A026741 -id:A026741 - cp's OEIS frontend

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431

Offset: 0

Ralf W. Grosse-Kunstleve, Dec 11 1999

Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
From Paul Curtz, Apr 17 2014: (Start)
Difference table of A026741/A045896:
   0,      1/6,    1/6,    3/20,     2/15,    5/42, ...
   1/6,      0,  -1/60,   -1/60,    -1/70,   -1/84, ... = 1/6, -A051712/A051713
  -1/6,  -1/60,      0,   1/420,    1/420,   1/504, ...
   3/20,  1/60,  1/420,       0,  -1/2520, -1/2520, ...
  -2/15, -1/70, -1/420, -1/2520,        0, 1/13860, ...
   5/42,  1/84,  1/504,  1/2520, -1/13860,       0, ...
Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Factor of A160466.

Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.

import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
-- Reinhard Zumkeller, Dec 12 2011

seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # Zerinvary Lajos, Apr 20 2008

Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *)
Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* Harvey P. Dale, Aug 15 2013 *)

Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ Charles R Greathouse IV, Mar 23 2016

G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - Reinhard Zumkeller, Dec 12 2011
Sum_{n>=0} 1/a(n) = 1 + log(2). - Amiram Eldar, Sep 11 2022
From Amiram Eldar, Sep 14 2022: (Start)
a(n) = lcm(2*n+2, n+2)/2.
a(n) = A045895(n+2)/2. (End)
E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024

A165367 Trisection a(n) = A026741(3n + 2).

Original entry on oeis.org

1, 5, 4, 11, 7, 17, 10, 23, 13, 29, 16, 35, 19, 41, 22, 47, 25, 53, 28, 59, 31, 65, 34, 71, 37, 77, 40, 83, 43, 89, 46, 95, 49, 101, 52, 107, 55, 113, 58, 119, 61, 125, 64, 131, 67, 137, 70, 143, 73, 149, 76, 155, 79, 161, 82, 167, 85, 173, 88, 179, 91, 185, 94, 191, 97, 197

Offset: 0

Paul Curtz, Sep 17 2009

The other trisections are A165351 and A165355.

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Cf. A000326, A016777, A016969, A022998, A026741, A045944, A165351, A165355.

A026741 := proc(n) if type(n,'odd') then n; else n/2 ; fi; end:
A165367 := proc(n) A026741(3*n+2) ; end: seq(A165367(n),n=0..100) ; # R. J. Mathar, Nov 22 2009

LinearRecurrence[{0, 2, 0, -1}, {1, 5, 4, 11}, 66] (* Jean-François Alcover, Nov 15 2017 *)

a(n) = (3*n+2)>>!(n%2); \\ Ruud H.G. van Tol, Oct 09 2023

a(n)*A022998(n) = A045944(n).
a(n)*A026741(n+1) = A000326(n+1).
a(2n) = A016777(n); a(2n+1) = A016969(n).
From R. J. Mathar Nov 22 2009: (Start)
a(n) = 2*a(n-2) - a(n-4).
G.f.: (1 + 5*x + 2*x^2 + x^3)/((1-x)^2*(1+x)^2). (End)

All comments rewritten as formulas by R. J. Mathar, Nov 22 2009

A051712 Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 25, 13, 9, 7, 29, 5, 31, 4, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 2, 49, 25, 17, 13, 53, 9, 55, 7, 19, 29, 59, 5, 61, 31, 21, 8, 65, 11, 67, 17, 23, 35, 71, 3, 73

Offset: 1

N. J. A. Sloane

			0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Cf. A026741, A045896, A051713.

Row 3 of table in A051714/A051715.

b[n_] := n/((n + 1) (n + 2)); Numerator[-Differences[Array[b, 100]]]
(* or *)
f[p_, e_] := p^e; f[2, e_] := If[e < 3, 1, 2^(e - 3)]; f[3, e_] := 3^(e - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n - 1]; Array[a, 100] (* Amiram Eldar, Nov 20 2022 *)

c(n) = a(n+1) is multiplicative with c(2^e) = 2^(e-3) if e > 2 and 1 otherwise, c(3^e) = 3^(e-1), and c(p^e) = p^e if p >= 5. [corrected by Amiram Eldar, Nov 20 2022]

Sum_{k=1..n} a(k) ~ (301/1152) * n^2. - Amiram Eldar, Nov 20 2022

A349341 Dirichlet inverse of A026741, which is defined as n if n is odd, n/2 if n is even.

