A042963 -id:A042963 - cp's OEIS frontend

A232096 a(n) = largest m such that m! divides 1+2+...+n; a(n) = A055881(A000217(n)).

1, 1, 3, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 5, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 2, 1, 1, 2, 4, 1, 1, 3, 3, 1, 1, 3, 2, 1, 1, 2, 3, 1, 1, 4, 4, 1, 1, 3, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 4, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 2, 1, 1, 2, 5, 1, 1, 3, 3, 1, 1, 3

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

A042963 gives the positions of ones and A014601 the positions of larger terms.

Cf. also A232097, A076733, A232094, A232095, A232098.

(define (A232096 n) (A055881 (A000217 n)))

a(n) = A055881(A000217(n)).

a(n) = A231719(A226061(n+1)). [Not a practical way to compute this sequence, but follows from the definitions]

A289841 Number of elements added at n-th stage to the structure of the complex square cross described in A289840.

Original entry on oeis.org

0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48

Offset: 0

Omar E. Pol, Jul 14 2017

For n = 0..17 the sequence is similar to the known toothpick sequences.
The surprising fact is that for n >= 18 the sequence has a periodic tail. More precisely, it has period 8: repeat [32, 80, 16, 16, 16, 64, 48, 48]. This tail is in accordance with the expansion of the four arms of the cross. The tail also can be written starting from the 20th stage, with period 8: repeat [16, 16, 16, 64, 48, 48, 32, 80], (see example).
This sequence is essentially the first differences of A289840. The behavior is similar to A290221 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

			For n = 0..17 the sequence is 0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32;
Terms 18 and beyond can be arranged in a rectangular array with eight columns as shown below:
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
...
On the other hand, in accordance with the periodic structure of the arms of the square cross, the terms 20 and beyond can be arranged in a rectangular array with eight columns as shown below:
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
...

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A004767, A008584, A042963, A139250, A139251, A220500, A220501, A289840, A290221, A294021.

concat(0, Vec(x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^100))) \\ Colin Barker, Nov 12 2017

G.f.: x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)). - Colin Barker, Nov 12 2017

A061857 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add 2 distinct integers from 1 to n in such a way that the sum is divisible by k.

Original entry on oeis.org

0, 1, 0, 3, 1, 1, 6, 2, 2, 1, 10, 4, 4, 2, 2, 15, 6, 5, 3, 3, 2, 21, 9, 7, 5, 4, 3, 3, 28, 12, 10, 6, 6, 4, 4, 3, 36, 16, 12, 8, 8, 5, 5, 4, 4, 45, 20, 15, 10, 9, 7, 6, 5, 5, 4, 55, 25, 19, 13, 11, 9, 8, 6, 6, 5, 5, 66, 30, 22, 15, 13, 10, 10, 7, 7, 6, 6, 5, 78, 36, 26, 18, 16, 12, 12, 9, 8, 7, 7

Offset: 1

Antti Karttunen, May 11 2001

Since the sum of two distinct integers from 1 to n can be as much as 2n-1, this triangular table cannot show all the possible cases. For larger triangles showing all solutions, see A220691 and A220693. - Antti Karttunen, Feb 18 2013 [based on Robert Israel's mail, May 07 2012]

			The second term on the sixth row is 6 because we have 6 solutions: {1+3, 1+5, 2+4, 2+6, 3+5, 4+6} and the third term on the same row is 5 because we have solutions {1+2,1+5,2+4,3+6,4+5}.
Triangle begins:
   0;
   1,  0;
   3,  1,  1;
   6,  2,  2,  1;
  10,  4,  4,  2,  2;
  15,  6,  5,  3,  3,  2;
  21,  9,  7,  5,  4,  3,  3;
  28, 12, 10,  6,  6,  4,  4,  3;
  36, 16, 12,  8,  8,  5,  5,  4,  4;
  45, 20, 15, 10,  9,  7,  6,  5,  5,  4;

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Stackexchange, Question 142323
Index entries for sequences related to subset sums modulo m

This is the lower triangular region of square array A220691. See A220693 for all nonzero solutions.

