A278910
Triangle of order m: C(n,k) = k*(n-k+1)^(k+m)+n-k, 0 <= k <= n, m = 0, read by rows.
Original entry on oeis.org
0, 1, 1, 2, 3, 2, 3, 5, 9, 3, 4, 7, 20, 25, 4, 5, 9, 35, 83, 65, 5, 6, 11, 54, 195, 326, 161, 6, 7, 13, 77, 379, 1027, 1217, 385, 7, 8, 15, 104, 653, 2504, 5123, 4376, 897, 8, 9, 17, 135, 1035, 5189, 15629, 24579, 15311, 2049, 9, 10, 19, 170, 1543, 9610, 38885, 93754, 114691, 52490, 4609, 10
Offset: 0
As an infinite triangular array:
0
1 1
2 3 2
3 5 9 3
4 7 20 25 4
5 9 35 83 65 5
As an infinite square array (matrix):
0 1 2 3 4 5
1 3 9 25 65 161
2 5 20 83 326 1217
3 7 35 195 1027 5123
4 9 54 379 2504 15629
5 11 77 653 5189 38885
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/* As triangle */ [[k*(n-k+1)^k+n-k: k in [0..n]]: n in [0..10]];
-
A278910 := proc(n,k)
k*(n-k+1)^k+n-k ;
end proc:
seq(seq(A278910(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Dec 02 2016
A281334
Triangle read by rows: T(n, k) = (n - k)*(k + 1)^3 + k, 0 <= k <= n.
Original entry on oeis.org
0, 1, 1, 2, 9, 2, 3, 17, 29, 3, 4, 25, 56, 67, 4, 5, 33, 83, 131, 129, 5, 6, 41, 110, 195, 254, 221, 6, 7, 49, 137, 259, 379, 437, 349, 7, 8, 57, 164, 323, 504, 653, 692, 519, 8, 9, 65, 191, 387, 629, 869, 1035, 1031, 737, 9, 10, 73, 218, 451, 754, 1085, 1378, 1543, 1466, 1009, 10
Offset: 1
Triangle begins:
0;
1, 1;
2, 9, 2;
3, 17, 29, 3;
4, 25, 56, 67, 4;
5, 33, 83, 131, 129, 5;
6, 41, 110, 195, 254, 221, 6;
7, 49, 137, 259, 379, 437, 349, 7;
8, 57, 164, 323, 504, 653, 692, 519, 8;
9, 65, 191, 387, 629, 869, 1035, 1031, 737, 9;
10, 73, 218, 451, 754, 1085, 1378, 1543, 1466, 1009, 10;
...
Cf. Triangle read by rows: T(n,k) = (n-k)*(k+1)^m+k:
A003056 (m = 0),
A059036 (m = 1),
A274602 (m = 2), this sequence (m = 3).
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/* As triangle */ [[(n-k)*(k+1)^3+k: k in [1..n]]: n in [0..10]];
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t[n_, k_] := (n - k)*(k + 1)^3 + k; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Feb 09 2017 *)
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for(n=0,10,for(k=0,n,print1((n-k)*(k+1)^3+k,", "))) \\ Derek Orr, Feb 26 2017
A274602
Triangle read by rows: T(n,k) = k*(n-k+1)^2 + n - k, 0 <= k <= n.
Original entry on oeis.org
0, 1, 1, 2, 5, 2, 3, 11, 9, 3, 4, 19, 20, 13, 4, 5, 29, 35, 29, 17, 5, 6, 41, 54, 51, 38, 21, 6, 7, 55, 77, 79, 67, 47, 25, 7, 8, 71, 104, 113, 104, 83, 56, 29, 8, 9, 89, 135, 153, 149, 129, 99, 65, 33, 9, 10, 109, 170, 199, 202, 185, 154, 115, 74, 37, 10
Offset: 1
0; 1,1; 2,5,2; 3,11,9,3; 4,19,20,13,4; 5,29,35,29,17,5; ...
As an infinite triangular array:
0
1 1
2 5 2
3 11 9 3
4 19 20 13 4
5 29 35 29 17 5
As an infinite square array (matrix) with comments:
0 1 2 3 4 5 A001477
1 5 9 13 17 21 A016813
2 11 20 29 38 47 A017185
3 19 35 51 67 83
4 29 54 79 104 129
5 41 77 113 149 185
Cf. Triangle read by rows: T(n,k) = k*(n-k+1)^m+n-k, 0 <= k <= n:
A003056 (m = 0),
A059036 (m = 1),
A278910 (m = k).
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/* As triangle */ [[k*(n-k+1)^2+n-k: k in [0..n]]: n in [0..10]];
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Table[k (n - k + 1)^(k + #) + n - k &[2 - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 02 2016 *)
A335984
Numbers m such that more than half the distinct positive terms of the sequence -k^2 + m*k - 1 are primes.
Original entry on oeis.org
4, 5, 7, 9, 11, 19, 21, 31, 33, 39, 49, 51, 81, 99, 101, 123, 129, 159, 171, 177, 189, 231, 291, 441, 879, 1011, 2751
Offset: 1
7 is in the sequence because with g(k) = -k^2+7*k-1, the positive terms of the sequence g(k) are 5=g(1), 9=g(2) and 11=g(3), and two out of the three (5 and 9) are prime.
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filter:= n -> nops(select(isprime, [seq(n*x-x^2-1,x=1..n/2)])) > 1/2*floor(n/2):
select(filter, [$1..10000]);
A127738
Triangle read by rows: the matrix product A004736 * A127701 of two triangular matrices.
Original entry on oeis.org
1, 3, 2, 5, 5, 3, 7, 8, 7, 4, 9, 11, 11, 9, 5, 11, 14, 15, 14, 11, 6, 13, 17, 19, 19, 17, 13, 7, 15, 20, 23, 24, 23, 20, 15, 8, 17, 23, 27, 29, 29, 27, 23, 17, 9, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10
Offset: 1
First few rows of the triangle:
1;
3, 2;
5, 5, 3;
7, 8, 7, 4;
9, 11, 11, 9, 5;
11, 14, 15, 14, 11, 6;
13, 17, 19, 19, 17, 13, 7;
...
Showing 1-5 of 5 results.
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