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Displaying 1-10 of 361 results found. page 1 2 3 4 5 6 7 8 9 10 ... 37
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A366487 First differences of "commas" sequence A121805. +0
0
11, 23, 59, 41, 51, 62, 83, 13, 43, 74, 14, 55, 5, 55, 5, 56, 16, 77, 47, 18, 99, 89, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 94, 34, 74, 14, 54, 94, 34, 74, 14, 54, 94 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..100000 (terms 1..1000 from N. J. A. Sloane)
Michael S. Branicky, Table of n, a(n) for n = 1..20000 (terms 1..1000 from N. J. A. Sloane)
PROG
(Python)
from itertools import islice
def agen(): # generator of terms
an, y = 1, 1
while y < 10:
prevan = an
an, y = an + 10*(an%10), 1
while y < 10:
if str(an+y)[0] == str(y):
an += y
break
y += 1
yield an - prevan
print(list(islice(agen(), 99))) # Michael S. Branicky, Nov 12 2023
CROSSREFS
Cf. A121805.
KEYWORD
nonn,fini,new
AUTHOR
N. J. A. Sloane, Nov 12 2023
STATUS
approved
A367252 a(n) is the number of ways to tile an n X n square as explained in comments. +0
0
1, 0, 1, 4, 88, 3939, 534560, 185986304, 175655853776, 437789918351688, 2898697572048432368, 50698981110982431863735, 2342038257118692026082013568, 285250169294740386915765591840768, 91531011920509198679773321121428857296, 77312253225939431362091700178995800855209496 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Draw a Dyck path from (0,0) to (n,n) so the path always stays above the diagonal. Now section the square into horizontal rows of height one to the left of the path and tile these rows using 1 X 2 and 1 X 1 tiles. Similarly, section the part to the right of the path into columns with width one and tile these using 2 X 1 and 1 X 1 tiles. Furthermore, no 1 X 1 tiles are allowed in the bottom row.
LINKS
Table of n, a(n) for n=0..15.
Anna Tscharre, A possible 6 X 6 tiling
FORMULA
a(n) == 1 (mod 2) <=> n in { A055010 }. - Alois P. Heinz, Nov 11 2023
MAPLE
b:= proc(x, y) option remember; (F->
`if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-1), 0)+
`if`(y>x, b(x, y-1)*F(x+1), 0)))(combinat[fibonacci])
end:
a:= n-> b(n$2):
seq(a(n), n=0..15); # Alois P. Heinz, Nov 11 2023
CROSSREFS
Special case of A003150.
Cf. A055010, A307082.
KEYWORD
nonn,new
AUTHOR
Anna Tscharre, Nov 11 2023
STATUS
approved
A366928 a(n) is the smallest k such that A301573(k) = n. +0
0
1, 0, 6, 12, 20, 41, 42, 56, 72, 90, 110, 155, 156, 182, 270, 271, 272, 306, 379, 380, 420, 462, 551, 552, 600, 650, 702, 756, 812, 870, 930, 1055, 1056, 1122, 1190, 1260, 1405, 1406, 1482, 1560, 1640, 1805, 1806, 1892, 1980, 2254, 2255, 2256, 2352, 2450, 2550, 2652, 2861, 2862, 2970 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..54.
PROG
(PARI) ispp(n) = {ispower(n) || n==1}; \\ A001597
f(n) = my(k=0); while(!ispp(n+k) && !ispp(n-k), k++); k; \\ A301573
a(n) = my(k=0); while (f(k) != n, k++); k; \\ Michel Marcus, Oct 29 2023
CROSSREFS
Cf. A001597, A301573.
KEYWORD
nonn,new
AUTHOR
Dmitry Kamenetsky, Oct 28 2023
EXTENSIONS
More terms from Michel Marcus, Oct 29 2023
STATUS
approved
A366600 a(n) = (1 + A033264(n))*a(A053645(n)) for n > 0 with a(0) = 1. +0
0
1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 6, 2, 4, 4, 8, 1, 2, 2, 6, 2, 6, 6, 12, 2, 4, 4, 18, 4, 8, 8, 16, 1, 2, 2, 6, 2, 6, 6, 12, 2, 6, 6, 24, 6, 12, 12, 24, 2, 4, 4, 18, 4, 18, 18, 36, 4, 8, 8, 54, 8, 16, 16, 32, 1, 2, 2, 6, 2, 6, 6, 12, 2, 6, 6, 24, 6, 12, 12, 24, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..79.
FORMULA
a(2n + 1) = a(n).
a(4n) = a(2n) with a(0) = 1.
a(4n + 2) = 2*b(n), b(2n + 1) = 2*b(n), b(2n) = 3*с(n - 1, 1) with b(0) = 1.
c(2n + 1, k) = c(n, k), c(4n + 2, k) = (k + 2)*c(2n, k), c(4n, k) = (k + 3)*c(n - 1, k + 1) with c(0, k) = 1.
