Displaying 1-10 of 34 results found.
Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.
+20
20
1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 30, 42, 25, 15, 8, 75, 105, 62, 37, 20, 9, 187, 262, 155, 92, 50, 22, 11, 467, 655, 387, 230, 125, 55, 27, 13, 1167, 1637, 967, 575, 312, 137, 67, 32, 14, 2917, 4092, 2417, 1437, 780, 342, 167, 80, 35, 16, 7292, 10230, 6042
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....2....5....12....30
3....7....17...42....105
4....10...25...62....155
6....15...37...92....230
8....20...50...125...312
9....22...55...137...342
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=2; b=5; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047215 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191722 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722 *)
Quintuple factorial numbers: a(n) = Product_{k=0..n-1} (5*k + 2).
+10
27
1, 2, 14, 168, 2856, 62832, 1696464, 54286848, 2008613376, 84361761792, 3965002804224, 206180145819648, 11752268311719936, 728640635326636032, 48818922566884614144, 3514962424815692218368, 270652106710808300814336, 22193472750286280666775552
FORMULA
E.g.f. (1-5*x)^(-2/5).
a(n) ~ sqrt(2*Pi)/Gamma(2/5)*n^(-1/10)*(5n/e)^n*(1 - (11/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = A084940(n)/ A000142(n)* A000079(n) = 5^n*Pochhammer(2/5, n) = 5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003 G.f.: 1/(1-2x/(1-5x/(1-7x/(1-10x/(1-12x/(1-15x/(1-17x/(1-20x/(1-22x/(1-25x/(1-.../(1- A047215(n+1)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009 a(n) = (-3)^n*Sum_{k=0..n} (5/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012 D-finite with recurrence: a(n) +(-5*n+3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012 G.f.: 1/G(0) where G(k) = 1 - x*(5*k+2)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013 Sum_{n>=0} 1/a(n) = 1 + (e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). - Amiram Eldar, Dec 19 2022
MAPLE
a := n->product(5*i+2, i=0..n-1); [seq(a(j), j=0..30)];
MATHEMATICA
Table[5^n*Pochhammer[2/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 17 2019 *)
PROG
(PARI) vector(20, n, n--; prod(k=0, n-1, 5*k+2)) \\ G. C. Greubel, Aug 17 2019 (Magma) [1] cat [(&*[(5*k+2): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019 (Sage) [product((5*k+2) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019 (GAP) List([0..20], n-> Product([0..n-1], k-> (5*k+2) )); # G. C. Greubel, Aug 17 2019
AUTHOR
Joe Keane (jgk(AT)jgk.org)
Anti-Fibonacci numbers: start with a(0) = 0, and extend by the rule that the next term is the sum of the two smallest numbers that are not in the sequence nor were used to form an earlier sum.
+10
22
0, 3, 9, 13, 18, 23, 29, 33, 39, 43, 49, 53, 58, 63, 69, 73, 78, 83, 89, 93, 98, 103, 109, 113, 119, 123, 129, 133, 138, 143, 149, 153, 159, 163, 169, 173, 178, 183, 189, 193, 199, 203, 209, 213, 218, 223, 229, 233, 238, 243, 249, 253, 258, 263, 269, 273, 279, 283
COMMENTS
In more detail, the sequence is constructed as follows: Start with a(0) = 0. The missing numbers are 1 2 3 4 5 6 ... Add the first two, and we get 3, which is therefore a(1). Cross 1, 2, and 1+2=3 off the missing list. The first two missing numbers are now 4 and 5, so a(2) = 4+5 = 9. Cross off 4,5,9 from the missing list. Repeat.
In other words, this is the sum of consecutive pairs in the sequence 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, ..., ( A249031) the complement to the present one in the natural numbers. For example, a(1)=1+2=3, a(2)=4+5=9, a(3)=6+7=13, ... - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008 The new definition is due to Philippe Lalloue (philip.lallouet(AT)orange.fr), May 08 2008, while the name "anti-Fibonacci numbers" is due to D. R. Hofstadter, Oct 23 2014.
