A042948 -id:A042948 - cp's OEIS frontend

A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.

1, 4, 2, 9, 5, 3, 20, 12, 8, 6, 41, 25, 17, 13, 7, 84, 52, 36, 28, 16, 10, 169, 105, 73, 57, 33, 21, 11, 340, 212, 148, 116, 68, 44, 24, 14, 681, 425, 297, 233, 137, 89, 49, 29, 15, 1364, 852, 596, 468, 276, 180, 100, 60, 32, 18, 2729, 1705, 1193, 937, 553

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1: A084639.
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...4...9....20...41
2...5...12...25...52
3...8...17...36...73
6...13..28...57...116
7...16..33...68...137

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

Cf. A042948, A042964, A191426.

(* 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] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042948: (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}]]
(* A191663 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191663 *)

A329649 Let D = A042948(n) be the n-th positive integer congruent to 0 or 1 mod 4, then a(n) = b(D) := Sum_{i=1..D} Kronecker(D,i)*i^2, where Kronecker(D,i) is the Kronecker symbol.

Original entry on oeis.org

1, 10, 4, 16, 159, 48, 52, 680, 136, 48, 168, 288, 4150, 448, 348, 64, 792, 5196, 740, 1120, 1312, 1232, 144, 192, 33565, 624, 1484, 2240, 3192, 2880, 2684, 43680, 4160, -544, 3312, 576, 6424, 5776, 3696, 192, 118071, 2016, 6120, 8096, 9256, 7360, 6696, 1152, 13192

Offset: 1

Jianing Song, Nov 18 2019

sign

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).
Conjecture (a): There are no 0's in this sequence.
Conjecture (b): if D is not of the form k^2 or 5*k^2 or 2^t (t odd, t > 1), then b(D) is divisible by 4*D; if D = 5*k^2, then b(D) is divisible by 4*k^2 = 4*(D/5).
Conjecture (c): let D be a fundamental discriminant > 1 (A003658), then Sum_{k>=1} Kronecker(D,k)/k^2 = Pi^2*b(D)/D^(5/2) = (4*Pi^2)/D^(3/2) * (b(D)/(4*D)), which is an integer times (4*Pi^2)/D^(3/2) if D > 8 by the conjecture (b). Here Sum_{k>=1} Kronecker(D,k)/k^2 is the value of the Dirichlet L-series of a non-principal character modulo D at s=2.

			For n = 3, D = 5, b(5) = 1^2 - 2^2 - 3^2 + 4^2 = 4. Here 5 = 5*1^2, we have b(5)/(4*1^2) = 1 is an integer. Also, Sum_{k>=1} Kronecker(5,k)/k^2 = Pi^2/(25*sqrt(5)) = Pi^2*b(5)/5^(5/2).
For n = 4, D = 8, b(8) = 1^2 - 3^2 - 5^2 + 7^2 = 16. We have Sum_{k>=1} Kronecker(8,k)/k^2 = Pi^2/(8*sqrt(2)) = Pi^2*b(8)/8^(5/2).
For n = 6, D = 12, b(12) = 1^2 - 5^2 - 7^2 + 11^2 = 48. Here 12 is not of the form k^2 or 5*k^2 or 2^t (t odd, t > 1), we have b(12)/(4*12) = 1 is an integer. Also, Sum_{k>=1} Kronecker(12,k)/k^2 = Pi^2/(6*sqrt(3)) = Pi^2*b(12)/12^(5/2).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A042948, A003658, A053818, A316860.

b[n_] = Sum[KroneckerSymbol[n, i]*i^2, {i, 1, n}];
a[n_] = b[2 n - Mod[n, 2]]

b(n) = sum(i=1, n, kronecker(n,i)*i^2)
a(n) = b(2*n - (n%2))

If D is a square, then b(D) = A053818(D), so b(D) is divisible by D if and only if sqrt(D) is not in A316860.

If D is not a square, D is divisible by p^e, where e >= 4 if p = 2, e >= 3 if p > 2, then it is easy to see that b(D) = p^2*b(D/p^2). So we only need to consider the value of b(D) where D is a cubefree number or 8 times a cubefree odd number. Specially, for odd t, we have b(2^t) = b(8)*(2^(t-3)) = 2^(t+1) for t > 1; b(5^t) = b(5)*(5^(t-1)) = 4*5^(t-1); b(13^t) = b(13)*(13^(t-1)) = 4*13^t and so on.

A104571 Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 5, 4, 1, 8, 5, 4, 1, 9, 8, 5, 4, 1, 12, 9, 8, 5, 4, 1, 13, 12, 9, 8, 5, 4, 1, 16, 13, 12, 9, 8, 5, 4, 1

Offset: 0

Gary W. Adamson, Mar 16 2005

			The first few rows are:
1;
4, 1;
5, 4, 1;
8, 5, 4, 1;
9, 8, 5, 4, 1;
...

Cf. A042948, A035608 (row sums), A104570, A104569, A074377.

The triangle is extracted from the product of lower triangular matrices (with the rest of the terms all zeros): G * R (or R * G); G = [1; 3, 1; 1, 3, 1; 3, 1, 3, 1;...]; R = [1; 1, 1; 1, 1, 1;...].

A367269 Triangle T(n, k) read by rows and based on A042948 yields a permutation of the natural numbers.