Another way to compute a(4n + 2):
a(2*(4^n - 1)/3) = (n + 1)!.
a(2^(2m)*(2k + 1) + 2*(4^m - 1)/3) = (m + 1)*a(2^(2m)*k + 2*(4^m - 1)/3).
a(2^(2m + 1)*(2k + 1) + 2*(4^(m + 1) - 1)/3) = a(2^(2m + 1)*k + 2*(4^(m + 1) - 1)/3).
Note that a(4n + 2) is completely defined by these 3 last formulas. However, it looks like that it is not so easy to identify m and k for a given n, which makes these formulas useless for computing this sequence.
EXAMPLE
a(6) = 4 because the binary expansion of 6 is 110 and we have [(10), 1(10)] -> [1, 1]. Increasing these values by 1 gives us 2*2 = 4.
a(18) = 6 because the binary expansion of 18 is 10010 and we have [(10), (10)0(10)] -> [1, 2]. Increasing these values by 1 gives us 2*3 = 6.
a(26) = 18 because the binary expansion of 26 is 11010 and we have [(10), (10)(10), 1(10)(10)] -> [1, 2, 2]. Increasing these values by 1 gives us 2*3*3 = 18.
For n=482, the bits of n and the resulting product for a(n) are
n = 482 = binary 1 1 1 1 0 0 0 1 0
a(n) = 162 = 3*3*3*3 *2
n=3863 = binary 111100010111 is the same a(n) = 162 since its final trailing "111" has no effect.
PROG
(PARI) a(n) = my(A = 1, B = 1); if(n, for(i=1, logint(n, 2), if(bittest(n, i), A *= (B += !bittest(n, i-1))))); A
CROSSREFS
Cf. A033264, A053645, A346422.
KEYWORD
nonn,base,new
AUTHOR
Mikhail Kurkov, Oct 14 2023
STATUS
approved
A366195 Integers whose binary expansion has the property that there exists a length-k substring of bits in the expansion that is strictly lexicographically later than the first k bits. +0
0
11, 19, 22, 23, 35, 37, 38, 39, 43, 44, 45, 46, 47, 55, 67, 69, 70, 71, 74, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 103, 110, 111, 131, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These are numbers whose binary expansion corresponds to an invalid prefix of a Lyndon word on a two-letter alphabet. If the alphabet is {x, y}, where x < y, then taking the binary expansion of a(n) and mapping 1 to x and 0 to y results in a string that is not a prefix to any Lyndon word. Moreover, this sequence enumerates all strings starting with x that are not prefixes of Lyndon words on this alphabet.
A328870 is a subsequence of this sequence.
For k>=4, the number of k-bit terms in this sequence is 1,3,10,24,58,130,287,613,1302,2720,5655,11665,23969...
LINKS
Table of n, a(n) for n=1..59.
Wikipedia, Lyndon word
EXAMPLE
The binary expansion of a(3) = 22 is 10110, which has a length-2 substring ("11") which is strictly lexicographically later than the first 2 bits ("10"). This also means that xyxxy is not a prefix of any Lyndon word over the alphabet {x,y}.
PROG
(Python)
def ok(n):
w = bin(n)[2:]
return any(any(w[:k] < w[i:i+k] for i in range(1, len(w)-k+1)) for k in range(2, len(w)))
print([k for k in range(157) if ok(k)]) # Michael S. Branicky, Nov 09 2023
CROSSREFS
Cf. A328870.
KEYWORD
nonn,base,new
AUTHOR
Peter Kagey, Nov 05 2023
STATUS
approved
A364842 Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s. +0
0
1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 8, 5, 1, 2, 5, 10, 14, 6, 1, 2, 5, 10, 18, 20, 7, 1, 2, 5, 10, 20, 30, 30, 8, 1, 2, 5, 10, 20, 34, 49, 40, 9, 1, 2, 5, 10, 20, 36, 59, 74, 55, 10, 1, 2, 5, 10, 20, 36, 63, 94, 110, 70, 11, 1, 2, 5, 10, 20, 36, 65, 104, 149, 158, 91, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..78.
EXAMPLE
Table begins:
| 0 1 2 3 4 5 6 7 8 9 10
--+----------------------------------
1 | 1 2 3 4 5 6 7 8 9 10 11
2 | 1 2 5 8 14 20 30 40 55 70 91
3 | 1 2 5 10 18 30 49 74 110 158 221
4 | 1 2 5 10 20 34 59 94 149 224 334
5 | 1 2 5 10 20 36 63 104 169 264 405
6 | 1 2 5 10 20 36 65 108 179 284 445
7 | 1 2 5 10 20 36 65 110 183 294 465
8 | 1 2 5 10 20 36 65 110 185 298 475
9 | 1 2 5 10 20 36 65 110 185 300 479
MATHEMATICA
Seed[i_, n_] := ConstantArray[2, i]~Join~ConstantArray[0, n - i];
A364842Table[n_] := Table[Seed[i, n] // EulerTransform, {i, 1, n}]
(*EulerTransform is defined in A005195*)
CROSSREFS
Cf. A000027 (row 1), A006918 (row 2), A117485 (row 3), A117486 ( row 4), A117487 (row 5), A160647 (row 6), A000712 (main diagonal).