Original definition: second members of pairs in A075325. If instead we take the sum of the last used non-term and the most recent (i.e., 1+2, 2+4, 4+5, 5+7, etc.), we get A008585. - Jon Perry, Nov 01 2014 The sequences a = A075325, b = A047215, and c = A075326 are the solutions of the system of complementary equations defined recursively as follows: a(n) = least new,
b(n) = least new,
c(n) = a(n) + b(n),
where "least new k" means the least positive integer not yet placed. For anti-tribonacci numbers, see A265389; for anti-tetranacci, see A299405. - Clark Kimberling, May 01 2018 We see the Fibonacci numbers 3, 13, 89 and 233 occur in this sequence of anti-Fibonacci numbers. Are there infinitely many Fibonacci numbers occurring in (a(n))? The answer is yes: at least 13% of the Fibonacci numbers occur in (a(n)). This follows from Thomas Zaslavsky's formula, which implies that the sequence A017305 = (10n+3) is a subsequence of (a(n)). The Fibonacci sequence A000045 modulo 10 equals A003893, and has period 60. In this period, the number 3 occurs 8 times. - Michel Dekking, Feb 14 2019
FORMULA
See Zaslavsky (2016) link.
MAPLE
c:=0; a:=[c]; t:=0; M:=100;
for n from 1 to M do
s:=t+1; if s in a then s:=s+1; fi;
t:=s+1; if t in a then t:=t+1; fi;
c:=s+t;
a:=[op(a), c];
od:
[seq(a[n], n=1..nops(a))];
MATHEMATICA
(* Three sequences a, b, c as in Comments *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
AppendTo[c, Last[a] + Last[b]], {z}];
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".",
Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
********
(* Sequence "a" via A035263 substitutions *) Accumulate[Prepend[Flatten[Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] /. Thread[{0, 1} -> {{5, 5}, {6, 4}}]], 3]]
********
(* Sequence "a" via Hofstadter substitutions; see his 2014 link *)
morph = Rest[Nest[Flatten[#/.{1->{3}, 3->{1, 1, 3}}]&, {1}, 6]]
hoff = Accumulate[Prepend[Flatten[morph/.Thread[{1, 3}->{{6, 4, 5, 5}, {6, 4, 6, 4, 6, 4, 5, 5}}]], 3]]
PROG
(Haskell)
import Data.List ((\\))
a075326 n = a075326_list !! n
a075326_list = 0 : f [1..] where
f ws@(u:v:_) = y : f (ws \\ [u, v, y]) where y = u + v
(Python)
def aupton(nn):
alst, disallowed, mink = [0], {0}, 1
for n in range(1, nn+1):
nextk = mink + 1
while nextk in disallowed: nextk += 1
an = mink + nextk
alst.append(an)
disallowed.update([mink, nextk, an])
mink = nextk + 1
while mink in disallowed: mink += 1
return alst
(Python)
def A075326(n): return 5*n-1-int((n|(~((m:=n-1>>1)+1)&m).bit_length())&1) if n else 0 # Chai Wah Wu, Sep 11 2024
Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.
+10
20
1, 5, 2, 15, 6, 3, 40, 16, 10, 4, 101, 41, 26, 11, 7, 255, 105, 66, 30, 20, 8, 640, 265, 166, 76, 51, 21, 9, 1601, 665, 416, 191, 130, 55, 25, 12, 4005, 1665, 1041, 480, 326, 140, 65, 31, 13, 10015, 4165, 2605, 1201, 816, 351, 165, 80, 35, 14, 25040, 10415
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
...
...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....5....15...40...101
2....6....16...41...105
3....10...26...66...166
4....11...30...76...191
7....20...51...130..326
8....21...55...140..351
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=5; b=6; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A008851 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722 *)
Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.