Original entry on oeis.org

1, 4, 3, 6, 5, 2, 13, 12, 9, 8, 15, 14, 11, 10, 7, 26, 25, 22, 21, 18, 17, 28, 27, 24, 23, 20, 19, 16, 43, 42, 39, 38, 35, 34, 31, 30, 45, 44, 41, 40, 37, 36, 33, 32, 29, 64, 63, 60, 59, 56, 55, 52, 51, 48, 47, 66, 65, 62, 61, 58, 57, 54, 53, 50, 49, 46, 89, 88, 85, 84, 81, 80, 77, 76, 73, 72, 69, 68

Offset: 0

Werner Schulte, Dec 06 2023

Compare this triangle to A364390.

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14
=================================================================
 0  :   1
 1  :   4   3
 2  :   6   5   2
 3  :  13  12   9   8
 4  :  15  14  11  10   7
 5  :  26  25  22  21  18  17
 6  :  28  27  24  23  20  19  16
 7  :  43  42  39  38  35  34  31  30
 8  :  45  44  41  40  37  36  33  32  29
 9  :  64  63  60  59  56  55  52  51  48  47
10  :  66  65  62  61  58  57  54  53  50  49  46
11  :  89  88  85  84  81  80  77  76  73  72  69  68
12  :  91  90  87  86  83  82  79  78  75  74  71  70  67
13  : 118 117 114 113 110 109 106 105 102 101  98  97  94  93
14  : 120 119 116 115 112 111 108 107 104 103 100  99  96  95  92
etc.

Cf. A006003, A042948, A364390.

T[n_, k_]:= (n+1) * (n+2) / 2 + n * Mod[n,2] - 2 * k + Mod[k,2]; Table[T[n,k],{n,0,11},{k,0,n}]//Flatten (* Stefano Spezia, Dec 06 2023 *)

T(n, k) = (n+1)*(n+2)/2+n*(n%2)-2*k+(k%2)

T(n, k) = (n+1) * (n+2) / 2 + n * (n mod 2) - 2 * k + (k mod 2) for 0 <= k <= n.
T(n, k) = T(n, 0) + A042948(k) for 0 <= k <= n.
T(n, 0) = (n+1) * (n+2) / 2 + n * (n mod 2) for n >= 0.
T(n, n) = (n^2 - n + 2) / 2 + (n+1) * (n mod 2) for n >= 0.
T(2*n, n) = 2 * n^2 + n + 1 + (n mod 2) for n >= 0.
T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-1) for 0 < k < n.
Row sums: A006003(n+1) - 2 * (-1)^n * (floor((n+1)/2))^2 for n >= 0.
G.f. of column k = 0: F(t, 0) = Sum_{n>=0} T(n, 0) * t^n = (1 + 3*t + t^3 - t^4) / ((1-t)^3 * (1+t)^2).
G.f.: F(t, x) = Sum_{n>=0, k=0..n} T(n, k) * x^k * t^n = (F(t, 0) - x * F(x*t, 0)) / (1-x) - 2*x*t / ((1-t) * (1-x*t)^2) + x*t / ((1-t) * (1-x^2*t^2)).
Alt. row sums: (n^(2 - n mod 2) + 2 - n mod 2) / 2 for n >= 0.