Analogous for initial 1s sequence A008284.
Cf. A115994.
KEYWORD
nonn,tabl,new
AUTHOR
Peter Kagey, Nov 09 2023
STATUS
approved
A367250 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 9. +0
0
0, 1, 35, 703, 11231, 158311, 2062655, 25466743, 302423471, 3487593511, 39314599775, 435241463383, 4748453693711, 51186327429511, 546278900354495, 5781325731101623, 60750456603203951, 634502309615150311, 6592506388026870815, 68188442304165981463, 702543059232886986191 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) is the number of n-digit numbers in A366966.
LINKS
Table of n, a(n) for n=1..21.
Index entries for linear recurrences with constant coefficients, signature (27,-242,720).
FORMULA
a(n) = 9*10^(n-1) - 17*9^(n-1) + 8^n.
a(n) = 27*a(n-1) - 242*a(n-2) + 720*a(n-3) for n > 3.
O.g.f.: x^2*(1 + 8*x)/((1 - 8*x)*(1 - 9*x)*(1 - 10*x)).
E.g.f.: (81*exp(10*x) - 170*exp(9*x) + 90*exp(8*x) - 1)/90.
MATHEMATICA
LinearRecurrence[{27, -242, 720}, {0, 1, 35}, 21]
CROSSREFS
Cf. A366966.
Cf. A001018, A001019, A011557.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367249.
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Nov 11 2023
STATUS
approved
A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8. +0
0
0, 3, 79, 1323, 18175, 223323, 2555119, 27828363, 292407775, 2990349243, 29943991759, 294872615403, 2864776362175, 27525734996763, 262061152909999, 2475899571994443, 23240879960425375, 216963121865909883, 2015960236625789839, 18656492902684557483, 172056837889322101375 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is the number of n-digit numbers in A366965.
LINKS
Table of n, a(n) for n=1..21.
Index entries for linear recurrences with constant coefficients, signature (24,-191,504).
FORMULA
a(n) = 17*9^(n-1) - 31*8^(n-1) + 2*7^n.
a(n) = 24*a(n-1) - 191*a(n-2) + 504*a(n-3) for n > 3.
O.g.f.: x^2*(3 + 7*x)/((1 - 7*x)*(1 - 8*x)*(1 - 9*x)).
E.g.f.: (136*exp(9*x) - 279*exp(8*x) + 144*exp(7*x) - 1)/72.
MATHEMATICA
LinearRecurrence[{24, -191, 504}, {0, 3, 79}, 21]
CROSSREFS
Cf. A366965.
Cf. A000420, A001018, A001019.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367250.
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Nov 11 2023
STATUS
approved
A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7. +0
0
0, 5, 111, 1601, 19095, 204545, 2045511, 19508081, 179752215, 1613908385, 14202967911, 123028446161, 1052237271735, 8907026785025, 74758478722311, 623053865857841, 5162154289325655, 42558224511290465, 349394287423788711, 2858263098464575121, 23311522539676521975 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is the number of n-digit numbers in A366964.
LINKS
Table of n, a(n) for n=1..21.
Index entries for linear recurrences with constant coefficients, signature (21,-146,336).
FORMULA
a(n) = 23*8^(n-1) - 41*7^(n-1) + 3*6^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(5 + 6*x)/((1 - 6*x)*(1 - 7*x)*(1 - 8*x)).
E.g.f.: (161*exp(8*x) - 328*exp(7*x) + 168*exp(6*x) - 1)/56.
MATHEMATICA
LinearRecurrence[{21, -146, 336}, {0, 5, 111}, 21]
CROSSREFS
Cf. A366964.
Cf. A000400, A000420, A001018.
Cf. A367243, A367244, A367245, A367246, A367247, A367249, A367250.
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Nov 11 2023
STATUS
approved
A367247 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 6. +0
0
0, 7, 131, 1609, 16415, 150817, 1296191, 10641169, 84520175, 654958177, 4980233951, 37312922929, 276288797135, 2026564724737, 14750977566911, 106695818055889, 767748717541295, 5500729672814497, 39270143125479071, 279511731951144049, 1984459091985376655, 14059238393314971457 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is the number of n-digit numbers in A366963.
LINKS
Table of n, a(n) for n=1..22.
Index entries for linear recurrences with constant coefficients, signature (18,-107,210).
FORMULA
a(n) = 27*7^(n-1) - 47*6^(n-1) + 4*5^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(7 + 5*x)/((1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
E.g.f.: (162*exp(7*x) - 329*exp(6*x) + 168*exp(5*x) - 1)/42.
MATHEMATICA
LinearRecurrence[{18, -107, 210}, {0, 7, 131}, 22]
CROSSREFS
Cf. A366963.
Cf. A000351, A000400, A000420.
Cf. A367243, A367244, A367245, A367246, A367248, A367249, A367250.
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Nov 11 2023
STATUS
approved
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Last modified November 12 19:50 EST 2023. Contains 366998 sequences. (Running on oeis4.)