Search: a210954 -id:a210954
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,12
|
|
COMMENTS
|
Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
|
|
LINKS
|
|
|
FORMULA
|
Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020
|
|
MATHEMATICA
|
nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)
|
|
PROG
|
(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 67: For j = 1 to n
40 Next j: Print a(n-1); : Next n
50 End
|
|
CROSSREFS
|
Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A195825
|
|
Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.
|
|
+10
38
|
|
|
1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
. Column k of
. this square
. Generalized Triangle Triangle array A195825
k m m-gonal "A" "B" [row sums of
. numbers triangle "B"
. with a(0)=1]
========================================================
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012
|
|
LINKS
|
|
|
FORMULA
|
Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017
|
|
EXAMPLE
|
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
...
Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.
|
|
PROG
|
(GW-BASIC)' A program (with two A-numbers) for the table of example section.
20 FOR k = 1 TO 10 'Column 1-10
30 T(0, k) = 1 'Row 0
40 FOR n = 1 TO 15 'Rows 1-15
50 FOR j = 1 TO n
70 NEXT j
80 NEXT n
90 NEXT k
100 FOR n = 0 TO 15
110 FOR j = 1 TO 10
120 PRINT T(n, k);
130 NEXT k
140 PRINT
150 NEXT n
160 END
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A211970
|
|
Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.
|
|
+10
13
|
|
|
1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
. Column k of
. this square
. Generalized Triangle Triangle array A211970
k m m-gonal "A" "B" [row sums of
. numbers triangle "B"
. (if k>=1) with a(0)=1,
. if k >= 0]
========================================================
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
6, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
10, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
16, 11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
24, 15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
36, 22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
54, 30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
78, 42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
112, 56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
160, 77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
224, 101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
312, 135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
432, 176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
...
|
|
PROG
|
(GW-BASIC)' A program (with two A-numbers) for the square array of the example section.
20 FOR K = 0 TO 10 'Column 0-10
30 T(0, K) = 1 'Row 0
40 FOR N = 1 TO 15 'Rows 1-15
50 FOR J = 1 TO N
70 NEXT J
80 NEXT N
90 NEXT K
100 FOR N = 0 TO 15: FOR K = 0 TO 10
110 PRINT T(N, K);
120 NEXT K: PRINT: NEXT N
130 END
|
|
CROSSREFS
|
Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
|
|
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 0, 14, 4, 1, 15, 5, 2, 16, 6, 3, 17, 7, 4, 18, 8, 5, 19, 9, 6, 20, 10, 7, 21, 11, 8, 22, 12, 9, 23, 13, 10, 24, 14, 11, 25, 15, 12, 26, 16, 13, 27, 17, 14, 28, 18, 15, 29, 19, 16, 30, 20, 17, 31, 21, 18
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195818(k).
|
|
LINKS
|
|
|
EXAMPLE
|
Written as an irregular triangle:
0;
1;
2;
3;
4;
5;
6;
7;
8;
9;
10, 0;
11, 1;
12, 2;
13, 3, 0;
14, 4, 1;
15, 5, 2;
16, 6, 3;
17, 7, 4;
18, 8, 5;
19, 9, 6;
|
|
CROSSREFS
|
Cf. A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195818, A210954, A210964.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
Search completed in 0.010 seconds
|