+10
20
1, 3, 2, 8, 5, 4, 20, 13, 10, 6, 50, 33, 25, 15, 7, 125, 83, 63, 38, 18, 9, 313, 208, 158, 95, 45, 23, 11, 783, 520, 395, 238, 113, 58, 28, 12, 1958, 1300, 988, 595, 283, 145, 70, 30, 14, 4895, 3250, 2470, 1488, 708, 363, 175, 75, 35, 16, 12238, 8125, 6175
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....3....8....20....50
2....5....13...33....83
4....10...25...63....158
6....15...38...95....238
7....18...45...113...283
9....23...58...145...363
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=3; b=5; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047218 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191724 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191724 *)
Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.
+10
20
1, 4, 2, 10, 5, 3, 25, 14, 9, 6, 64, 35, 24, 15, 7, 160, 89, 60, 39, 19, 8, 400, 224, 150, 99, 49, 20, 11, 1000, 560, 375, 249, 124, 50, 29, 12, 2500, 1400, 939, 624, 310, 125, 74, 30, 13, 6250, 3500, 2349, 1560, 775, 314, 185, 75, 34, 16, 15625, 8750, 5874
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....4....10....25....64
2....5....14....35...89
3....9....24...60...150
6....15...39...99...249
7....19...49...124..310
8....20...50...125...314
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=4; b=5; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047208 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191725 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191725 *)
Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.
+10
20
1, 2, 3, 6, 7, 4, 16, 17, 11, 5, 41, 42, 27, 12, 8, 102, 106, 67, 31, 21, 9, 256, 266, 167, 77, 52, 22, 10, 641, 666, 417, 192, 131, 56, 26, 13, 1602, 1666, 1042, 481, 327, 141, 66, 32, 14, 4006, 4166, 2606, 1202, 817, 352, 166, 81, 36, 15, 10016, 10416
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....2....6....16....41
3....7....17...42....106
4....11...27...67....167
5....12...31...77....192
8....21...52...131...327
9....22...56...141...352
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=2; b=6; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047216 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191726 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191726 *)
Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.
+10
20
1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 53, 41, 28, 13, 7, 133, 103, 71, 33, 18, 9, 333, 258, 178, 83, 46, 23, 10, 833, 646, 446, 208, 116, 58, 26, 12, 2083, 1616, 1116, 521, 291, 146, 66, 31, 14, 5208, 4041, 2791, 1303, 728, 366, 166, 78, 36, 15, 13021, 10103
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....3....8....21....53
2....6....16...41....103
4....11...28...71....178
5....13...33...83....208
7....18...46...116...291
9....23...58...146...366
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=3; b=6; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047219 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191727 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191727 *)
Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.
+10
20
1, 4, 2, 11, 6, 3, 29, 16, 9, 5, 74, 41, 24, 14, 7, 186, 104, 61, 36, 19, 8, 466, 261, 154, 91, 49, 21, 10, 1166, 654, 386, 229, 124, 54, 26, 12, 2916, 1636, 966, 574, 311, 136, 66, 31, 13, 7291, 4091, 2416, 1436, 779, 341, 166, 79, 34, 15, 18229, 10229
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....4....11...29....74
2....6....16...41....104
3....9....24...61....154
5....14...36...91....229
7....19...49...124...311
8....21...54...136...341
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=4; b=6; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047209 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191728 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191728 *)
Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.
+10
20
1, 2, 4, 3, 8, 5, 7, 18, 12, 6, 17, 43, 28, 13, 9, 42, 107, 68, 32, 22, 10, 103, 267, 168, 78, 53, 23, 11, 257, 667, 418, 193, 132, 57, 27, 14, 642, 1667, 1043, 482, 328, 142, 67, 33, 15, 1603, 4167, 2607, 1203, 818, 353, 167, 82, 37, 16, 4007, 10417, 6517
COMMENTS
For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702. ...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:
...
...
...
For further information about these 20 dispersions, see A191722. ...
Regarding the dispersions A191722- A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
EXAMPLE
Northwest corner:
1....2....3....7.....17
4....8....18...43....107
5....12...28...68....168
6....13...32...78....193
9....22...53...132...328
10...23...57...142...353
MATHEMATICA
(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a=2; b=3; m[n_]:=If[Mod[n, 2]==0, 1, 0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}] (* A047221 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191729 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191729 *)
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