A164284 -id:A164284 - cp's OEIS frontend

A269044 a(n) = 13*n + 7.

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Bruno Berselli, Feb 18 2016

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Magma
```
[13*n+7: n in [0..60]];
```

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima
```
makelist(13*n+7, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+7)
```
Sage
```
[13*n+7 for n in (0..60)]
```

G.f.: (7 + 6*x)/(1 - x)^2.
a(n) =  A088227(4*n+3).
a(n) = -A186113(-n-1).
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

cp's OEIS Frontend

A269044 a(n) = 13*n + 7.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

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cp's OEIS Frontend

A269044 a(n) = 13*n + 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.