A144437 -id:A144437 - cp's OEIS frontend

A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

0, -1, -1, -4, -5, -2, -7, -8, -3, -10, -11, -4, -13, -14, -5, -16, -17, -6, -19, -20, -7, -22, -23, -8, -25, -26, -9, -28, -29, -10, -31, -32, -11, -34, -35, -12, -37, -38, -13, -40, -41, -14, -43, -44, -15, -46, -47, -16, -49, -50, -17, -52, -53, -18, -55, -56, -19, -58, -59, -20

Offset: 0

Paul Curtz, Sep 07 2011

0,  -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5,    = a(n)/b(n),
1,    0,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
1,   -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
3,   -2,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
5,   -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
11, -10,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .
a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).
b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).
a(n+5)-a(n+2)=b(n+5)  (=-1,-3,-3,=-A169609(n)).

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Sep 18 2012 *)

a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.
From Chai Wah Wu, May 07 2024: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 7.
G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)

A256095 Triangle of greatest common divisors of two triangular numbers (A000217).

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 6, 1, 3, 6, 10, 1, 1, 2, 10, 15, 1, 3, 3, 5, 15, 21, 1, 3, 3, 1, 3, 21, 28, 1, 1, 2, 2, 1, 7, 28, 36, 1, 3, 6, 2, 3, 3, 4, 36, 45, 1, 3, 3, 5, 15, 3, 1, 9, 45, 55, 1, 1, 1, 5, 5, 1, 1, 1, 5, 55, 66, 1, 3, 6, 2, 3, 3, 2, 6, 3, 11, 66, 78, 1, 3, 6, 2, 3, 3, 2, 6, 3, 1, 6, 78, 91, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 13, 91, 105, 1, 3, 3, 5, 15, 21, 7, 3, 15, 5, 3, 3, 7, 105

Offset: 0

Wolfdieter Lang, Mar 17 2015

			The triangle T(n, m) begins:
n\m   0 1 2 3  4  5  6  7  8  9 10 11 12 13  14
0:    0
1:    1 1
2:    3 1 3
3:    6 1 3 6
4:   10 1 1 2 10
5:   15 1 3 3  5 15
6:   21 1 3 3  1  3 21
7:   28 1 1 2  2  1  7 28
8:   36 1 3 6  2  3  3  4 36
9:   45 1 3 3  5 15  3  1  9 45
10:  55 1 1 1  5  5  1  1  1  5 55
11:  66 1 3 6  2  3  3  2  6  3 11 66
12:  78 1 3 6  2  3  3  2  6  3  1  6 78
13:  91 1 1 1  1  1  7  7  1  1  1  1 13 91
14: 105 1 3 3  5 15 21  7  3 15  5  3  3  7 105
...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Cf. A000217, A144437.

T(2n,n) gives A026741.

T:= (i,j) -> igcd(i*(i+1)/2,j*(j+1)/2):
seq(seq(T(i,j),j=0..i),i=0..20); # Robert Israel, Jan 20 2020

tabl(nn) = {for (n=0, nn, trn = n*(n+1)/2; for (k=0, n, print1(gcd(trn, k*(k+1)/2), ", ");); print(););} \\ Michel Marcus, Mar 17 2015

T(n, m) = gcd(Tri(n), Tri(m)), 0 <= m <= n, with the triangular numbers Tri = A000217.
T(n, 0) = Tri(n) = T(n, n). T(n, 1) = 1, n >= 0.
Columns m=2: A144437(n-1), m=3: repeat(6, 2, 3, 3, 2, 6, 3, 1, 6, 6, 1, 3) (guess), m=4: repeat(10, 5, 1, 2, 2, 5, 5, 2, 2, 1, 5, 10, 2, 1, 1, 10, 10, 1, 1, 2) (guess), m=5 repeat(15, 3, 1, 3, 15, 5, 3, 3, 1, 15, 15, 1, 3, 3, 5) (guess), ...
From Robert Israel, Jan 21 2020: (Start) The guesses are correct. More generally, for each k>=1, T(n,k) is periodic in n with period 2*A000217(k) if k == 0 or 3 (mod 4), A000217(k) if k == 1 or 2 (mod 4). (End)

A168673 Binomial transform of A169609.