The left edge (first diagonal) of the triangle: A000217, the second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A058212, the fourth by A001971, the central column by A042963? trinv is given at A054425. Cf. A061865.

a061857 n k = length [()| i <- [2..n], j <- [1..i-1], mod (i + j) k == 0]
a061857_row n = map (a061857 n) [1..n]
a061857_tabl = map a061857_row [1..]
-- Reinhard Zumkeller, May 08 2012

[seq(DivSumChoose2Triangle(j),j=1..120)]; DivSumChoose2Triangle := (n) -> nops(DivSumChoose2(trinv(n-1),(n-((trinv(n-1)*(trinv(n-1)-1))/2))));
DivSumChoose2 := proc(n,k) local a,i,j; a := []; for i from 1 to (n-1) do for j from (i+1) to n do if(0 = ((i+j) mod k)) then a := [op(a),[i,j]]; fi; od; od; RETURN(a); end;

a[n_, 1] := n*(n-1)/2; a[n_, k_] := Module[{r}, r = Reduce[1 <= i < j <= n && Mod[i + j, k] == 0, {i, j}, Integers]; Which[Head[r] === Or, Length[r], Head[r] === And, 1, r === False, 0, True, Print[r, " not parsed"]]]; Table[a[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 04 2014 *)

(define (A061857 n) (A220691bi (A002024 n) (A002260 n))) ;; Antti Karttunen, Feb 18 2013. Needs A220691bi from A220691.

From Robert Israel, May 08 2012: (Start)
Let n+1 = b mod k with 0 <= b < k, q = (n+1-b)/k.  Let k = c mod 2, c = 0 or 1.
If b = 0 or 1 then a(n,k) = q^2*k/2 + q*b - 2*q - b + 1 + c*q/2.
If b >= (k+3)/2 then a(n,k) = q^2*k/2 + q*b - 2*q + b - 1 - k/2 + c*(q+1)/2.
Otherwise a(n,k) = q^2*k/2 + q*b - 2*q + c*q/2. (End)

Offset corrected by Reinhard Zumkeller, May 08 2012

A185952 Partial products of A002313, the primes that are 1 or 2 (mod 4).

Original entry on oeis.org

2, 10, 130, 2210, 64090, 2371330, 97224530, 5152900090, 314326905490, 22945864100770, 2042181904968530, 198091644781947410, 20007256122976688410, 2180790917404459036690, 246429373666703871145970

Offset: 1

Jonathan Vos Post, Feb 07 2011

Product of the first n primes which are natural primes which are not Gaussian primes. Product of the first n primes congruent to 1 or 2 modulo 4. Product of the first n primes of form x^2+y^2. Product of the first n primes p such that -1 is a square mod p. Factors of primorials (A002110) not divisible by natural primes which are also Gaussian primes.

Essentially twice A006278.

			a(10) = 2 * 5 * 13 * 17 * 29 * 37 * 41 * 53 * 61 * 73 = 22945864100770.

G. C. Greubel, Table of n, a(n) for n = 1..300

Cf. A000040, A002110, A002313, A042963, A103222.

Rest@ FoldList[#1*#2 &, 1, Select[ Prime@ Range@ 30, Mod[#, 4] != 3 &]] (* Robert G. Wilson v *)

pp(v)=my(t=1); vector(#v,i,t*=v[i])
pp(select(n->n%4<3, primes(20))) \\ Charles R Greathouse IV, Apr 21 2015

a(n) = Product_{i=1..n} A002313(i) = 2 * Product_{i=1..n} {p in A000040 but p not in A002145} = Product_{i=1..n} {A000040 intersection A042963}.

Terms corrected by Robert G. Wilson v, Feb 11 2011

A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.

Original entry on oeis.org

0, 1, 4, 13, 24, 41, 60, 85, 112, 145, 180, 221, 264, 313, 364, 421, 480, 545, 612, 685, 760, 841, 924, 1013, 1104, 1201, 1300, 1405, 1512, 1625, 1740, 1861, 1984, 2113, 2244, 2381, 2520, 2665, 2812, 2965, 3120, 3281, 3444, 3613, 3784, 3961, 4140, 4325, 4512

Offset: 0

Wesley Ivan Hurt, Mar 07 2015

Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. a(n) is the total area of each figure. The perimeter, P, of each figure is given by P(n) = 4*A042963(n), n>0 (see example).

For n>0, partial sums of a(n) are in A056640.

			                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290 (squares), A002620 (quarter-squares), A042963.

Cf. A056640, A085046, A255876.

[(4*n^2 - 4*n + 1 - (-1)^n)/2 : n in [0..100]];

A255840:=n->(4*n^2 - 4*n + 1 - (-1)^n)/2: seq(A255840(n), n=0..100);

CoefficientList[Series[x (1 + 2 x + 5 x^2)/((1 + x) (1 - x)^3), {x, 0, 50}], x]

vector(100,n,(4*(n-1)^2 - 4*(n-1) + 1 + (-1)^n)/2) \\ Derek Orr, Mar 09 2015

G.f.: x*(1+2*x+5*x^2)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = A000290(n) + 4*A002620(n).
a(n) - a(n-1) = A047471(n). - Wesley Ivan Hurt, Apr 28 2017

A365968 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). An infinite binary tree with root node 0 in row n = 0. Each node then has left child (2j) - k - 1 and right child (2j) - k + 1, where j and k are the values of the parent and grandparent nodes respectively.

