A057029 -id:A057029 - cp's OEIS frontend

A057027 Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form an increasing sequence and the others a decreasing sequence.

1, 2, 3, 4, 6, 5, 7, 10, 8, 9, 11, 15, 12, 14, 13, 16, 21, 17, 20, 18, 19, 22, 28, 23, 27, 24, 26, 25, 29, 36, 30, 35, 31, 34, 32, 33, 37, 45, 38, 44, 39, 43, 40, 42, 41, 46, 55, 47, 54, 48, 53, 49, 52, 50, 51, 56, 66, 57, 65, 58, 64, 59, 63, 60, 62, 61, 67, 78, 68, 77, 69, 76

Offset: 1

Clark Kimberling, Jul 28 2000

Arrange the quotients F(i)/F(j) of Fibonacci numbers, for 2<=i

			For n=6, the ordered quotients are 1/8, 1/5, 2/8, 1/3, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3; the positions of 1/5, 2/5, 3/5 are 2, 6, 8 (first terms of diagonal T(i, i-1)).
Triangle starts:
  1;
  2, 3;
  4, 6, 5;
  7,10, 8, 9;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Reflection of the array in A057028 about its central column, a permutation of the natural numbers.
Inverse permutation to A064578. Central column: A057029.
Column 1 is A000124, column 2 is A000217.
Row sums are A006003.

nn= 12; t = Table[Range[Binomial[n, 2] + 1, Binomial[n + 1, 2]], {n, nn}]; Table[t[[n, If[OddQ@ k, Ceiling[k/2], -k/2] ]], {n, nn}, {k, n}] // Flatten (* Michael De Vlieger, Jul 02 2016 *)

From Werner Schulte, Sep 09 2024: (Start)
T(n, k) = (n^2 + (-1)^k * (n - k) + (3 + (-1)^k) / 2) / 2.
T(n, 1) = (n^2 - n + 2) / 2 = A000124(n).
T(n, 2) = (n^2 + n) / 2 = A000217(n) for n >= 2.
T(n, k) = T(n, k-2) - (-1)^k  for 3 <= k <= n. (End)
G.f.: x*y*(1 + x*(y - 1) - x^4*(y - 1)*y^2 + x^5*y^3 + x^3*y*(y^2 - y - 1) - x^2*(y^2 + y - 1))/((1 - x)^3*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 10 2024

Corrected and extended by Vladeta Jovovic, Oct 18 2001

A274681 Numbers k such that 4*k + 1 is a triangular number.

Original entry on oeis.org

0, 5, 11, 26, 38, 63, 81, 116, 140, 185, 215, 270, 306, 371, 413, 488, 536, 621, 675, 770, 830, 935, 1001, 1116, 1188, 1313, 1391, 1526, 1610, 1755, 1845, 2000, 2096, 2261, 2363, 2538, 2646, 2831, 2945, 3140, 3260, 3465, 3591, 3806, 3938, 4163, 4301, 4536

Offset: 1

Colin Barker, Jul 02 2016

Also, numbers of the form m*(8*m + 3) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

			5 is in the sequence since 4*5 + 1 = 21 is a triangular number (21 = 1 + 2 + 3 + 4 + 5 + 6). - _Michael B. Porter_, Jul 03 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000096 (n+1), A074377 (2*n+1), A045943 (3*n+1), A085787 (5*n+1).
Cf. A057029.
Cf. similar sequences listed in A299645.

[(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: n in [1..80]]; // Wesley Ivan Hurt, Jul 02 2016

A274681:=n->(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A274681(n), n=1..100); # Wesley Ivan Hurt, Jul 02 2016

Rest@ CoefficientList[Series[x^2 (5 + 6 x + 5 x^2)/((1 - x)^3 (1 + x)^2), {x, 0, 48}], x] (* Michael De Vlieger, Jul 02 2016 *)
Select[Range[0,5000],OddQ[Sqrt[8(4#+1)+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,11,26,38},50] (* Harvey P. Dale, Apr 21 2018 *)

