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Search: a054119 -id:a054119
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Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.
+10
20
1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16
OFFSET
1,2
COMMENTS
The square array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...
FORMULA
A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).
EXAMPLE
The top left {1..9} x {1..18} corner of the array:
1, 2, 6, 24, 120, 720, 5040, 40320, 362880
3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680
4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600
5, 14, 54, 264, 1560, 10800, 85680, 766080, 7620480
7, 26, 126, 744, 5160, 41040, 367920, 3669120, 40279680
9, 32, 150, 864, 5880, 46080, 408240, 4032000, 43908480
10, 36, 168, 960, 6480, 50400, 443520, 4354560, 47174400
11, 38, 174, 984, 6600, 51120, 448560, 4394880, 47537280
13, 50, 246, 1464, 10200, 81360, 730800, 7297920, 80196480
15, 56, 270, 1584, 10920, 86400, 771120, 7660800, 83825280
16, 60, 288, 1680, 11520, 90720, 806400, 7983360, 87091200
17, 62, 294, 1704, 11640, 91440, 811440, 8023680, 87454080
18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880
PROG
(Scheme)
(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))
CROSSREFS
Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.
KEYWORD
nonn,base,tabl
AUTHOR
Antti Karttunen, Sep 22 2016
STATUS
approved
a(n) = n! + (n - 1)! + (n - 2)! + n - 3.
+10
4
3, 9, 33, 152, 867, 5884, 46085, 408246, 4032007, 43908488, 522547209, 6745939210, 93884313611, 1401079680012, 22317642547213, 377917892352014, 6778983923712015, 128403161542656016, 2560949482291200017, 53645489280294912018, 1177524571957493760019, 27027108408834293760020
OFFSET
2,1
COMMENTS
a(n) is a lower bound for the length of every superpermutation on n symbols (see links). An upper bound for the length of a minimal superpermutation is given by A341300(n).
LINKS
Anonymous 4chan poster, Robin Houston, Jay Pantone, and Vince Vatter, A lower bound on the length of the shortest superpattern, 2018.
Michael Engen and Vincent Vatter, Containing All Permutations, The American Mathematical Monthly, 128 (1), 2021, pp. 4-24 (preprint version).
James Grime and Brady Haran, Superpermutations, Numberphile video, 2018.
Wikipedia, Superpermutation.
FORMULA
a(n) = A054119(n) + n - 3.
E.g.f.: (3 - x - x^2 - exp(x)*(3 - 4*x + x^2) - (1 - x)*x*log(1 - x))/(1 - x). - Stefano Spezia, Sep 18 2024
a(n) = (n-2)!*n^2 + n - 3. - Chai Wah Wu, Sep 20 2024
D-finite with recurrence (-n+1)*a(n) +(n-2)*(n+2)*a(n-1) -(n-1)*(n-3)*a(n-2) -(4*n-7)*(n-4)=0. - R. J. Mathar, Sep 23 2024
MATHEMATICA
Table[n^2 * (n - 2)! + n - 3, {n, 2, 25}]
PROG
(Python)
from sympy import factorial
def A376269(n): return n**2*factorial(n-2)+n-3 # Chai Wah Wu, Sep 20 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paolo Xausa, Sep 18 2024
STATUS
approved
a(1) = 2; for n > 1, a(n) = (n-2)! * n^3.
+10
3
2, 8, 27, 128, 750, 5184, 41160, 368640, 3674160, 40320000, 482993280, 6270566400, 87697209600, 1314380390400, 21016195200000, 357082280755200, 6424604169984000, 122021710626816000, 2439660069310464000, 51218989645824000000, 1126555274886193152000, 25905540583064862720000, 621623493403188756480000, 15538186060797648568320000
OFFSET
1,1
COMMENTS
In factorial base representation (A007623) the terms are written as: 10, 110, 1011, 10110, 101100, 1011000, 10110000, ... From a(3) = 27 = "1011" onward each term begins always with "1011", followed by n-3 zeros. - Antti Karttunen, Sep 24 2016
FORMULA
a(1) = 2; for n > 1, a(n) = (n-2)! * n^3.
a(n) = n * A054119(n).
For n >= 3, a(n) = (n+1)! + (n-1)! + (n-2)!.
MATHEMATICA
Join[{2}, Table[(n-2)! n^3, {n, 2, 30}]] (* Harvey P. Dale, Apr 14 2017 *)
PROG
(Scheme, two alternatives)
(define (A276940 n) (if (= 1 n) 2 (* n n n (A000142 (- n 2)))))
(define (A276940 n) (cond ((= 1 n) 2) ((= 2 n) 8) (else (+ (A000142 (+ 1 n)) (A000142 (- n 1)) (A000142 (- n 2))))))
CROSSREFS
Row 20 of A276955 (from a(3) = 27 onward).
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Sep 24 2016
STATUS
approved
Decimal expansion of Sum_{k>=0} 1/(k! + (k+1)! + (k+2)!).
+10
1
4, 0, 0, 3, 7, 9, 6, 7, 7, 0, 0, 4, 6, 4, 1, 3, 4, 0, 5, 0, 0, 2, 7, 8, 6, 2, 7, 1, 0, 3, 4, 3, 0, 6, 5, 9, 7, 8, 2, 3, 4, 5, 8, 4, 7, 9, 0, 7, 1, 7, 5, 5, 8, 2, 1, 2, 6, 5, 0, 6, 4, 3, 0, 7, 2, 6, 4, 3, 0, 5, 2, 2, 5, 9, 7, 4, 0, 8, 1, 1, 1, 9, 5, 9, 4, 2, 8, 5, 3, 1
OFFSET
0,1
LINKS
Eric Weisstein's World of Mathematics, Exponential Integral.
FORMULA
Sum_{k>=0} 1/(k! + (k+1)! + (k+2)!) = exp(1) - 1 + gamma - ExpIntegralEi[1].
From Amiram Eldar, Jun 26 2021: (Start)
Equals Sum_{k>=2} 1/A054119(k).
Equals -Integral{x=0..1} x*log(x)*exp(x) dx. (End)
EXAMPLE
0.40037967700464134050027...
PROG
(PARI) exp(1) - 1 + Euler - real(-eint1(-1)) \\ Michel Marcus, Mar 09 2019
CROSSREFS
Cf. A001113 (exp(1)), A001620 (gamma), A054119, A091725 (ExpIntegralEi[1]).
KEYWORD
nonn,cons
AUTHOR
Seiichi Manyama, Mar 09 2019
STATUS
approved

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