A016885 -id:A016885 - cp's OEIS frontend

A016897 a(n) = 5*n + 4.

4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259, 264, 269, 274, 279, 284

Offset: 0

N. J. A. Sloane

Except for 1, 2, n such that Sum_{k=1..n} (k mod 5)*C(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
Numbers ending in 4 or 9. - Lekraj Beedassy, Jul 08 2006
The set of numbers congruent to 4 mod 5. - Gary Detlefs, Mar 07 2010
Also the number of (not necessarily maximal) cliques in the n-book graph and (n+1)-ladder graph. - Eric W. Weisstein, Nov 29 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 944.
Tanya Khovanova, Recursive Sequences.
Leo Tavares, Illustration: Mirror Triangles.
Eric Weisstein's World of Mathematics, Book Graph.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Ladder Graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001622, A008587, A016861, A016873, A016885.

[5*n+4: n in [0..70]]; // Vincenzo Librandi, May 02 2011

a[1]:=4:for n from 2 to 100 do a[n]:=a[n-1]+5 od: seq(a[n], n=1..57); # Zerinvary Lajos, Mar 16 2008

Range[4, 500, 5] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[5 n + 4, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
5 Range[0, 20] + 4 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {9, 14}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(4 + x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=5*n+4 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (4+x)/(1-x)^2. - Paul Barry, Feb 27 2003
a(n) = 2*a(n-1) - a(n-2), n>1. - Philippe Deléham, Nov 03 2008
a(n) = A131098(n+2) + n + 1. - Jaroslav Krizek, Aug 15 2009
a(n) = 10*n - a(n-1) + 3, n>0. - Vincenzo Librandi, Nov 20 2010
A000041(a(n)) == 0 mod 5 is the first of Ramanujan's congruences. - Ivan N. Ianakiev, Dec 29 2014
a(n) = (n+2)^2 - 2*A000217(n-1). See Mirror Triangles illustration. - Leo Tavares, Aug 18 2021
Sum_{n>=0} (-1)^n/a(n) = sqrt(10*(5+sqrt(5)))*Pi/50 - log(2)/5 - sqrt(5)*log(phi)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: exp(x)*(4 + 5*x). - Elmo R. Oliveira, Mar 08 2024

A047219 Numbers that are congruent to {1, 3} mod 5.

Original entry on oeis.org

1, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28, 31, 33, 36, 38, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 73, 76, 78, 81, 83, 86, 88, 91, 93, 96, 98, 101, 103, 106, 108, 111, 113, 116, 118, 121, 123, 126, 128, 131, 133, 136, 138, 141, 143, 146, 148

Offset: 1

N. J. A. Sloane

A001844(N) = N^2 + (N+1)^2 is divisible by 5 if and only if N=a(n), n>=1. E.g., n=2: 5|(3^2 + 4^2). But 7^2 + 8^2 is not congruent to 0 (mod 5). - Wolfdieter Lang, May 09 2012
The number of partitions of 5*(n-1) into at most 2 parts. - Colin Barker, Mar 31 2015
The maximum possible number of 6-cycles in an outerplanar graph on n+5 vertices. - Stephen Bartell, Jul 10 2025

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000566, A001622, A212160, A212161, A001844.

Bisections give: A016861 (odd part), A016885 (even part).

Flatten[#+{1,3}&/@(5*Range[0,30])] (* Harvey P. Dale, Dec 22 2013 *)

a(n)=(5*n-3)\2 \\ Charles R Greathouse IV, Oct 28 2011

a(n) = floor((5*n-3)/2). - Santi Spadaro, Jul 24 2001, corrected by Gary Detlefs, Oct 28 2011
G.f.: x*(1 + 2*x + 2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 07 2011
a(n) = 2*n + floor((n-1)/2) - 1. - Arkadiusz Wesolowski, Sep 19 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1/2 + sqrt(5)/10)*Pi/5 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: 2 + ((10*x - 7)*exp(x) - exp(-x))/4. - David Lovler, Aug 23 2022
a(n) = a(n-2) + 5 for n >= 3. - Rémi Guillaume, Nov 23 2024

A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

Original entry on oeis.org

1, 5, 2, 25, 10, 3, 125, 50, 15, 4, 625, 250, 75, 20, 6, 3125, 1250, 375, 100, 30, 7, 15625, 6250, 1875, 500, 150, 35, 8, 78125, 31250, 9375, 2500, 750, 175, 40, 9, 390625, 156250, 46875, 12500, 3750, 875, 200, 45, 11, 1953125, 781250, 234375, 62500, 18750

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type
"(a or b or c or d mod m)", used in Mathematica programs for A191707-A191711):  if f(n)=(n mod 3), then
  (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by
   a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,
"(a, b, c, d mod m)" is given by
   a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
  1...5....25....125...625
  2...10...50....250...1250
  3...15...75....375...1875
  4...20...100...500...2500
  6...30...150...750...3750

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047201, A008587, A191707, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n
Table[f[n], {n, 1, 30}]  (* A008587 *)
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}]]
(* A191702 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702  *)

T(i,j) = T(i,1)*T(1,j) = (i-1+floor((i+3)/4))*5^(j-1), i>=1, j>=1.

A017305 a(n) = 10*n + 3.

Original entry on oeis.org

3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 433, 443, 453, 463, 473, 483, 493, 503, 513, 523, 533

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(38).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016885, A017198, A017281, A017293, A156677.

[10*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

Range[3, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n) = A017198(n) - A156677(n+2). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
G.f.: (3+7*x)/(x-1)^2. - R. J. Mathar, Apr 11 2016
E.g.f.: exp(x)*(3 + 10*x). - Stefano Spezia, Aug 22 2023
a(n) = A016885(2*n). - Elmo R. Oliveira, Apr 10 2025

A017341 a(n) = 10*n + 6.