A002620 Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812

Offset: 0

N. J. A. Sloane

b(n) = a(n+2) is the number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also number of 2-covers of an n-set; also number of 2 X n binary matrices with no zero columns up to row and column permutation. - Vladeta Jovovic, Jun 08 2000
a(n) is also the maximal number of edges that a triangle-free graph of n vertices can have. For n = 2m, the maximum is achieved by the bipartite graph K(m, m); for n = 2m + 1, the maximum is achieved by the bipartite graph K(m, m + 1). - Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001
a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1). - Santi Spadaro, Jul 13 2001
This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n-1} (the symmetric group S_n). - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Let M_n denote the n X n matrix m(i,j) = 2 if i = j; m(i, j) = 1 if (i+j) is even; m(i, j) = 0 if i + j is odd, then a(n+2) = det M_n. - Benoit Cloitre, Jun 19 2002
Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy, Aug 19 2002
Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd, Sep 17 2002
For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd, Sep 17 2002
Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy, Oct 17 2002
Maximum product of two integers whose sum is n. - Matthew Vandermast, Mar 04 2003
a(n+2) gives number of non-symmetric partitions of n into at most 3 parts, with zeros used as padding. E.g., a(7) = 12 because we can write 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. - Jon Perry, Jul 08 2003
a(n-1) gives number of distinct elements greater than 1 of non-symmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g., 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. Of these, 050, 140, 320, 230, 221, 131 qualify and a(4) = 6. - Jon Perry, Jul 08 2003
Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj Beedassy, Oct 02 2003
Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8). - Richard Bean, Jun 12 2003 and Nov 18 2003
a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be assigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n = 3, two strokes can exist, "x -> y -> z" and " x -> z", so a(3) = 2. n = 4, four maximal strokes exist, "u -> x -> z" and "u -> y" and "u -> z" and "x -> y -> z", so a(4) = 4. - Yasutoshi Kohmoto, Dec 20 2003
Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g., a(4) = 4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD and UUU*DU*DU*DDD, where U = (1, 1), D = (1, -1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0. - Rick L. Shepherd, Feb 27 2004
See A092186 for another application.
Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
a(n+1) is the transform of n under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens. - Alonso del Arte, Jun 17 2005
a(n+1) is the number of noncongruent integer-sided triangles with largest side n. - David W. Wilson [Comment corrected Sep 26 2006]
A quarter-square table can be used to multiply integers since n*m = a(n+m) - a(n-m) for all integer n, m. - Michael Somos, Oct 29 2006
The sequence is the size of the smallest strong critical set in a Latin square of order n. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
Maximal number of squares (maximal area) in a polyomino with perimeter 2n. - Tanya Khovanova, Jul 04 2007
For n >= 3 a(n-1) is the number of bracelets with n+3 beads, 2 of which are red, 1 of which is blue. - Washington Bomfim, Jul 26 2008
Equals row sums of triangle A122196. - Gary W. Adamson, Nov 29 2008
Also a(n) is the number of different patterns of a 2-colored 3-partition of n. - Ctibor O. Zizka, Nov 19 2014
Also a(n-1) = C(((n+(n mod 2))/2), 2) + C(((n-(n mod 2))/2), 2), so this is the second diagonal of A061857 and A061866, and each even-indexed term is the average of its two neighbors. - Antti Karttunen
Equals triangle A171608 * ( 1, 2, 3, ...). - Gary W. Adamson, Dec 12 2009
a(n) gives the number of nonisomorphic faithful representations of the Symmetric group S_3 of dimension n. Any faithful representation of S_3 must contain at least one copy of the 2-dimensional irrep, along with any combination of the two 1-dimensional irreps. - Andrew Rupinski, Jan 20 2011
a(n+2) gives the number of ways to make change for "c" cents, letting n = floor(c/5) to account for the 5-repetitive nature of the task, using only pennies, nickels and dimes (see A187243). - Adam Sasson, Mar 07 2011
a(n) belongs to the sequence if and only if a(n) = floor(sqrt(a(n))) * ceiling(sqrt(a(n))), that is, a(n) = k^2 or a(n) = k*(k+1), k >= 0. - Daniel Forgues, Apr 17 2011
a(n) is the sum of the positive integers < n that have the opposite parity as n.
Deleting the first 0 from the sequence results in a sequence b = 0, 1, 2, 4, ... such that b(n) is sum of the positive integers <= n that have the same parity as n. The sequence b(n) is the additive counterpart of the double factorial. - Peter Luschny, Jul 06 2011
Third outer diagonal of Losanitsch's Triangle, A034851. - Fred Daniel Kline, Sep 10 2011
Written as a(1) = 1, a(n) = a(n-1) + ceiling (a(n-1)) this is to ceiling as A002984 is to floor, and as A033638 is to round. - Jonathan Vos Post, Oct 08 2011
a(n-2) gives the number of distinct graphs with n vertices and n regions. - Erik Hasse, Oct 18 2011
Construct the n-th row of Pascal's triangle (A007318) from the preceding row, starting with row 0 = 1. a(n) counts the total number of additions required to compute the triangle in this way up to row n, with the restrictions that copying a term does not count as an addition, and that all additions not required by the symmetry of Pascal's triangle are replaced by copying terms. - Douglas Latimer, Mar 05 2012
a(n) is the sum of the positive differences of the parts in the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 27 2013
a(n) is the maximum number of covering relations possible in an n-element graded poset. For n = 2m, this bound is achieved for the poset with two sets of m elements, with each point in the "upper" set covering each point in the "lower" set. For n = 2m+1, this bound is achieved by the poset with m nodes in an upper set covering each of m+1 nodes in a lower set. - Ben Branman, Mar 26 2013
a(n+2) is the number of (integer) partitions of n into 2 sorts of 1's and 1 sort of 2's. - Joerg Arndt, May 17 2013
Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1). - Ivan N. Ianakiev, May 23 2013. [This conjecture was mentioned in A220492, A222030. - Omar E. Pol, Oct 25 2013]
For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2 - m; a(2m*p + 1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p - 3. - Richard R. Forberg, Jun 09 2013
Apart from the initial term this is the elliptic troublemaker sequence R_n(1,2) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013
a(n) is also the total number of twin hearts patterns (6c4c) packing into (n+1) X (n+1) coins, the coins left is A042948 and the voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 24 2013
Partitions of 2n into parts of size 1, 2 or 4 where the largest part is 4, i.e., A073463(n,2). - Henry Bottomley, Oct 28 2013
a(n+1) is the minimum length of a sequence (of not necessarily distinct terms) that guarantees the existence of a (not necessarily consecutive) subsequence of length n in which like terms appear consecutively. This is also the minimum cardinality of an ordered set S that ensures that, given any partition of S, there will be a subset T of S so that the induced subpartition on T avoids the pattern ac/b, where a < b < c. - Eric Gottlieb, Mar 05 2014
Also the number of elements of the list 1..n+1 such that for any two elements {x,y} the integer (x+y)/2 lies in the range ]x,y[. - Robert G. Wilson v, May 22 2014
Number of lattice points (x,y) inside the region of the coordinate plane bounded by x <= n, 0 < y <= x/2. For a(11)=30 there are exactly 30 lattice points in the region below:
6|                                 .
.|                              .  |
5|                           .+__+
.|                        .  |  |  |
4|                     .+__++__+
.|                  .  |  |  |  |  |
3|               .+__++__++__+
.|            .  |  |  |  |  |  |  |
2|         .+__++__++__++__+
.|      .  |  |  |  |  |  |  |  |  |
1|   .+__++__++__++__++__+
.|.  |  |  |  |  |  |  |  |  |  |  |
0|.+__++__++__++__++__++_________
  0  1  2  3  4  5  6  7  8  9 10 11 .. n
  0  0  1  2  4  6  9 12 16 20 25 30 .. a(n) - Wesley Ivan Hurt, Oct 26 2014
a(n+1) is the greatest integer k for which there exists an n x n matrix M of nonnegative integers with every row and column summing to k, such that there do not exist n entries of M, all greater than 1, and no two of these entries in the same row or column. - Richard Stanley, Nov 19 2014
In a tiling of the triangular shape T_N with row length k for row k = 1, 2, ..., N >= 1 (or, alternatively row length N = 1-k for row k) with rectangular tiles, there can appear rectangles (i, j), N >= i >= j >= 1, of a(N+1) types (and their transposed shapes obtained by interchanging i and j). See the Feb 27 2004 comment above from Rick L. Shepherd. The motivation to look into this came from a proposal of Kival Ngaokrajang in A247139. - Wolfdieter Lang, Dec 09 2014
Every positive integer is a sum of at most four distinct quarter-squares; see A257018. - Clark Kimberling, Apr 15 2015
a(n+1) gives the maximal number of distinct elements of an n X n matrix which is symmetric (w.r.t. the main diagonal) and symmetric w.r.t. the main antidiagonal. Such matrices are called bisymmetric. See the Wikipedia link. - Wolfdieter Lang, Jul 07 2015
For 2^a(n+1), n >= 1, the number of binary bisymmetric n X n matrices, see A060656(n+1) and the comment and link by Dennis P. Walsh. - Wolfdieter Lang, Aug 16 2015
a(n) is the number of partitions of 2n+1 of length three with exactly two even entries (see below example). - John M. Campbell, Jan 29 2016
a(n) is the sum of the asymmetry degrees of all 01-avoiding binary words of length n. The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. a(6) = 9 because the 01-avoiding binary words of length 6 are 000000, 100000, 110000, 111000, 111100, 111110, and 111111, and the sum of their asymmetry degrees is 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9. Equivalently, a(n) = Sum_{k>=0} k*A275437(n,k). - Emeric Deutsch, Aug 15 2016
a(n) is the number of ways to represent all the integers in the interval [3,n+1] as the sum of two distinct natural numbers. E.g., a(7)=12 as there are 12 different ways to represent all the numbers in the interval [3,8] as the sum of two distinct parts: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 3+4=7, 3+5=8. - Anton Zakharov, Aug 24 2016
a(n+2) is the number of conjugacy classes of involutions (considering the identity as an involution) in the hyperoctahedral group C_2 wreath S_n. - Mark Wildon, Apr 22 2017
a(n+2) is the maximum number of pieces of a pizza that can be made with n cuts that are parallel or perpendicular to each other. - Anton Zakharov, May 11 2017
Also the matching number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
The answer to a question posed by W. Mantel: a(n) is the maximum number of edges in an n-vertex triangle-free graph. Also solved by H. Gouwentak, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. - Charles R Greathouse IV, Feb 01 2018
Number of nonisomorphic outer planar graphs of order n >= 3, size n+2, and maximum degree 4. - Christian Barrientos and Sarah Minion, Feb 27 2018
Maximum area of a rectangle with perimeter 2n and sides of integer length. - André Engels, Jul 29 2018
Also the crossing number of the complete bipartite graph K_{3,n+1}. - Eric W. Weisstein, Sep 11 2018
a(n+2) is the number of distinct genotype frequency vectors possible for a sample of n diploid individuals at a biallelic genetic locus with a specified major allele. Such vectors are the lists of nonnegative genotype frequencies (n_AA, n_AB, n_BB) with n_AA + n_AB + n_BB = n and n_AA >= n_BB. - Noah A Rosenberg, Feb 05 2019
a(n+2) is the number of distinct real spectra (eigenvalues repeated according to their multiplicity) for an orthogonal n X n matrix. The case of an empty spectrum list is logically counted as one of those possibilities, when it exists. Thus a(n+2) is the number of distinct reduced forms (on the real field, in orthonormal basis) for elements in O(n). - Christian Devanz, Feb 13 2019
a(n) is the number of non-isomorphic asymmetric graphs that can be created by adding a single edge to a path on n+4 vertices. - Emma Farnsworth, Natalie Gomez, Herlandt Lino, and Darren Narayan, Jul 03 2019
a(n+1) is the number of integer triangles with largest side n. - James East, Oct 30 2019
a(n) is the number of nonempty subsets of {1,2,...,n} that contain exactly one odd and one even number.  For example, for n=7, a(7)=12 and the 12 subsets are {1,2}, {1,4}, {1,6}, {2,3}, {2,5}, {2,7}, {3,4}, {3,6}, {4,5}, {4,7}, {5,6}, {6,7}. - Enrique Navarrete, Dec 16 2019
a(n+1) is also the n-th term of the Saind sequence (w_n){n>=1}, i.e., the infinite sequence caused by the entries of the queue of the degree sequences associated with the Saind arrays, as n increases. - _Giulia Palma, Jun 24 2020
Aside from the first two terms, a(n) enumerates the number of distinct normal ordered terms in the expansion of the differential operator (x + d/dx)^m associated to the Hermite polynomials and the Heisenberg-Weyl algebra. It also enumerates the number of distinct monomials in the bivariate polynomials corresponding to the partial sums of the series for cos(x+y) and sin(x+y). Cf. A344678. - Tom Copeland, May 27 2021
a(n) is the maximal number of negative products a_i * a_j (1 <= i <= j <= n), where all a_i are real numbers. - Logan Pipes, Jul 08 2021
From Allan Bickle, Dec 20 2021: (Start)
a(n) is the maximum product of the chromatic numbers of a graph of order n-1 and its complement.  The extremal graphs are characterized in the papers of Finck (1968) and Bickle (2023).
a(n) is the maximum product of the degeneracies of a graph of order n+1 and its complement.  The extremal graphs are characterized in the paper of Bickle (2012). (End)
a(n) is the maximum number m such that m white rooks and m black rooks can coexist on an n-1 X n-1 chessboard without attacking each other. - Aaron Khan, Jul 13 2022
Partial sums of A004526. - Bernard Schott, Jan 06 2023
a(n) is the number of 231-avoiding odd Grassmannian permutations of size n. - Juan B. Gil, Mar 10 2023
a(n) is the number of integer tuples (x,y) satisfying n + x + y >= 0, 25*n + x - 11*y >=0, 25*n - 11*x + y >=0, n + x + y == 0 (mod 12) , 25*n + x - 11*y == 0 (mod 5), 25*n - 11*x + y == 0 (mod 5) . For n=2, the sole solution is (x,y) = (0,0) and so a(2) = 1. For n = 3, the a(3) = 2 solutions are (-3, 2) and (2, -3). - Jeffery Opoku, Feb 16 2024
Let us consider triangles whose vertices are the centers of three squares constructed on the sides of a right triangle. a(n) is the integer part of the area of these triangles, taken without repetitions and in ascending order. See the illustration in the links. - Nicolay Avilov, Aug 05 2024
For n>=2, a(n) is the indendence number of the 2-token graph F_2(P_n) of the path graph P_n on n vertices. (Alternatively, as noted by Peter Munn, F_2(P_n) is the nXn square lattice, or grid, graph diminished by a cut across the diagonal.) - Miquel A. Fiol, Oct 05 2024
For n >= 1, also the lower matching number of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Dec 14 2024
a(n-1) is also the minimal number of edges that a graph of n vertices must have such that any 3 vertices share at least one edge. - Ruediger Jehn, May 20 2025
a(n) is the number of edges of the antiregular graph A_n.  This is the unique connected graph with n vertices and degrees 1 to n-1 (floor(n/2) repeated). - Allan Bickle, Jun 15 2025