Original entry on oeis.org

0, -1, 1, -3, -1, 1, 3, -6, -4, -2, 0, 0, 2, 4, 6, -10, -8, -6, -4, -4, -2, 0, 2, -2, 0, 2, 4, 4, 6, 8, 10, -15, -13, -11, -9, -9, -7, -5, -3, -7, -5, -3, -1, -1, 1, 3, 5, -5, -3, -1, 1, 1, 3, 5, 7, 3, 5, 7, 9, 9, 11, 13, 15, -21, -19, -17, -15, -15, -13, -11, -9

Offset: 0

John Tyler Rascoe, Sep 23 2023

For n in A014601 row n will contain all even numbers from 0 to A000217(n).

For n in A042963 row n will contain all odd numbers from 1 to A000217(n).

			Triangle begins:
        k=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  n=0:    0;
  n=1:   -1,  1;
  n=2:   -3, -1,  1,  3;
  n=3:   -6, -4, -2,  0,  0,  2,  4,  6;
  n=4:  -10, -8, -6, -4, -4, -2,  0,  2, -2,  0,  2,  4,  4,  6,  8, 10;
  ...
The binary tree starts with root 0 in row n = 0. For rows n < 2, k = 0.
In row n = 3, the parent node -3 has left child -6 = 2*(-3) - (-1) - 1.
The tree begins:
row
[n]
[0]                   ______0______
                     /             \
[1]              __-1__           __1__
                /      \         /     \
[2]           -3       -1       1       3
              / \      / \     / \     / \
[3]         -6  -4   -2   0   0   2   4   6
.

John Tyler Rascoe, Rows n = 0..12, flattened

Cf. A004718, A274575, A360173, A363718.

T(n,k) = sum(i=0,n-1, if(bittest(k,i), i+1, -(i+1))); \\ Kevin Ryde, Nov 14 2023

def A365968(n, k):
    b, x = bin(k)[2:].zfill(n), 0
    for i in range(0, n):
        x += (-1)**(int(b[n-(i+1)])+1)*(i+1)
    return(x) # John Tyler Rascoe, Nov 12 2023

T(n,k) = - Sum_{i=0..n-1} (i+1)*(-1)^b[i] where the binary expansion of k is k = Sum_{i=0..n-1} b[i]*2^i. - Kevin Ryde, Nov 14 2023

A046711 From the Bruck-Ryser theorem: numbers n == 1 or 2 (mod 4) which are also the sum of 2 squares.

Original entry on oeis.org

1, 2, 5, 9, 10, 13, 17, 18, 25, 26, 29, 34, 37, 41, 45, 49, 50, 53, 58, 61, 65, 73, 74, 81, 82, 85, 89, 90, 97, 98, 101, 106, 109, 113, 117, 121, 122, 125, 130, 137, 145, 146, 149, 153, 157, 162, 169, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 225

Offset: 1

N. J. A. Sloane

Intersection of A001481 and A042963; A000161(a(n)) > 0. - Reinhard Zumkeller, Feb 14 2012

M. Hall, Jr., Combinatorial Theory, Theorem 12.3.2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. H. Bruck and H. J. Ryser, The nonexistence of certain projective planes, Canad. J. Math., 1 (1949), 88-93.
Index entries for sequences related to sums of squares

Cf. A000161, A001481, A042963.

a046711 n = a046711_list !! (n-1)
a046711_list = [x | x <- a042963_list, a000161 x > 0]
-- Reinhard Zumkeller, Aug 16 2011

max = 225; Flatten[ Table[ a^2 + b^2, {a, 0, Sqrt[max]}, {b, a, Sqrt[max - a^2]}], 1] // Union // Select[#, (1 <= Mod[#, 4] <= 2)& ]& (* Jean-François Alcover, Sep 13 2012 *)
With[{max=15},Select[Select[Total/@Tuples[Range[0,max]^2,2], MemberQ[ {1,2}, Mod[ #,4]]&]//Union,#<=max^2&]] (* Harvey P. Dale, Jan 14 2017 *)

from itertools import count, islice
from sympy import factorint
def A046711_gen(): # generator of terms
    return filter(lambda n:0 < n & 3 < 3 and all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A046711_list = list(islice(A046711_gen(),30)) # Chai Wah Wu, Jun 28 2022

More terms from James Sellers

A056699 First differences are 2,1,-2,3 (repeated).