Original entry on oeis.org

1, 4, 10, 20, 38, 74, 148, 298, 598, 1196, 2390, 4778, 9556, 19114, 38230, 76460, 152918, 305834, 611668, 1223338, 2446678, 4893356, 9786710, 19573418, 39146836, 78293674, 156587350, 313174700, 626349398, 1252698794, 2505397588, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 02 2009

Sequence and successive differences are identical to their third differences. See principal sequence A024495. Main diagonal of the array of successive differences is A083595 (1,6,8,20,36,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A169609, A144437, A168615, A024495, A083595, A130772, A062092, A130151.

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)- 3*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 30 2016

LinearRecurrence[{3,-3,2},{1,4,10},25] (* G. C. Greubel, Jul 29 2016 *)
RecurrenceTable[{a[0] == 1, a[1] == 4, a[2] == 10, a[n] == 3 a[n-1] - 3 a[n-2] + 2 a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Jul 30 2016 *)

a(n)=([0,1,0; 0,0,1; 2,-3,3]^n*[1;4;10])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

a(n+1) - 2a(n) = A130772(n).
a(n) = A062092(n) - A130151(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n > 2; a(0) = 1, a(1) = 4, a(2) = 10.
G.f.: (1 + x + x^2)/(1 -3*x +3*x^2 -2*x^3). - Philippe Deléham, Dec 03 2009

Edited and extended by Klaus Brockhaus, Dec 03 2009

A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.

Original entry on oeis.org

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5

Offset: 0

Paul Curtz, Dec 07 2009

The diagonal of a table of numerators of the Rydberg-Ritz spectrum of hydrogen:
  1,  1,  1,   1,  1,   1,  1,   1,  1,   1,   1, ... A000012
  0,  5,  3,  21,  2,  45, 15,  77,  6, 117,  35, ... A061037
  0,  9,  5,  33,  3,  65, 21, 105,  1, 153,  45, ... A061041
  0, 13,  7,   5,  4,  85,  1, 133, 10,   7,  55, ... A061045
  0, 17,  9,  57,  5, 105, 33, 161,  3, 225,  65, ... A061049
  0, 21, 11,  69,  6,   1, 39, 189, 14, 261,   3, ...
  0, 25, 13,   1,  7, 145,  5, 217,  1,  11,  85, ...
  0, 29, 15,  93,  8, 165, 51,   5, 18, 333,  95, ...
  0, 33, 17, 105,  9, 185, 57, 273,  5, 369, 105, ...
  0, 37, 19,  13, 10, 205,  7, 301, 22,   5, 115, ...
  0, 41, 21, 129, 11,   9, 69, 329,  3, 441,   1, ...
In that respect, constructed similar to A144437.

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

Cf. A171373 (binomial transform), A171408, A105371.

[1] cat [Numerator(5/(6*n)^2): n in [1..100]]; // G. C. Greubel, Sep 20 2018

Table[If[n==0,1,Numerator[5/(6*n)^2]], {n,0,100}] (* G. C. Greubel, Sep 20 2018 *)

concat([1], vector(100, n, numerator(5/(6*n)^2))) \\ G. C. Greubel, Sep 20 2018

a(n) = numerator of 5/(6*n)^2 .
Period 5: repeat [1,5,5,5,5].
G.f.: (1 + 5*x + 5*x^2 + 5*x^3 + 5*x^4)/((1-x)*(1 + x + x^2 + x^3 + x^4)).
a(n) = 1 + 4*sign(n mod 5). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (21-8*cos(2*n*Pi/5)-8*cos(4*n*Pi/5))/5. - Wesley Ivan Hurt, Sep 27 2018

Edited by R. J. Mathar, Dec 15 2009

cp's OEIS Frontend

A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

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A256095 Triangle of greatest common divisors of two triangular numbers (A000217).

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A168673 Binomial transform of A169609.

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A171372 a(n) = Numerator of 1/(2*n)^2 - 1/(3*n)^2 for n > 0, a(0) = 1.

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A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.