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Search: keyword:new
Displaying 1-10 of 397 results found. page 1 2 3 4 5 6 7 8 9 10 ... 40
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A374553 Number of length n inversion sequences avoiding the patterns 010 and 102. +0
0
1, 1, 2, 5, 15, 51, 186, 707, 2763, 11024, 44714, 183830, 764374, 3209031, 13584217, 57918257, 248502212, 1072159593, 4648747281, 20245772943, 88524364619, 388469248937, 1710304847176, 7552480937589, 33442335151831, 148456424569164, 660560252794208 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Benjamin Testart, Table of n, a(n) for n = 0..700
Jay Pantone, The enumeration of inversion sequences avoiding the patterns 201 and 210, arXiv:2310.19632 [math.CO], 2023.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.
FORMULA
Conjecture: G.f. F(x) is algebraic with minimal polynomial x * (x^2 - x + 1)*(x - 1)^2 * F(x)^3 + 2*x*(x - 1)*(2*x^2 - 2*x + 1)*F(x)^2 - (x^4 - 8*x^3 + 11*x^2 - 6*x + 1)*F(x) - (2*x - 1)*(x - 1)^2.
CROSSREFS
Cf. A200753, A263779, A279559, A279566, A374541, A374542.
KEYWORD
nonn,new
AUTHOR
Benjamin Testart, Jul 17 2024
STATUS
approved
A374554 Number of length n inversion sequences avoiding the patterns 100 and 102. +0
0
1, 1, 2, 6, 21, 80, 318, 1305, 5487, 23535, 102603, 453400, 2026408, 9144361, 41607161, 190675552, 879318056, 4077566276, 19001732690, 88940105945, 417948841012, 1971086634986, 9326180071850, 44258248464408, 210605264950063, 1004694354945863, 4804017049287049 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Benjamin Testart, Table of n, a(n) for n = 0..400
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.
CROSSREFS
Cf. A200753, A263780, A279564, A279566, A374541, A374542, A374544, A374545, A374553.
KEYWORD
nonn,new
AUTHOR
Benjamin Testart, Jul 17 2024
STATUS
approved
A374510 Sum of those numbers t which have a unique representation as the sum of floor(n/2) distinct squares from among 1^2,...,n^2. +0
0
0, 5, 14, 90, 220, 910, 1976, 5100, 8336, 15785, 22331, 31850, 40925, 49735, 58848, 74800, 86011, 107559, 123964, 152110, 181504, 220110, 293366, 357700, 393982, 458874, 497123, 570836, 755393, 888770, 987508, 1121120, 1239126, 1395870, 1461465, 1620600 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A000330(n-1) <= a(n).
LINKS
Table of n, a(n) for n=1..36.
Project Euler, Subsets with a Unique Sum
EXAMPLE
a(3) = 14 because: k = 1 and
14 = 4+9 and
4 = 4+0
9 = 9+0
and 9 <= n^2.
a(4) = 90 because: k = 2 and
90 = 5+10+13+17+20+25 and
5 = 1+4
10 = 1+9
13 = 4+9
17 = 1+16
20 = 4+16
25 = 9+16 and 16 <= n^2.
PROG
(Python)
from collections import defaultdict
def a(n):
k = n >> 1
dp = [defaultdict(int) for _ in range(k + 1)]
dp[0][0] = 1
for s in [i**2 for i in range(1, n + 1)]:
for j in range(k, 0, -1):
for m in list(dp[j - 1].keys()):
dp[j][m + s] += dp[j - 1][m]
return sum(t for t, v in dp[k].items() if v == 1)
print([a(n) for n in range(1, 37)])
(Python)
from itertools import combinations
from collections import Counter
def A374510(n): return sum(d for d, e in Counter(sum(s) for s in combinations((m**2 for m in range(1, n+1)), n>>1)).items() if e == 1) # Chai Wah Wu, Jul 17 2024
CROSSREFS
Cf. A000290, A000330.
KEYWORD
nonn,new
AUTHOR
Darío Clavijo, Jul 09 2024
STATUS
approved
A374600 If n = i^2 * A005117(j) for some i, j > 0 then a(n) = j^2 * A005117(i). +0
0
1, 4, 9, 2, 16, 25, 36, 8, 3, 49, 64, 18, 81, 100, 121, 5, 144, 12, 169, 32, 196, 225, 256, 50, 6, 289, 27, 72, 324, 361, 400, 20, 441, 484, 529, 7, 576, 625, 676, 98, 729, 784, 841, 128, 48, 900, 961, 45, 10, 24, 1024, 162, 1089, 75, 1156, 200, 1225, 1296 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(A249327(n, k)) = A249327(k, n).
a(n) = n iff n = k^2 * A005117(k) for some k > 0.
EXAMPLE
For n = 84: 84 = 2^2 * A005117(14), so a(84) = 14^2 * A005117(2) = 392.
PROG
(PARI) \\ See Links section.
CROSSREFS
See A374611 for a similar sequence.
Cf. A000188, A005117, A007913, A249327.
KEYWORD
nonn,new
AUTHOR
Rémy Sigrist, Jul 13 2024
STATUS
approved
A374611 If n = A066207(i) * A066208(j) for some i, j > 0 then a(n) = A066207(j) * A066208(i). +0
0
1, 3, 2, 7, 9, 6, 4, 13, 5, 19, 21, 14, 8, 12, 18, 27, 29, 15, 10, 37, 11, 39, 43, 26, 49, 24, 16, 28, 17, 38, 53, 57, 42, 61, 36, 35, 20, 30, 22, 63, 71, 33, 23, 79, 45, 81, 87, 54, 25, 89, 58, 56, 31, 48, 91, 52, 32, 51, 101, 74, 34, 107, 40, 111, 72, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers
FORMULA
a(A360646(n, k)) = A360646(k, n).
a(n) = n iff n = A066207(k) * A066208(k) for some k > 0.