Original entry on oeis.org

1, 3, 4, 2, 5, 7, 8, 6, 9, 11, 12, 10, 13, 15, 16, 14, 17, 19, 20, 18, 21, 23, 24, 22, 25, 27, 28, 26, 29, 31, 32, 30, 33, 35, 36, 34, 37, 39, 40, 38, 41, 43, 44, 42, 45, 47, 48, 46, 49, 51, 52, 50, 53, 55, 56, 54, 57, 59, 60, 58, 61, 63, 64, 62, 65, 67, 68, 66

Offset: 1

Michael Knauth (knauth_jur(AT)yahoo.de), Nov 21 2003

Second quadrisection of natural numbers shifted right two places. - Ralf Stephan, Jun 10 2005

A permutation of the natural numbers partitioned into quadruples [4k-3,4k-1,4k,4k-2] for k > 0. Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the third and fourth elements; repeat for all quadruples. - Guenther Schrack, Oct 18 2017

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Inverse: A284307.
Sequence of fixed points: A016813(n-1) for n > 0.
Odd elements: A005408(n-1) for n > 0.
Indices of odd elements: A042963(n) for n > 0.
Even elements: 2*A103889(n) for n > 0.
Indices of even elements: A014601(n) for n > 0.
Cf. A067060, A092486, A116966, A132400, A292576, A263449, A042938 (bisection)

a = [1 3 4 2];
max = 10000;  % Generation of a b-file
for n := 5:max
   a(n) = a(n-4) + 4;
end;
% Guenther Schrack, Oct 18 2017

[Floor((n - ((-1)^n + (-1)^(n*(n-1)/2)*(2+(-1)^n)) / 2)): n in [1..100]]; // Vincenzo Librandi, Feb 05 2018

LinearRecurrence[{1,0,0,1,-1},{1,3,4,2,5},70] (* Harvey P. Dale, May 10 2014 *)
Table[Floor[(n - ((-1)^n + (-1)^(n (n - 1) / 2) (2 + (-1)^n)) / 2)], {n, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

for(n=1, 10000, print1(n - ((-1)^n + (-1)^(n*(n-1)/2)*(2+(-1)^n))/2, ", ")) \\ Guenther Schrack, Oct 18 2017

G.f.: x*(2*x^4 - 2*x^3 + x^2 + 2*x + 1)/((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Nov 08 2012
From Guenther Schrack, Oct 18 2017: (Start)
a(n) = a(n-4) + 4 for n > 4.
a(n) = n + periodic[0,1,1,-2].
a(n) = A092486(A067060(n) - 1) for n > 0.
a(n) = A292576(n) - 2*((-1)^floor(n/2)) for n > 0.
a(A116966(n-1)) = A263449(n-1) for n > 0.
A263449(a(n) - 1) = A116966(n-1) for n > 0.
a(n+2) - a(n) = (-1)^floor(n^2/4)*A132400(n+1) for n > 0.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. (End)
a(n) = A298364(n-1) + 1 for n > 1. - Guenther Schrack, Feb 04 2018

A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 11, 6, 31, 16, 23, 12, 9, 63, 32, 47, 24, 19, 10, 127, 64, 95, 48, 39, 20, 13, 255, 128, 191, 96, 79, 40, 27, 14, 511, 256, 383, 192, 159, 80, 55, 28, 17, 1023, 512, 767, 384, 319, 160, 111, 56, 35, 18, 2047, 1024, 1535, 768, 639

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000225 (-1+2^n)
Row 2:  A000079 (2^n)
Row 3:  A055010
Row 4:  3*A000079
Row 5:  A153894
Row 6:  5*A000079
Row 7:  A086224
Row 8:  A005009
Row 9:  A052996
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
This sequence is a permutation of the natural numbers.  - L. Edson Jeffery, Aug 13 2014

			Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159

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

Cf. A042963, A014601, A191665, A191426.

Cf. A241957.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
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}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 14, 10, 8, 11, 27, 19, 15, 9, 22, 54, 38, 30, 18, 12, 43, 107, 75, 59, 35, 23, 13, 86, 214, 150, 118, 70, 46, 26, 16, 171, 427, 299, 235, 139, 91, 51, 31, 17, 342, 854, 598, 470, 278, 182, 102, 62, 34, 20, 683, 1707, 1195, 939, 555, 363

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A005578
Row 2:  A160113
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2...3....6...11
4...7...14....27...54
5...10...19...38...75
8...15..30...59...118
8...18..35...70...139

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

Cf. A042963, A014601, A191426, A191663.

(* 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] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042964: (2+4k,3+4k) *)
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}]] (* A191666 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666  *)

cp's OEIS Frontend

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A255840 a(n) = (4*n^2 - 4*n + 1 - (-1)^n)/2.

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A046711 From the Bruck-Ryser theorem: numbers n == 1 or 2 (mod 4) which are also the sum of 2 squares.

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A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.

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