A375764
a(n) is the sum of distinct sums of all subsets with two or more elements of {1, 2, ..., n}.
Original entry on oeis.org
0, 0, 3, 18, 52, 117, 228, 403, 663, 1032, 1537, 2208, 3078, 4183, 5562, 7257, 9313, 11778, 14703, 18142, 22152, 26793, 32128, 38223, 45147, 52972, 61773, 71628, 82618, 94827, 108342, 123253, 139653, 157638, 177307, 198762, 222108, 247453, 274908, 304587
Offset: 0
For n = 3 the starting set is {1,2,3} and there are subsets {1,2}{1,3}{2,3}{1,2,3} that sum to 3,4,5 and 6 and the sum of distinct sums (3+4+5+6) is 18.
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Join[{0, 0}, Nest[PolygonalNumber, Range[2, 50], 2] - 3] (* Paolo Xausa, Sep 13 2024 *)
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a = lambda n: max(0,(n**4+2*n**3+3*n**2+2*n-24)//8)
print([a(n) for n in range(1,40)])
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def A375764(n): return (m:=n*(n+1)-4)*(m+10)>>3 if n>1 else 0 # Chai Wah Wu, Aug 30 2024
A376205
Prime indices of triangular numbers k such that k + 1 and k - 2 are not prime.
Original entry on oeis.org
2, 31, 59, 67, 79, 83, 101, 103, 109, 113, 127, 137, 139, 151, 173, 181, 191, 199, 227, 229, 239, 241, 263, 283, 293, 311, 317, 331, 347, 349, 353, 367, 373, 379, 383, 397, 421, 433, 449, 461, 463, 479, 487, 491, 499, 523, 547, 557, 563, 569, 577, 587, 593, 599
Offset: 1
31 is a term since it is prime and both A000217(31) + 1 = 497 and A000217(31) - 2 = 494 are not prime.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 496 at p. 155.
A376206
Prime indices of triangular numbers k such that either k + 1 or k - 2 is prime.
Original entry on oeis.org
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 61, 71, 73, 89, 97, 107, 131, 149, 157, 163, 167, 179, 193, 197, 211, 223, 233, 251, 257, 269, 271, 277, 281, 307, 313, 337, 359, 389, 401, 409, 419, 431, 439, 443, 457, 467, 503, 509, 521, 541, 571, 613, 641, 673, 691
Offset: 1
29 is a term since it is prime and A000217(29) - 2 = 433 is prime.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 496 at p. 155.
Original entry on oeis.org
1, 4, 2, 8, 6, 3, 13, 11, 8, 4, 19, 17, 14, 10, 5, 26, 24, 21, 17, 12, 6, 34, 32, 29, 25, 20, 14, 7, 43, 41, 38, 34, 29, 23, 16, 8, 53, 51, 48, 44, 39, 33, 26, 18, 9, 64, 62, 59, 55, 50, 44, 37, 29, 20, 10
Offset: 1
First few rows of the triangle:
1;
4, 2;
8, 6, 3;
13, 11, 8, 4;
19, 17, 14, 10, 5;
26, 24, 21, 17, 12, 6;
34, 32, 29, 25, 20, 14, 7;
...
A246982
Triangle read by rows: T(n,k) = maximal number of inversions among all k-sorted permutations on n letters (n>=2, 1<=k<=n-1).
Original entry on oeis.org
1, 1, 3, 2, 4, 6, 2, 4, 8, 10, 3, 6, 9, 13, 15, 3, 7, 9, 15, 19, 21, 4, 8, 12, 16, 22, 26, 28, 4, 9, 14, 16, 24, 30, 34, 36, 5, 10, 16, 20, 25, 33, 39, 43, 45, 5, 11, 17, 23, 25, 35, 43, 49, 53, 55, 6, 12, 18, 26, 30, 36, 46, 54, 60, 64, 66, 6, 13, 20
Offset: 2
Triangle begins:
1,
1,3,
2,4,6,
2,4,8,10,
3,6,9,13,15,
3,7,9,15,19,21,
4,8,12,16,22,26,28,
4,9,14,16,24,30,34,36,
5,10,16,20,25,33,39,43,45,
5,11,17,23,25,35,43,49,53,55,
...
A341768
a(n) = n * (binomial(n,2) - 2).