EXAMPLE
For n = 42: 42 = 21 * 2 = A066207(7) * A066208(2), so a(42) = A066207(2) * A066208(7) = 3 * 11 = 33.
PROG
(PARI) \\ See Links section.
CROSSREFS
See A374600 for a similar sequence.
Cf. A066207, A066208, A247503, A248101, A360646.
KEYWORD
nonn,new
AUTHOR
Rémy Sigrist, Jul 14 2024
STATUS
approved
A374715 Number of distinct sums i^2 + j^2 + k^2 for 1<=i<=j<=k<=n. +0
0
1, 4, 10, 20, 33, 51, 69, 94, 122, 157, 187, 233, 273, 316, 373, 432, 485, 558, 614, 694, 770, 849, 915, 1019, 1108, 1205, 1304, 1410, 1504, 1640, 1742, 1872, 1997, 2121, 2245, 2410, 2534, 2678, 2821, 2994, 3136, 3320, 3472, 3647, 3820, 3993, 4157, 4393, 4558, 4757, 4963, 5186, 5360, 5593 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..54.
PROG
(PARI) a(n) = my(v=vector(3*n^2)); for(i=1, n, for(j=i, n, for(k=j, n, v[i^2+j^2+k^2]+=1))); sum(i=1, #v, v[i]>0);
(Python)
def A374715(n): return len({i**2+j**2+k**2 for i in range(1, n+1) for j in range(i, n+1) for k in range(j, n+1)}) # Chai Wah Wu, Jul 17 2024
CROSSREFS
Cf. A061786, A374716.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jul 17 2024
STATUS
approved
A374718 Number of maximal matchings in the n-Sierpinski gasket graph. +0
0
3, 11, 897, 498314475 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..4.
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set.
Eric Weisstein's World of Mathematics, Sierpinski Gasket Graph.
KEYWORD
nonn,more,new
AUTHOR
Eric W. Weisstein, Jul 17 2024
STATUS
approved
A374716 Number of distinct sums i^2 + j^2 + k^2 + l^2 for 1<=i<=j<=k<=l<=n. +0
0
1, 5, 15, 34, 58, 93, 128, 175, 227, 289, 349, 429, 504, 592, 685, 791, 891, 1014, 1124, 1262, 1394, 1543, 1676, 1851, 2006, 2185, 2356, 2554, 2733, 2948, 3143, 3374, 3585, 3824, 4045, 4313, 4549, 4818, 5064, 5363, 5632, 5934, 6216, 6538, 6834, 7161, 7466, 7838, 8160, 8515, 8852, 9248, 9587, 9989 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..54.
PROG
(PARI) a(n) = my(v=vector(4*n^2)); for(i=1, n, for(j=i, n, for(k=j, n, for(l=k, n, v[i^2+j^2+k^2+l^2]+=1)))); sum(i=1, #v, v[i]>0);
(Python)
def A374716(n): return len({i**2+j**2+k**2+l**2 for i in range(1, n+1) for j in range(i, n+1) for k in range(j, n+1) for l in range(k, n+1)}) # Chai Wah Wu, Jul 17 2024
CROSSREFS
Cf. A061786, A374715.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jul 17 2024
STATUS
approved
A374714 Number of distinct sums i^3 + j^3 + k^3 + l^3 for 1<=i<=j<=k<=l<=n. +0
0
1, 5, 15, 35, 70, 119, 202, 317, 473, 671, 902, 1138, 1515, 2008, 2521, 3039, 3758, 4592, 5539, 6657, 7879, 9209, 10797, 12304, 14243, 16371, 18348, 21006, 23816, 26563, 29848, 33046, 36698, 40190, 44885, 49068, 54040, 59479, 64762, 70420, 76810, 83414, 90659, 98158, 105838, 114127, 123048 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..47.
PROG
(PARI) a(n) = my(v=vector(4*n^3)); for(i=1, n, for(j=i, n, for(k=j, n, for(l=k, n, v[i^3+j^3+k^3+l^3]+=1)))); sum(i=1, #v, v[i]>0);
CROSSREFS
Cf. A061791, A374713.
Cf. A374711, A374716.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jul 17 2024
STATUS
approved
A374713 Number of distinct sums i^3 + j^3 + k^3 for 1<=i<=j<=k<=n. +0
0
1, 4, 10, 20, 35, 55, 83, 119, 164, 218, 280, 343, 431, 535, 648, 760, 903, 1064, 1241, 1442, 1659, 1891, 2151, 2409, 2714, 3044, 3369, 3754, 4160, 4582, 5044, 5499, 6015, 6500, 7094, 7669, 8308, 8990, 9683, 10394, 11180, 12010, 12876, 13773, 14720, 15693, 16721, 17705, 18845, 20010 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..50.
PROG
(PARI) a(n) = my(v=vector(3*n^3)); for(i=1, n, for(j=i, n, for(k=j, n, v[i^3+j^3+k^3]+=1))); sum(i=1, #v, v[i]>0);
CROSSREFS
Cf. A061791, A374714.
Cf. A374715.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jul 17 2024
STATUS
approved
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Last modified July 17 21:15 EDT 2024. Contains 374377 sequences. (Running on oeis4.)