Original entry on oeis.org

1, -1, -3, -1, -5, 3, -7, -1, 0, 5, -11, 3, -13, 7, 15, -1, -17, 0, -19, 5, 21, 11, -23, 3, 0, 13, 0, 7, -29, -15, -31, -1, 33, 17, 35, 0, -37, 19, 39, 5, -41, -21, -43, 11, 0, 23, -47, 3, 0, 0, 51, 13, -53, 0, 55, 7, 57, 29, -59, -15, -61, 31, 0, -1, 65, -33, -67, 17, 69, -35, -71, 0, -73, 37, 0, 19, 77, -39, -79

Offset: 1

Antti Karttunen, Nov 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Agrees with A349343 on odd numbers.
Cf. A026741, A349342.
Cf. also A328203, A349134, A349344, A349346, A349353.

a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,n/(2#)]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

A349341(n) = { my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1], -1, if(1==f[i,2], -f[i,1], 0))); };

from sympy import prevprime, factorint, prod
def f(p, e):
    return -1 if p == 2 else 0 if e > 1 else -p
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

a(1) = 1; a(n) = -Sum_{d|n, d < n} A026741(n/d) * a(d).
a(n) = A349342(n) - A026741(n).
a(2n+1) = A349343(2n+1) for all n >= 1.
Multiplicative with a(2^e) = -1, a(p) = -p and a(p^e) = 0 if e > 1. - Sebastian Karlsson, Nov 15 2021

A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 60, 60, 70, 84, 504, 120, 990, 165, 572, 1092, 2730, 280, 4080, 2448, 1938, 855, 7980, 1540, 10626, 3036, 4600, 7800, 17550, 819, 21924, 12180, 8990, 7440, 32736, 5984, 39270, 5355, 15540, 25308, 54834, 4940, 63960, 34440

Offset: 1

N. J. A. Sloane

			0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A051712. Row 3 of table in A051714/A051715.

Denominator[#[[1]]-#[[2]]&/@(Partition[#[[1]]/(#[[2]]#[[3]])&/@Partition[ Range[50],3,1],2,1])] (* Harvey P. Dale, Nov 15 2014 *)

A214281 Triangle by rows, row n contains the ConvOffs transform of the first n terms of 1, 1, 3, 2, 5, 3, 7, ... (A026741 without leading zero).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 6, 2, 1, 1, 5, 10, 10, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 28, 28, 70, 28, 28, 4, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 5, 45, 60, 210, 126, 210, 60, 45, 5, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Gary W. Adamson, Jul 09 2012

The ConvOffs transform of a sequence s(0), s(1), ..., s(t-1) is defined by a(0)=1 and a(n) = a(n-1)*s(t-n)/s(n-1) for 1 <= n < t. An example of this process is also shown in the Narayana triangle, A001263. By increasing the length t of the input sequence (here: A026741) we create more and more rows of the triangle.

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,   1;
  1,  2,  6,   2,   1;
  1,  5, 10,  10,   5,   1;
  1,  3, 15,  10,  15,   3,   1;
  1,  7, 21,  35,  35,  21,   7,   1;
  1,  4, 28,  28,  70,  28,  28,   4,   1;
  1,  9, 36,  84, 126, 126,  84,  36,   9,  1;
  1,  5, 45,  60, 210, 126, 210,  60,  45,  5,  1;
  1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1;
  ...

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A134683 (row sums), A026741, A001263.

T(n,k) = binomial(n,k) if n is odd.

A092530 a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.

Original entry on oeis.org

0, 2, 4, 9, 12, 20, 24, 35, 40, 54, 60, 77, 84, 104, 112, 135, 144, 170, 180, 209, 220, 252, 264, 299, 312, 350, 364, 405, 420, 464, 480, 527, 544, 594, 612, 665, 684, 740, 760, 819, 840, 902, 924, 989, 1012, 1080, 1104, 1175, 1200, 1274, 1300, 1377, 1404, 1484

Offset: 0

N. J. A. Sloane, Apr 08 2004

Colin Barker, Table of n, a(n) for n = 0..1000
A. W. Vyawahare and K. M. Purohit, Near pseudo Smarandache function, Smarandache Notions, 14 (2004), 42-61.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Equals A000217 + A026741.

seq(n*(1+ceil(n/2)), n=0..53); # Zerinvary Lajos and Klaus Brockhaus, Apr 10 2007

{0}~Join~Array[Block[{k = 1}, While[GCD[#1, #2 + k] < #1, k++]; #2 + k] & @@ {#, (#^2 + #)/2} &, 53] (* or *)
CoefficientList[Series[x (2 + 2 x + x^2 - x^3)/((1 - x)^3*(1 + x)^2), {x, 0, 53}], x] (* Michael De Vlieger, Feb 03 2019 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,4,9,12},60] (* Harvey P. Dale, Sep 21 2024 *)

for(n=0,53,print1(n*(1+ceil(n/2)),",")); \\ Klaus Brockhaus, Apr 10 2007

concat(0, Vec(x*(2 + 2*x + x^2 - x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Feb 03 2019

a(0) = 0, a(2n) = a(2n-1) + n, a(2n-1) = a(2n-2) + 3n-1. - Amarnath Murthy, Jul 04 2004
From Colin Barker, Feb 03 2019: (Start)
G.f.: x*(2 + 2*x + x^2 - x^3) / ((1 - x)^3*(1 + x)^2).
a(n) = (n*(2 + n)) / 2 for n even.
a(n) = (n*(3 + n)) / 2 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)

