Displaying 1-10 of 28 results found.
Dispersion of A047222, (numbers >1 and congruent to 0 or 2 or 3 mod 5), by antidiagonals.
+20
20
1, 2, 4, 3, 7, 6, 5, 12, 10, 9, 8, 20, 17, 15, 11, 13, 33, 28, 25, 18, 14, 22, 55, 47, 42, 30, 23, 16, 37, 92, 78, 70, 50, 38, 27, 19, 62, 153, 130, 117, 83, 63, 45, 32, 21, 103, 255, 217, 195, 138, 105, 75, 53, 35, 24, 172, 425, 362, 325, 230, 175, 125, 88
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....5....8
4....7....12...20...33
6....10...17...28...47
9....15...25...42...70
11...18...30...50...83
14...23...38...63...105
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; c2=5; m[n_]:=If[Mod[n, 3]==0, 1, 0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}] (* A047222 *) 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}]] (* A191738 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191738 *)
Nonnegative integers not congruent to 2 mod 4.
+10
77
0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92
COMMENTS
Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence (starting at 3) gives values of AUB, sorted and duplicates removed. Values of AUBUC give same sequence. - David W. WilsonThese are the nonnegative integers that can be written as a difference of two squares, i.e., n = x^2 - y^2 for integers x,y. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 25 2002. Equivalently, nonnegative numbers represented by the quadratic form x^2-y^2 of discriminant 4. The primes in this sequence are all the odd primes. - N. J. A. Sloane, May 30 2014 Numbers n such that Kronecker(4,n) == mu(gcd(4,n)). - Jon Perry, Sep 17 2002 Count, sieving out numbers of the form 2*(2*n+1) ( A016825, "nombres pair-impairs"). A generalized Chebyshev transform of the Jacobsthal numbers: apply the transform g(x) -> (1/(1+x^2)) g(x/(1+x^2)) to the g.f. of A001045(n+2). Partial sums of 1,2,1,1,2,1,.... - Paul Barry, Apr 26 2005 The sequence 1,1,3,4,5,... is the image of A001045(n+1) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005 With offset 0 starting (1, 3, 4,...) = INVERT transform of A009531 starting (1, 2, -1, -4, 1, 6,...) with offset 0. Apparently these are the regular numbers modulo 4 [Haukkanan & Toth]. - R. J. Mathar, Oct 07 2011 Numbers of the form x*y in nonnegative integers x,y with x+y even. - Michael Somos, May 18 2013 Numbers that are the sum of zero or more consecutive odd positive numbers. - Gionata Neri, Sep 01 2015 Nonnegative integers of the form (2+(3*m-2)/4^j)/3, j,m >= 0. - L. Edson Jeffery, Jan 02 2017 This is { x^2 - y^2; x >= y >= 0 }; with the restriction x > y one gets the same set without zero; with the restriction x > 0 (i.e., differences of two nonzero squares) one gets the set without 1. An odd number 2n-1 = n^2 - (n-1)^2, a number 4n = (n+1)^2 - (n-1)^2. - M. F. Hasler, May 08 2018
REFERENCES
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.
FORMULA
Recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = n - 1 + (3n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. Partial sums of the period-3 sequence 0, 1, 1, 2, 1, 1, 2, 1, 1, 2, ... ( A101825). - Ralf Stephan, May 19 2013 G.f.: A(x) = x^2*(1+x)^2/((1-x)^2*(1+x+x^2)); a(n)=Sum{k=0..floor(n/2)}, binomial(n-k-1, k)* A001045(n-2*k), n>0. - Paul Barry, Jan 16 2005, R. J. Mathar, Dec 09 2009 Euler transform of length 3 sequence [3, -2, 1].
a(2-n) = -a(n). (End)
a(n) = (12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 4k-1, a(3k-1) = 4k-3, a(3k-2) = 4k-4. (End)
The g.f. A(x) satisfies (A(x)/x)^2 + A(x)/x = x*B(x)^2, where B(x) is the o.g.f. of A042968. - Peter Bala, Apr 12 2017 Sum_{n>=2} (-1)^n/a(n) = log(sqrt(2)+2)/sqrt(2) - (sqrt(2)-1)*log(2)/4. - Amiram Eldar, Dec 05 2021 a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 2, with a(2) = 1 and a(3) = 3.
a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A047222 and A006165. (End) E.g.f.: (9 + 12*exp(x)*(x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023
EXAMPLE
G.f. = x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 11*x^9 + 12*x^10 + ...
