A047370
Numbers that are congruent to {2, 3, 5} mod 7.
Original entry on oeis.org
2, 3, 5, 9, 10, 12, 16, 17, 19, 23, 24, 26, 30, 31, 33, 37, 38, 40, 44, 45, 47, 51, 52, 54, 58, 59, 61, 65, 66, 68, 72, 73, 75, 79, 80, 82, 86, 87, 89, 93, 94, 96, 100, 101, 103, 107, 108, 110, 114, 115, 117, 121, 122, 124, 128, 129, 131, 135, 136, 138, 142
Offset: 1
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[7*Floor((n-1)/3)+2^((n-1) mod 3)+1: n in [1..50]]; // Wesley Ivan Hurt, May 25 2014
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A047370:=n->7*floor((n-1)/3) + 2^((n-1) mod 3)+1; seq(A047370(n), n=1..50); # Wesley Ivan Hurt, May 25 2014
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Select[Range[200], MemberQ[{2,3,5}, Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,1,-1}, {2,3,5,9}, 60] (* Harvey P. Dale, Apr 29 2013 *)
Table[7*Floor[(n - 1)/3] + 2^Mod[n - 1, 3] + 1, {n, 50}] (* Wesley Ivan Hurt, May 25 2014 *)
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x='x + O('x^50); Vec(x*(2+x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2)) \\ G. C. Greubel, Feb 21 2017
A154611
Numbers n such that 7*n+3 is not prime.
Original entry on oeis.org
1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79
Offset: 1
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[n: n in [0..80] | not IsPrime(7*n+3)]; // Vincenzo Librandi, Apr 05 2013
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With[{nn=100},Sort[Join[Range[1,nn-1,2],Select[Range[0,nn,2], !PrimeQ[ 7#+3]&]]]] (* Harvey P. Dale, Aug 22 2012 *)
Select[Range[0, 100], !PrimeQ[7 # + 3]&] (* Vincenzo Librandi, Apr 05 2013 *)
A361226
Square array T(n,k) = k*((1+2*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 0, 2, 5, 3, 0, 3, 9, 12, 6, 0, 4, 13, 21, 22, 10, 0, 5, 17, 30, 38, 35, 15, 0, 6, 21, 39, 54, 60, 51, 21, 0, 7, 25, 48, 70, 85, 87, 70, 28, 0, 8, 29, 57, 86, 110, 123, 119, 92, 36, 0, 9, 33, 66, 102, 135, 159, 168, 156, 117, 45
Offset: 0
The rows are
0, 0, 1, 3, 6, 10, 15, 21, ... = A161680
0, 1, 5, 12, 22, 35, 51, 70, ... = A000326
0, 2, 9, 21, 38, 60, 87, 119, ... = A005476
0, 3, 13, 30, 54, 85, 123, 168, ... = A022264
0, 4, 17, 39, 70, 110, 159, 217, ... = A022266
... .
Columns: A000004, A001477, A016813, A017197=3*A016777, 2*A017101, 5*A016873, 3*A017581, 7*A017017, ... (coefficients from A026741).
Difference between two consecutive rows: A000290. Hence A143844.
This square array read by antidiagonals leads to the triangle
0
0 0
0 1 1
0 2 5 3
0 3 9 12 6
0 4 13 21 22 10
0 5 17 30 38 35 15
... .
Cf.
A000004,
A000290,
A000326,
A001477,
A002414,
A005476,
A016777,
A016813,
A016873,
A017017,
A017101,
A017197,
A017581,
A022264,
A022266,
A026741,
A034827,
A160378,
A161680,
A360962.
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T[n_, k_] := k*((2*n + 1)*k - 1)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 05 2023 *)
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a(n) = { my(row = (sqrtint(8*n+1)-1)\2, column = n - binomial(row + 1, 2)); binomial(column, 2) + column^2 * (row - column) } \\ David A. Corneth, Mar 05 2023
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# Seen as a triangle:
from functools import cache
@cache
def Trow(n: int) -> list[int]:
if n == 0: return [0]
r = Trow(n - 1)
return [r[k] + k * k if k < n else r[n - 1] + n - 1 for k in range(n + 1)]
for n in range(7): print(Trow(n)) # Peter Luschny, Mar 05 2023
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