			a(3) = 2, floor(3/2)*ceiling(3/2) = 2.
[ n] a(n)
---------
[ 2] 1
[ 3] 2
[ 4] 1 + 3
[ 5] 2 + 4
[ 6] 1 + 3 + 5
[ 7] 2 + 4 + 6
[ 8] 1 + 3 + 5 + 7
[ 9] 2 + 4 + 6 + 8
From _Wolfdieter Lang_, Dec 09 2014: (Start)
Tiling of a triangular shape T_N, N >= 1 with rectangles:
N=5, n=6: a(6) = 9 because all the rectangles (i, j) (modulo transposition, i.e., interchange of i and j) which are of use are:
  (5, 1)                ;  (1, 1)
  (4, 2), (4, 1)        ;  (2, 2), (2, 1)
                        ;  (3, 3), (3, 2), (3, 1)
That is (1+1) + (2+2) + 3 = 9 = a(6). Partial sums of 1, 1, 2, 2, 3, ... (A004526). (End)
Bisymmetric matrices B: 2 X 2, a(3) = 2 from B[1,1] and B[1,2]. 3 X 3, a(4) = 4 from B[1,1], B[1,2], B[1,3], and B[2,2]. - _Wolfdieter Lang_, Jul 07 2015
From _John M. Campbell_, Jan 29 2016: (Start)
Letting n=5, there are a(n)=a(5)=6 partitions of 2n+1=11 of length three with exactly two even entries:
(8,2,1) |- 2n+1
(7,2,2) |- 2n+1
(6,4,1) |- 2n+1
(6,3,2) |- 2n+1
(5,4,2) |- 2n+1
(4,4,3) |- 2n+1
(End)
From _Aaron Khan_, Jul 13 2022: (Start)
Examples of the sequence when used for rooks on a chessboard:
.
A solution illustrating a(5)=4:
  +---------+
  | B B . . |
  | B B . . |
  | . . W W |
  | . . W W |
  +---------+
.
A solution illustrating a(6)=6:
  +-----------+
  | B B . . . |
  | B B . . . |
  | B B . . . |
  | . . W W W |
  | . . W W W |
  +-----------+
(End)

