n - i, i], {i, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} A108087(n-k,k).
a(n) = Sum_{k=1..n} k * A259691(n-1,k).
a(n) = Sum_{k=1..n} A259691(n,k)/k.
a(n) = A347420(n) - A000110(n).
a(n) = 1 + A005490(n) - A000110(n).
a(n) mod 2 = A088911(n+5).

A056829 Nearest integer to n^2/6.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 17, 20, 24, 28, 33, 38, 43, 48, 54, 60, 67, 74, 81, 88, 96, 104, 113, 122, 131, 140, 150, 160, 171, 182, 193, 204, 216, 228, 241, 254, 267, 280, 294, 308, 323, 338, 353, 368, 384, 400, 417, 434, 451, 468, 486, 504

Offset: 0

N. J. A. Sloane, Sep 02 2000

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Round[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1},{0,0,1,2,3,4,6,8},60] (* Harvey P. Dale, Oct 11 2020 *)

G.f.: -x^2*(1+x^4) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ). - R. J. Mathar, Jul 10 2015

a(n) = A056827(n)+A088911(n+3). - R. J. Mathar, Jul 10 2015

A117907 Expansion of x + (1-x)^2/(1-x^6).

Original entry on oeis.org

1, -1, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -2, 1

Offset: 0

Paul Barry, Apr 01 2006

Diagonal sums of A117906.

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

Cf. A088911, A117906.

R:=PowerSeriesRing(Integers(), 90); Coefficients(R!( x + (1-x)^2/(1-x^6) )); // G. C. Greubel, Oct 20 2021

(* From Harvey P. Dale, Nov 29 2013 *)
CoefficientList[Series[x+(1-x)^2/(1-x^6), {x,0,90}], x]
Join[{1,-1}, LinearRecurrence[{-1,-1,-1,-1,-1}, {1,0,0,0,1}, 90]]
PadRight[{1,-1}, 90, {1,-2,1,0,0,0}] (* End *)

def A117907(n): return (-1)^n if (n<2) else (((5*n-1)//3)%2) - 3*bool(n%6==1)
[A117907(n) for n in (0..90)] # G. C. Greubel, Oct 20 2021

G.f.: (1 +x^2 +x^3 +x^4 +x^5 +x^6)/(1 +x +x^2 +x^3 +x^4 +x^5).

a(n) = floor((5*n-1)/3) mod 2 - 3*[(n mod 6) = 1], n >= 2, with a(0) = 1, a(1) = -1. - G. C. Greubel, Oct 20 2021

cp's OEIS Frontend

A131030 Period 6: repeat [16, 7, 7, 16, 25, 25].

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A291962 Decimal repunits written in base 2.

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A153234 a(n) = floor(2^n/9).

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A131025 Antidiagonal sums of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

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A112713 Expansion of x/(1 - x + x^5 - x^6).

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A131023 First subdiagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

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A131024 Row sums of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.

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