A100454
a(n) = sum of n-th column in array in A100452.
Original entry on oeis.org
1, 7, 24, 58, 109, 188, 307, 444, 641, 885, 1149, 1493, 1936, 2358, 2975, 3645, 4267, 5102, 6057, 6941, 8124, 9395, 10458, 12140, 13561, 15336, 17110, 19204, 21124, 23596, 26219, 28587, 31254, 34593, 37252, 40545, 44524, 47451, 51724, 55853, 60068, 64152, 69801, 73657, 79372
Offset: 1
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function t(n, k) // t = A100452
if k eq 1 then return n^2;
else return (n-k+1)*Floor((t(n, k-1) -1)/(n-k+1));
end if;
end function;
A100454:= func< n | (&+[t(n,n-k+1): k in [1..n]]) >;
[A100454(n): n in [1..60]]; // G. C. Greubel, Apr 07 2023
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t[1, n_]:= n^2; (* t = A100452 *)
t[m_, n_]/; 1, ]=0;
A100454[n_]:= A100454[n]= Sum[t[n-k+1,n], {k,n}];
Table[A100454[n], {n, 60}] (* G. C. Greubel, Apr 07 2023 *)
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def t(n, k): # t = A100452
if (k==1): return n^2
else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
def A100454(n): return sum(t(n,n-k+1) for k in range(1,n+1))
[A100454(n) for n in range(1,61)] # G. C. Greubel, Apr 07 2023
A100453
a(n) = smallest number to appear exactly n times in the array A100452.
Original entry on oeis.org
1, 49, 224, 720, 960
Offset: 1
A100461
Triangle read by rows, based on array described below.
Original entry on oeis.org
1, 1, 2, 1, 2, 4, 3, 4, 6, 8, 7, 8, 9, 12, 16, 25, 26, 27, 28, 30, 32, 49, 50, 51, 52, 55, 60, 64, 109, 110, 111, 112, 115, 120, 126, 128, 229, 230, 231, 232, 235, 240, 245, 248, 256, 481, 482, 483, 484, 485, 486, 490, 496, 504, 512, 1003, 1004, 1005, 1008, 1010, 1014, 1015, 1016, 1017, 1020, 1024
Offset: 1
Array begins:
1 2 4 8 16 32 ...
* 1 2 6 12 30 ...
* * 1 4 9 28 ...
* * * 3 8 27 ...
* * * * 7 26 ...
* * * * * 25 ...
and triangle begins:
1;
1, 2;
1, 2, 4;
3, 4, 6, 8;
7, 8, 9, 12, 16;
25, 26, 27, 28, 30, 32;
49, 50, 51, 52, 55, 60, 64;
109, 110, 111, 112, 115, 120, 126, 128;
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function t(n,k) // t = A100461
if k eq 1 then return 2^(n-1);
else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
end if;
end function;
[t(n,n-k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 07 2023
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t[n_, k_]:= t[n, k]= If[k==1, 2^(n-1), (n-k+1)*Floor[(t[n, k-1] -1)/(n-k+1)]];
Table[t[n, n-k+1], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 07 2023 *)
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def t(n,k): # t = A100461
if (k==1): return 2^(n-1)
else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n,n-k+1) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 07 2023
A100451
a(n) = 0 for n <= 2; for n >= 3, a(n) = (n-2)*floor((n^2-2)/(n-2)).
Original entry on oeis.org
0, 0, 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597, 2700
Offset: 1
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[0, 0] cat [(n-2)*Floor((n^2-2)/(n-2)): n in [3..30]]; // Vincenzo Librandi, Oct 04 2011
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Join[{0,0,7,14},Table[(n-2)(n+2),{n,5,60}]] (* or *) Join[{0,0,7,14}, LinearRecurrence[{3,-3,1},{21,32,45},60]] (* Harvey P. Dale, Oct 03 2011 *)
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a(n)=if(n<3,0,(n^2-2)\(n-2)*(n-2)) \\ Charles R Greathouse IV, Oct 16 2015
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def A100451(n):
return 7 * (n - 2) * ((n - 1) // 2) if n < 5 else (n - 2) * (n + 2)
print([A100451(n) for n in range(1, 61)]) # G. C. Greubel, Apr 07 2023
A101224
Triangle, read by rows, where T(n,1) = n^2-n+1 for n>=1 and T(n,k) = (n-k+1)*floor( (T(n,k-1)-1)/(n-k+1) ) for 1A000960).
Original entry on oeis.org
1, 3, 2, 7, 6, 5, 13, 12, 10, 9, 21, 20, 18, 16, 15, 31, 30, 28, 27, 26, 25, 43, 42, 40, 36, 33, 32, 31, 57, 56, 54, 50, 48, 45, 44, 43, 73, 72, 70, 66, 65, 64, 63, 62, 61, 91, 90, 88, 84, 78, 75, 72, 69, 68, 67, 111, 110, 108, 104, 98, 96, 95, 92, 90, 88, 87, 133, 132, 130, 126
Offset: 1
T(4,4) = 9 since we start with T(4,1)=4^2-4+1=13 and then
T(4,2)=(4-2+1)*floor((T(4,1)-1)/(4-2+1))=3*floor((13-1)/3)=12,
T(4,3)=(4-3+1)*floor((T(4,2)-1)/(4-3+1))=2*floor((12-1)/2)=10,
T(4,4)=(4-4+1)*floor((T(4,3)-1)/(4-4+1))=1*floor((10-1)/1)=9.
Rows begin:
[1],
[3,2],
[7,6,5],
[13,12,10,9],
[21,20,18,16,15],
[31,30,28,27,26,25],
[43,42,40,36,33,32,31],
[57,56,54,50,48,45,44,43],
[73,72,70,66,65,64,63,62,61],...
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T(n,k)=if(k==1,n^2-n+1,(n-k+1)*floor((T(n,k-1)-1)/(n-k+1)))
Showing 1-5 of 5 results.
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