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A087811 is another version of this sequence.
Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013, A128174, A000601, A115514, A189151, A063657, A171608, A005044, A030179, A275437, A004526.
Differences of A002623. Complement of A049068.
a(n) = A014616(n-2) + 2 = A033638(n) - 1 = A078126(n) + 1. Cf. A055802, A055803.
Antidiagonal sums of array A003983.
Cf. A033436 - A033444. - Reinhard Zumkeller, Nov 30 2009
Cf. A008217, A014980, A197081, A197122.
Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A077043, A060656 (2^a(n)), A344678.
Cf. A000290, A056834, A130519, A056838, A056865.
Cf. A250000 (queens on a chessboard), A176222 (kings on a chessboard), A355509 (knights on a chessboard).
Maximal product of k positive integers with sum n, for k = 2..10: this sequence (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

# using the formula by Paul Barry
A002620 := List([1..10^4], n-> (2*n^2 - 1 + (-1)^n)/8); # Muniru A Asiru, Feb 01 2018

a002620 = (`div` 4) . (^ 2) -- Reinhard Zumkeller, Feb 24 2012

[ Floor(n/2)*Ceiling(n/2) : n in [0..40]];

A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)),x,60);
with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=1)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=0..57) ; # Zerinvary Lajos, Mar 09 2007