Original entry on oeis.org
0, -2, -2, 3, 16, 40, 78, 133, 208, 306, 430, 583, 768, 988, 1246, 1545, 1888, 2278, 2718, 3211, 3760, 4368, 5038, 5773, 6576, 7450, 8398, 9423, 10528, 11716, 12990, 14353, 15808, 17358, 19006, 20755, 22608, 24568, 26638, 28821, 31120, 33538, 36078, 38743, 41536, 44460
Offset: 0
a(7) = A147875(7) = A000566(-7) = 133.
Cf.
A005449,
A005564,
A006002,
A014105,
A033954,
A034856,
A045944,
A060354,
A062728,
A135705,
A147875,
A179986,
A292551.
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Table[n (Binomial[n, 2] - 2), {n, 0, 45}]
LinearRecurrence[{4, -6, 4, -1}, {0, -2, -2, 3}, 46]
CoefficientList[Series[-x (2 - 6 x + x^2)/(1 - x)^4, {x, 0, 45}], x]
A347976
Triangle T(n,k) read by rows: the rows list volumes of rank 2 Schubert matroid polytopes.
Original entry on oeis.org
1, 2, 4, 3, 8, 11, 4, 13, 22, 26, 5, 19, 38, 52, 57, 6, 26, 60, 94, 114, 120, 7, 34, 89, 158, 213, 240, 247, 8, 43, 126, 251, 376, 459, 494, 502, 9, 53, 172, 381, 632, 841, 960, 1004, 1013, 10, 64, 228, 557, 1018, 1479, 1808, 1972, 2026, 2036, 11, 76, 295, 789, 1580, 2503, 3294, 3788, 4007, 4072, 4083
Offset: 3
The triangle T(n,k) starts as follows:
[n\k] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[3] 1;
[4] 2, 4;
[5] 3, 8, 11;
[6] 4, 13, 22, 26;
[7] 5, 19, 38, 52, 57;
[8] 6, 26, 60, 94, 114, 120;
[9] 7, 34, 89, 158, 213, 240, 247;
[10] 8, 43, 126, 251, 376, 459, 494, 502;
[11] 9, 53, 172, 381, 632, 841, 960, 1004, 1013;
[12] 10, 64, 228, 557, 1018, 1479, 1808, 1972, 2026, 2036;
[13] 11, 76, 295, 789, 1580, 2503, 3294, 3788, 4007, 4072, 4083;
[14] 12, 89, 374, 1088, 2374, 4089, 5804, 7090, 7804, 8089, 8166, 8178;
...
A384993
Triangle read by rows: T(n,k) is the number of compositions (p_0,p_1,...,p_m) of n such that max(i + p_i) = k for 0 <= i <= m.
Original entry on oeis.org
1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 4, 4, 0, 0, 0, 3, 8, 5, 0, 0, 0, 1, 12, 13, 6, 0, 0, 0, 0, 12, 26, 19, 7, 0, 0, 0, 0, 8, 40, 46, 26, 8, 0, 0, 0, 0, 4, 48, 88, 73, 34, 9, 0, 0, 0, 0, 1, 47, 140, 163, 108, 43, 10, 0, 0, 0, 0, 0, 38, 190, 307, 273, 152, 53, 11
Offset: 0
Triangle begins:
k=0 1 2 3 4 5 6 7 8
n=0 [1]
n=1 [0, 1]
n=2 [0, 0, 2]
n=3 [0, 0, 1, 3]
n=4 [0, 0, 0, 4, 4]
n=5 [0, 0, 0, 3, 8, 5]
n=6 [0, 0, 0, 1, 12, 13, 6]
n=7 [0, 0, 0, 0, 12, 26, 19, 7]
n=8 [0, 0, 0, 0, 8, 40, 46, 26, 8]
...
The composition of n = 8, (2,1,3,1,1) has values of i + p_i : 2,2,5,4,5 that have a maximum value of 5 so this composition is counted under T(8,5) = 40.
T(4,3) = 4 counts: (1,2,1), (2,1,1), (2,2), (3,1).
T(4,4) = 4 counts: (1,1,1,1), (1,1,2), (1,3), (4).
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r(i) = {sum(k=1,i, x^k)}
P(n) = {1 + sum(i=1,n, prod(j=0,i-1, r(n-j)))}
C(k) = {sum(i=1,k, x^k * P(k-i) * prod(j=0,i-2, 1 + r(k-j-2)))}
T_rowlist(max_row) = {my(N = max_row+1, h = 1 + sum(k=1,N, y^k * C(k))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
Comments