PARI
```
isok(n) = ispolygonal(4*n+1, 3)
```

select(n->ispolygonal(4*n+1, 3), vector(10000, n, n-1))

concat(0, Vec(x^2*(5+6*x+5*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

def A274681(n): return (n>>1)*((n<<2)+(-1 if n&1 else -3)) # Chai Wah Wu, Mar 11 2025

G.f.: x^2*(5 + 6*x + 5*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = A057029(n) - 1.
a(n) = (1 - (-1)^n + 2*(-4 + (-1)^n)*n + 8*n^2)/4.
a(n) = (4*n^2 - 3*n)/2 for n even, a(n) = (4*n^2 - 5*n + 1)/2 for n odd.

A236267 a(n) = 8n^2 + 3n + 1.

Original entry on oeis.org

1, 12, 39, 82, 141, 216, 307, 414, 537, 676, 831, 1002, 1189, 1392, 1611, 1846, 2097, 2364, 2647, 2946, 3261, 3592, 3939, 4302, 4681, 5076, 5487, 5914, 6357, 6816, 7291, 7782, 8289, 8812, 9351, 9906, 10477, 11064, 11667, 12286, 12921, 13572, 14239, 14922, 15621, 16336

Offset: 0

Vladimir Shevelev, Jan 21 2014

Positions a(n) of hexagonal numbers such that h(a(n)) = h(a(n)-1) + h(4*n+1), where h = A000384.

First bisection of A057029. The sequence contains infinitely many squares: 1, 676, 779689, 899760016, ... [Bruno Berselli, Jan 24 2014]

			For n=5, A000384(a(5)) = 93096 = A000384(a(5)-1) + A000384(4*5+1) = 92235 + 861.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A057029, A064225, A152948, A236257.

[8*n^2+3*n+1: n in [0..50]]; // Bruno Berselli, Jan 24 2014

Table[8 n^2 + 3 n + 1, {n, 0, 50}] (* Bruno Berselli, Jan 24 2014 *)
LinearRecurrence[{3,-3,1},{1,12,39},50] (* Harvey P. Dale, May 26 2019 *)

Vec(-(6*x^2+9*x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jan 21 2014

From Colin Barker, Jan 21 2014: (Start)
G.f.: -(6*x^2 + 9*x + 1)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(1 + 11*x + 8*x^2). - Elmo R. Oliveira, Oct 19 2024

More terms from Colin Barker, Jan 21 2014

a(44)-a(45) from Elmo R. Oliveira, Oct 19 2024

A373900 a(n) is the permanent of the n X n matrix M(n) whose generic element is given by M_{i,j} = A057027(i+j-1,j) with i,j in [n].

Original entry on oeis.org

1, 1, 12, 540, 75872, 20955144, 11384126656, 9651484407168, 13211097362836992, 25194877206154652160, 69640283454545443829760, 250781830072455488420118528, 1222842630390899923255269335040, 7431235824692236144506864480645120, 58351873068720341047993109561429852160

Offset: 0

Stefano Spezia, Sep 10 2024

nonn

The matrix M(n) is singular for n = 2 and n > 3.

			a(4) = 75872:
  [1,  3,  5,  9]
  [2,  6,  8, 14]
  [4, 10, 12, 20]
  [7, 15, 17, 27]

Cf. A057027, A057029.

A057027[n_, k_]:=(n^2 + (-1)^k*(n - k) + (3 + (-1)^k)/2)/2; a[n_]:=Permanent[Table[A057027[i+j-1,j],{i,n},{j,n}]]; Join[{1},Array[a,14]]

Showing 1-4 of 4 results.

cp's OEIS Frontend

A057027 Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form an increasing sequence and the others a decreasing sequence.

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A274681 Numbers k such that 4*k + 1 is a triangular number.

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A236267 a(n) = 8*n^2 + 3*n + 1.

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Magma

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A373900 a(n) is the permanent of the n X n matrix M(n) whose generic element is given by M_{i,j} = A057027(i+j-1,j) with i,j in [n].

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Mathematica

A236267 a(n) = 8n^2 + 3n + 1.