Table[Ceiling[n/2] Floor[n/2], {n, 0, 56}] (* Robert G. Wilson v, Jun 18 2005 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 1, 2}, 60] (* Harvey P. Dale, Oct 05 2012 *)
Table[Floor[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
Floor[Range[0, 20]^2/4] (* Eric W. Weisstein, Sep 11 2018 *)
CoefficientList[Series[-(x^2/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)
Table[Floor[n^2/2]/2, {n, 0, 56}] (* Clark Kimberling, Dec 05 2021 *)

makelist(floor(n^2/4),n,0,50); /* Martin Ettl, Oct 17 2012 */

PARI
```
a(n)=n^2\4
```

(t(n)=n*(n+1)/2);for(i=1,50,print1(",",(-1)^i*sum(k=1,i,(-1)^k*t(k))))

a(n)=n^2>>2 \\ Charles R Greathouse IV, Nov 11 2009

x='x+O('x^100); concat([0, 0], Vec(x^2/((1-x)^2*(1-x^2)))) \\ Altug Alkan, Oct 15 2015

def A002620(n): return (n**2)>>2 # Chai Wah Wu, Jul 07 2022

def A002620():
     x, y = 0, 1
     yield x
     while true:
         yield x
         x, y = x + y, x//y + 1
a = A002620(); print([next(a) for i in range(58)]) # Peter Luschny, Dec 17 2015

a(n) = (2*n^2-1+(-1)^n)/8. - Paul Barry, May 27 2003
G.f.: x^2/((1-x)^2*(1-x^2)) = x^2 / ( (1+x)*(1-x)^3 ). - Simon Plouffe in his 1992 dissertation, leading zeros dropped
E.g.f.: exp(x)*(2*x^2+2*x-1)/8 + exp(-x)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Jaume Oliver Lafont, Dec 05 2008
a(-n) = a(n) for all n in Z.
a(n) = a(n-1) + floor(n/2), n > 0. Partial sums of A004526. - Adam Kertesz, Sep 20 2000
a(n) = a(n-1) + a(n-2) - a(n-3) + 1 [with a(-1) = a(0) = a(1) = 0], a(2k) = k^2, a(2k-1) = k(k-1). - Henry Bottomley, Mar 08 2000
0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
a(n) = Sum_{k=1..n} floor(k/2). - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
a(n) = n*floor((n-1)/2) - floor((n-1)/2)*(floor((n-1)/2)+ 1); a(n) = a(n-2) + n-2 with a(1) = 0, a(2) = 0. - Santi Spadaro, Jul 13 2001
Also: a(n) = binomial(n, 2) - a(n-1) = A000217(n-1) - a(n-1) with a(0) = 0. - Labos Elemer, Apr 26 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k, 2). - Paul Barry, Jul 01 2003
a(n) = (-1)^n * partial sum of alternating triangular numbers. - Jon Perry, Dec 30 2003
a(n) = A024206(n+1) - n. - Philippe Deléham, Feb 27 2004
a(n) = a(n-2) + n - 1, n > 1. - Paul Barry, Jul 14 2004
a(n+1) = Sum_{i=0..n} min(i, n-i). - Marc LeBrun, Feb 15 2005
a(n+1) = Sum_{k = 0..floor((n-1)/2)} n-2k; a(n+1) = Sum_{k=0..n} k*(1-(-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = A108561(n+1,n-2) for n > 2. - Reinhard Zumkeller, Jun 10 2005
1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + ...))))))) = 6/(Pi^2 - 6) = 1.550546096730... - Philippe Deléham, Jun 20 2005
a(n) = Sum_{k=0..n} Min_{k, n-k}, sums of rows of the triangle in A004197. - Reinhard Zumkeller, Jul 27 2005
For n > 2 a(n) = a(n-1) + ceiling(sqrt(a(n-1))). - Jonathan Vos Post, Jan 19 2006
Sequence starting (2, 2, 4, 6, 9, ...) = A128174 (as an infinite lower triangular matrix) * vector [1, 2, 3, ...]; where A128174 = (1; 0,1; 1,0,1; 0,1,0,1; ...). - Gary W. Adamson, Jul 27 2007
a(n) = Sum_{i=k..n} P(i, k) where P(i, k) is the number of partitions of i into k parts. - Thomas Wieder, Sep 01 2007
a(n) = sum of row (n-2) of triangle A115514. - Gary W. Adamson, Oct 25 2007
For n > 1: gcd(a(n+1), a(n)) = a(n+1) - a(n). - Reinhard Zumkeller, Apr 06 2008
a(n+3) = a(n) + A000027(n) + A008619(n+1) = a(n) + A001651(n+1) with a(1) = 0, a(2) = 0, a(3) = 1. - Yosu Yurramendi, Aug 10 2008
a(2n) = A000290(n). a(2n+1) = A002378(n). - Gary W. Adamson, Nov 29 2008
a(n+1) = a(n) + A110654(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = Sum_{k=0..n} (k mod 2)*(n-k); Cf. A000035, A001477. - Reinhard Zumkeller, Nov 05 2009
a(n-1) = (n*n - 2*n + n mod 2)/4. - Ctibor O. Zizka, Nov 23 2009
a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceiling((n^2-1)/4). - Mircea Merca, Nov 29 2010
n*a(n+2) = 2*a(n+1) + (n+2)*a(n). Holonomic Ansatz with smallest order of recurrence. - Thotsaporn Thanatipanonda, Dec 12 2010
a(n+1) = (n*(2+n) + n mod 2)/4. - Fred Daniel Kline, Sep 11 2011
a(n) = A199332(n, floor((n+1)/2)). - Reinhard Zumkeller, Nov 23 2011
a(n) = floor(b(n)) with b(n) = b(n-1) + n/(1+e^(1/n)) and b(0)= 0. - Richard R. Forberg, Jun 08 2013
a(n) = Sum_{i=1..floor((n+1)/2)} (n+1)-2i. - Wesley Ivan Hurt, Jun 09 2013
a(n) = floor((n+2)/2 - 1)*(floor((n+2)/2)-1 + (n+2) mod 2). - Wesley Ivan Hurt, Jun 09 2013
Sum_{n>=2} 1/a(n) = 1 + zeta(2) = 1+A013661. - Enrique Pérez Herrero, Jun 30 2013
Empirical: a(n-1) = floor(n/(e^(4/n)-1)). - Richard R. Forberg, Jul 24 2013
a(n) = A007590(n)/2. - Wesley Ivan Hurt, Mar 08 2014
A237347(a(n)) = 3; A235711(n) = A003415(a(n)). - Reinhard Zumkeller, Mar 18 2014
A240025(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
0 = a(n)*a(n+2) + a(n+1)*(-2*a(n+2) + a(n+3)) for all integers n. - Michael Somos, Nov 22 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n-1)/2). - Wesley Ivan Hurt, Mar 12 2015
a(4n+1) = A002943(n) for all n>=0. - M. F. Hasler, Oct 11 2015
a(n+2)-a(n-2) = A004275(n+1). - Anton Zakharov, May 11 2017
a(n) = floor(n/2)*floor((n+1)/2). - Bruno Berselli, Jun 08 2017
a(n) = a(n-3) + floor(3*n/2) - 2. - Yuchun Ji, Aug 14 2020
a(n)+a(n+1) = A000217(n). - R. J. Mathar, Mar 13 2021
a(n) = A004247(n,floor(n/2)). - Logan Pipes, Jul 08 2021
a(n) = floor(n^2/2)/2. - Clark Kimberling, Dec 05 2021
Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 1. - Amiram Eldar, Mar 10 2022