Original entry on oeis.org

6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 106, 116, 126, 136, 146, 156, 166, 176, 186, 196, 206, 216, 226, 236, 246, 256, 266, 276, 286, 296, 306, 316, 326, 336, 346, 356, 366, 376, 386, 396, 406, 416, 426, 436, 446, 456, 466, 476, 486, 496, 506, 516, 526, 536

Offset: 0

N. J. A. Sloane

Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004

Numbers k such that k and (4^h)^k end with the same digit, where h > 0. - Bruno Berselli, Dec 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000400 (powers of 6), A008592, A016861, A016885, A017329.

[10*n+6: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[6, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=10*n+6 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2) with n>1, a(0)=6, a(1)=16. - Vincenzo Librandi, May 29 2011
a(n) = (n+1)*A016861(n+1) - n*A016861(n). - Bruno Berselli, Jan 18 2013
From Stefano Spezia, May 31 2021: (Start)
G.f.: 2*(3 + 2*x)/(1 - x)^2.
E.g.f.: 2*(3 + 5*x)*exp(x). (End)
a(n) = 2*A016885(n) = A016861(2*n+1). - Elmo R. Oliveira, Apr 10 2025

A269044 a(n) = 13*n + 7.

Original entry on oeis.org

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Bruno Berselli, Feb 18 2016

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Magma
```
[13*n+7: n in [0..60]];
```

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima
```
makelist(13*n+7, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+7)
```
Sage
```
[13*n+7 for n in (0..60)]
```

G.f.: (7 + 6*x)/(1 - x)^2.
a(n) =  A088227(4*n+3).
a(n) = -A186113(-n-1).
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021

A017365 a(n) = 10*n + 8.

Original entry on oeis.org

8, 18, 28, 38, 48, 58, 68, 78, 88, 98, 108, 118, 128, 138, 148, 158, 168, 178, 188, 198, 208, 218, 228, 238, 248, 258, 268, 278, 288, 298, 308, 318, 328, 338, 348, 358, 368, 378, 388, 398, 408, 418, 428, 438, 448, 458, 468, 478, 488, 498, 508, 518, 528, 538

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016885, A016897.

[10*n+8: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[8, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{8,18},60] (* Harvey P. Dale, Aug 31 2015 *)

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 29 2011
From R. J. Mathar, Nov 26 2014: (Start)
G.f.: 2*(4+x)/(x-1)^2.
a(n) = 2*A016897(n). (End)
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 2*exp(x)*(4 + 5*x).
a(n) = A016885(2*n+1). (End)

A134490 a(n) = Fibonacci(5n + 3).

Original entry on oeis.org

2, 21, 233, 2584, 28657, 317811, 3524578, 39088169, 433494437, 4807526976, 53316291173, 591286729879, 6557470319842, 72723460248141, 806515533049393, 8944394323791464, 99194853094755497, 1100087778366101931

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (11,1)

Cf. A000045, A102312, A099100, A134491.

[Fibonacci(5*n+3): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[5n + 3], {n, 0, 30}]
```

a(n)=fibonacci(5*n+3) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = 11a(n-1) + a(n-2). - Paul Curtz, May 07 2008
From R. J. Mathar, Nov 02 2008: (Start)
G.f.: (2-x)/(1 - 11x - x^2).
a(n) = |A122574(n+2)| + A049666(n+1). (End)
a(n) = A000045(A016885(n)). - Michel Marcus, Nov 08 2013

Offset changed to 0 by R. J. Mathar, Jul 28 2008

A191703 Dispersion of A016861, (5k+1), by antidiagonals.

Original entry on oeis.org

1, 6, 2, 31, 11, 3, 156, 56, 16, 4, 781, 281, 81, 21, 5, 3906, 1406, 406, 106, 26, 7, 19531, 7031, 2031, 531, 131, 36, 8, 97656, 35156, 10156, 2656, 656, 181, 41, 9, 488281, 175781, 50781, 13281, 3281, 906, 206, 46, 10, 2441406, 878906, 253906, 66406, 16406

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...6... 31....156...781
2...11...56....281...1406
3...16...81....406...2031
4...21...106...531...2656
5...26...131...656...3281
7...36...181...906...4531

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047202, A016861, A191708, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n+1
Table[f[n], {n, 1, 30}]  (* A016861 *)
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}]] (* A191703 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191703 *)

A191704 Dispersion of A016873, (5k+2), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 7, 12, 4, 32, 57, 17, 5, 157, 282, 82, 22, 6, 782, 1407, 407, 107, 27, 8, 3907, 7032, 2032, 532, 132, 37, 9, 19532, 35157, 10157, 2657, 657, 182, 42, 10, 97657, 175782, 50782, 13282, 3282, 907, 207, 47, 11, 488282, 878907, 253907, 66407, 16407, 4532

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...2....7.....32....157
3...12...57....282...1407
4...17...82....407...2032
5...22...107...532...2657
6...27...132...657...3282
6...37...182...907...4532

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047207, A016873, A191709, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-3
Table[f[n], {n, 1, 30}] (* A016873 *)
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}]] (* A191704 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191704 *)

cp's OEIS Frontend

A016897 a(n) = 5*n + 4.

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Magma

Maple

Mathematica

PARI

Formula

A047219 Numbers that are congruent to {1, 3} mod 5.

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Mathematica

PARI

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A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

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Mathematica

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A017305 a(n) = 10*n + 3.

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Magma

Mathematica

Formula

A017341 a(n) = 10*n + 6.

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Magma

Mathematica

PARI

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A269044 a(n) = 13*n + 7.

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Magma

Mathematica

Maxima

PARI

Sage

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A017365 a(n) = 10*n + 8.

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