MAPLE
a_list := proc(len) local rec; rec := proc(n) option remember;
ifelse(n <= 4, [0, 1, 3, 4][n], rec(n-1) + rec(n-3) - rec(n-4)) end:
seq(rec(n), n=1..len) end: a_list(76); # Peter Luschny, Aug 06 2022
MATHEMATICA
nn=100; Complement[Range[0, nn], Range[2, nn, 4]] (* Harvey P. Dale, May 21 2011 *) LinearRecurrence[{1, 0, 1, -1}, {0, 1, 3, 4}, 70] (* L. Edson Jeffery, Jan 21 2015 *) Select[Range[0, 100], ! MemberQ[{2}, Mod[#, 4]] &] (* Vincenzo Librandi, Sep 03 2015 *)
PROG
(Haskell)
a042965 = (`div` 3) . (subtract 3) . (* 4)
a042965_list = 0 : 1 : 3 : map (+ 4) a042965_list
CROSSREFS
Essentially the complement of A016825. See A267958 for these numbers multiplied by 4.
Numbers that are congruent to {0, 1, 4} mod 5.
+10
32
0, 1, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 69, 70, 71, 74, 75, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 95, 96, 99, 100, 101, 104, 105, 106, 109
COMMENTS
n^3 and n have the same last digit.
Partial sums of (0, 1, 3, 1, 1, 3, 1, 1, 3, 1, ...). - Gary W. Adamson, Jun 19 2008 Row sum of a triangle where every "triple" contains 1,2,2. - Craig Knecht, Jul 30 2015 Nonnegative m such that floor(k*m^2/5) = k*floor(m^2/5), where k = 2, 3 or 4. - Bruno Berselli, Dec 03 2015
REFERENCES
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 459.
FORMULA
G.f.: x^2*(1+3*x+x^2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011 E.g.f: (5/3)*(x-1)*exp(x) + (2/3)*exp(-x/2)*cos(sqrt(3)*x/2) + (2/9)*exp(-x/2)*sin(sqrt(3)*x/2) + 1. - Robert Israel, Aug 04 2015 a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-15+6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k-1, a(3k-1) = 5k-4, a(3k-2) = 5k-5. (End)
a(n) = 5*n/3 - 2*(n mod 3)/3 - 1. - Ammar Khatab, Aug 26 2020 Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/5 - arccoth(3/sqrt(5))/sqrt(5). - Amiram Eldar, Dec 10 2021 a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 4 with a(1) = 0, a(2) = 1 and a(3) = 4.
a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A047222 and A042965. (End)
MAPLE
for n to 1000 do if n^3 - n mod 10 = 0 then print(n); fi; od;
MATHEMATICA
Select[Range[0, 150], MemberQ[{0, 1, 4}, Mod[#, 5]] &] (* or *) LinearRecurrence[{1, 0, 1, -1}, {0, 1, 4, 5}, 91] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *) CoefficientList[Series[x (1 + 3 x + x^2) / ((1 + x + x^2) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
PROG
(PARI) concat(0, Vec(x^2*(1+3*x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Dec 03 2015 (PARI) a(n) = vecsum(divrem(5*n-7, 3)); \\ Kevin Ryde, Aug 08 2022 (Magma) [n : n in [0..150] | n mod 5 in [0, 1, 4]]; // Wesley Ivan Hurt, Jun 14 2016 (Python)
def a(n): return sum(divmod(5*n-7, 3))
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 A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.
+10
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 *)
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 *)
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