A007696 Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

Original entry on oeis.org

1, 1, 5, 45, 585, 9945, 208845, 5221125, 151412625, 4996616625, 184874815125, 7579867420125, 341094033905625, 16713607661375625, 885821206052908125, 50491808745015763125, 3080000333445961550625, 200200021673987500790625, 13813801495505137554553125

Offset: 0

N. J. A. Sloane

a(n), n >= 1, enumerates increasing quintic (5-ary) trees. See David Callan's comment on A007559 (number of increasing quarterny trees).

Hankel transform is A169619. - Paul Barry, Dec 03 2009

			G.f. = 1 + x + 5*x^2 + 45*x^3 + 585*x^4 + 9945*x^5 + 208845*x^6 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq. 3 (2000), Article 00.2.4.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Integer Seq. 13 (2010), Article 10.6.7; see page 39.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integer Seq. 9 (2006), Article 06.1.1.

Cf. A001147, A001813, A004981, A007559, A008545, A034255, A047053, A051142, A264781.

a(n) = A049029(n, 1) for n >= 1 (first column of triangle).

a:=[1,1];; for n in [3..20] do a[n]:=(4*(n-1)-7)*(a[n-1]+4*a[n-2]); od; a; # G. C. Greubel, Aug 15 2019

[n le 2 select 1 else (4*(n-1)-7)*(Self(n-1) + 4*Self(n-2)): n in [1..20]]; // G. C. Greubel, Aug 15 2019

x:='x'; G(x):=(1-4*x)^(-1/4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: seq(eval(f[n],x=0),n=0..17);# Zerinvary Lajos, Apr 03 2009
A007696 := n -> mul(k, k = select(k-> k mod 4 = 1, [$ 1 .. 4*n])): seq(A007696(n), n=0..17); # Peter Luschny, Jun 23 2011

a[ n_]:= Pochhammer[ 1/4, n] 4^n; (* Michael Somos, Jan 17 2014 *)
a[ n_]:= If[n < 0, 1 / Product[ -k, {k, 3, -4n-1, 4}], Product[ k, {k, 1, 4n-3, 4}]]; (* Michael Somos, Jan 17 2014 *)
Range[0, 19]! CoefficientList[Series[((1-4x)^(-1/4)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)

A007696(n):=prod(4*k+1,k,0,n-1)$
makelist(A007696(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

{a(n) = if( n<0, 1 / prod(k=1, -n, 1 - 4*k), prod(k=1, n, 4*k - 3))}; /* Michael Somos, Jan 17 2014 */