A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

Original entry on oeis.org

1, 9, 11, 45, 39, 126, 94, 270, 185, 495, 321, 819, 511, 1260, 764, 1836, 1089, 2565, 1495, 3465, 1991, 4554, 2586, 5850, 3289, 7371, 4109, 9135, 5055, 11160, 6136, 13464, 7361, 16065, 8739, 18981, 10279, 22230, 11990, 25830, 13881

Offset: 1

Omar E. Pol, Sep 21 2011

nonn

The integers in same rows of the source triangle have a property related to Euler's Pentagonal Theorem.

Note that the column 1 of the mentioned triangle gives the positive terms of A001318 (see example).

			a(1) = 1
a(2) = 2+3+4 = 9
a(3) = 5+6 = 11
a(4) = 7+8+9+10+11 = 45
a(5) = 12+13+14 = 39
a(6) = 15+16+17+18+19+20+21 = 126
a(7) = 22+23+24+25 = 94
a(8) = 26+27+28+29+30+31+32+33+34 = 270
a(9) = 35+36+37+38+39 = 185

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A026741, A195310, A195311, A004188 (bisection).

A195309 := proc(n)
        (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32
end proc:
seq(A195309(n),n=1..60) ; # R. J. Mathar, Oct 08 2011

LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,9,11,45,39,126,94,270},80] (* Harvey P. Dale, Jun 22 2015 *)

a(n) = (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32 . - R. J. Mathar, Oct 08 2011

G.f. x*(1+9*x+7*x^2+9*x^3+x^4) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Oct 08 2011

A349342 Sum of A026741 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 6, 0, 3, 9, 10, 0, 9, 0, 14, 30, 7, 0, 9, 0, 15, 42, 22, 0, 15, 25, 26, 27, 21, 0, 0, 0, 15, 66, 34, 70, 18, 0, 38, 78, 25, 0, 0, 0, 33, 45, 46, 0, 27, 49, 25, 102, 39, 0, 27, 110, 35, 114, 58, 0, 15, 0, 62, 63, 31, 130, 0, 0, 51, 138, 0, 0, 36, 0, 74, 75, 57, 154, 0, 0, 45, 81, 82, 0, 21, 170, 86

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A026741, A349341.

A026741(n) = if(n%2,n,n/2);
A349341(n) = { my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1], -1, if(1==f[i,2], -f[i,1], 0))); };
A349342(n) = (A026741(n)+A349341(n));

a(n) = A026741(n) + A349341(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A026741(d) * A349341(n/d).

A139633 Triangle read by rows: binomial transform of a diagonalized matrix of A026741.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 2, 1, 4, 18, 8, 5, 1, 5, 30, 20, 25, 3, 1, 6, 45, 40, 75, 18, 7, 1, 7, 63, 70, 175, 63, 49, 4, 1, 8, 84, 112, 350, 168, 196, 32, 9, 1, 9, 108, 168, 630, 378, 588, 144, 81, 5, 1, 10, 135, 240, 1050, 756, 1470, 480, 405, 50, 11, 1, 11, 165, 330, 1650, 1386

Offset: 1

Gary W. Adamson and Roger L. Bagula, Apr 27 2008

Row sums = A084860: (1, 2, 6, 15, 36, 84, 192,...).

			First few rows of the triangle are:
1;
1, 1;
1, 2, 3;
1, 3, 9, 2;
1, 4, 18, 8, 5;
1, 5, 30, 20, 25, 3;
1, 6, 45, 40, 75, 18, 7;
1, 7, 63, 70, 175, 63, 49, 4;
...

Cf. A026741, A084860.

Let X = a diagonalized matrix of A026741: [1; 0,1; 0,0,3; 0,0,0,2;], where the first few nonzero terms of A026741 are (1, 1, 3, 2, 5, 3, 7,...). The triangle = A007318 * X.

cp's OEIS Frontend

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

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A165367 Trisection a(n) = A026741(3n + 2).

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A051712 Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

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A349341 Dirichlet inverse of A026741, which is defined as n if n is odd, n/2 if n is even.

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A051713 Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

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A214281 Triangle by rows, row n contains the ConvOffs transform of the first n terms of 1, 1, 3, 2, 5, 3, 7, ... (A026741 without leading zero).

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A092530 a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.

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A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

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