[4^n*rising_factorial(1/4, n) for n in (0..20)] # G. C. Greubel, Aug 15 2019

E.g.f.: (1 - 4*x)^(-1/4).
a(n) ~ 2^(1/2) * Pi^(1/2) * Gamma(1/4)^(-1) * n^(-1/4) * 2^(2*n) * e^(-n) * n^n * (1 - 1/96 * n^(-1) - ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001 [corrected by Vaclav Kotesovec, Jul 19 2025]
a(n) = Sum_{k = 0..n} (-4)^(n-k) * A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1/(1 - x/(1 - 4*x/(1 - 5*x/(1 - 8*x/(1 - 9*x/(1 - 12*x/(1 - 13*x/(1 - .../(1 - A042948(n+1)*x/(1 -... (continued fraction). - Paul Barry, Dec 03 2009
a(n) = (-3)^n * Sum_{k = 0..n} (4/3)^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
G.f.: 1/T(0), where T(k) =  1 - x * (4*k + 1)/(1 - x * (4*k + 4)/T(k+1)) (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1 + x/Q(0), where Q(k) = 1 + x + 2*(2*k - 1)*x - 4*x*(k+1)/Q(k+1) (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x * (4*k + 1)/(x * (4*k + 1) + 1/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013
0 = a(n) * (4*a(n+1) - a(n+2)) + a(n+1) * a(n+1) for all n in Z. - Michael Somos, Jan 17 2014
a(-n) = (-1)^n / A008545(n). - Michael Somos, Jan 17 2014
Let T(x) = 1/(1 - 3*x)^(1/3) be the e.g.f. for the sequence of triple factorial numbers A007559. Then the e.g.f. A(x) for the quartic factorial numbers satisfies T(Integral_{t = 0..x} A(t) dt) = A(x). (Cf. A007559 and A008548.) - Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/4], [], 4*x). - Peter Luschny, Oct 08 2015
a(n) = A264781(4*n+1, n). - Alois P. Heinz, Nov 24 2015
a(n) = 4^n * Gamma(n + 1/4)/Gamma(1/4). - Artur Jasinski, Aug 23 2016
D-finite with recurrence: a(n) +(-4*n+3)*a(n-1)=0, n>=1. - R. J. Mathar, Feb 14 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/4)*(Gamma(1/4) - Gamma(1/4, 1/4))/(2*sqrt(2)). - Amiram Eldar, Dec 18 2022

Better description from Wolfdieter Lang, Dec 11 1999

A121805 The "comma sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

Original entry on oeis.org

1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, 530, 535, 590, 595, 651, 667, 744, 791, 809, 908, 997, 1068, 1149, 1240, 1241, 1252, 1273, 1304, 1345, 1396, 1457, 1528, 1609, 1700, 1701, 1712, 1733, 1764, 1805, 1856, 1917, 1988, 2070

Offset: 1

Eric Angelini, Dec 11 2006

An equivalent, but more formal definition, is: a(1) = 1; for n > 1, let x be the least significant digit of a(n-1); then a(n) = a(n-1) + x*10 + y where y is the most significant digit of a(n) and is the smallest such y, if such a y exists. If no such y exists, stop.
The sequence contains exactly 2137453 terms, with a(2137453)=99999945. The next term does not exist. - W. Edwin Clark, Dec 11 2006
It is remarkable that the sequence persists for so long. - N. J. A. Sloane, Dec 15 2006
The similar sequence A139284, which starts at a(1)=2, persists even longer, ending at a(194697747222394) = 9999999999999918. - Giovanni Resta, Nov 30 2019
Conjecture: This sequence is finite, for any initial term. - N. J. A. Sloane, Nov 14 2023
The base 2 analog (suggested by William Cheswick) is 1, 4, 5, 8, 9, 12, 13, ..., (see A042948) with successive differences 3, 1, 3, 1, ... (repeat). - N. J. A. Sloane, Nov 15 2023
Does not satisfy Benford's Law. - Michael S. Branicky, Nov 16 2023
Using the notion of "comma transform" of a sequence, as defined in A367360, this is the lexicographically earliest sequence of positive integers with the property that its first differences and comma transform coincide. - N. J. A. Sloane, Nov 23 2023

			Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first differences between the terms of the sequence:
Sequence:   1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, ...
Differences: 11, 23, 59, 41 , 51 , 62 , 83 , 13 , 43 , 74 , 14 , ...
To illustrate the formula in the comment: a(6) = 186 and a(7) = 248 = 186 + 62.

Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Michael S. Branicky, Table of n, a(n) for n = 1..20000 (terms 1..1001 from Zak Seidov)
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
Lorenzo Angelini, Happy birthday Éric!!, Youtube video.
Michael S. Branicky, Graph of a(n)/n over entire sequence
Simon Demers, Table of n, a(n) for n = 1..2137453 (full sequence)
Robert Dougherty-Bliss, The Comma Sequence is WILD, 2024 video.
Robert Dougherty-Bliss and Natalya Ter-Saakov, The Comma Sequence is Finite in Other Bases, arXiv:2408.03434 [math.NT], 2024.
Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Index entries for sequences related to Benford's law

See A366487 and A367349 for first differences.
Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.
Cf. A330128, A330129, A367338 (comma-successor), A367360.
See also A260261, A042948.

digits:=n->ListTools:-Reverse(convert(n,base,10)):
nextK:=proc(K) local i,L; for i from 0 to 9 do L:=K+digits(K)[ -1]*10+i; if i = digits(L)[1] then return L; fi; od; FAIL; end:
A121805:=proc(n) option remember: if n = 1 then return 1; fi; return nextK(A121805(n-1)); end: # W. Edwin Clark

a[1] = 1; a[n_] := a[n] = For[x=Mod[a[n-1], 10]; y=0, y <= 9, y++, an = a[n-1] + 10*x + y; If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 45] (* Jean-François Alcover, Nov 25 2014 *)

a=1; for(n=1,1000, print1(a", "); a+=a%10*10; for(k=1, 9, digits(a+k)[1]==k&&(a+=k)&&next(2)); error("blocked at a("n")=",a-a%10*10)) \\ M. F. Hasler, Jul 21 2015

from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 45))) # Michael S. Branicky, Apr 08 2022

A121805 <- data.frame(n=seq(from=1,to=2137453),a=integer(2137453)); A121805$a[1]=1; for (i in seq(from=2,to=2137453)){LSD=A121805$a[i-1] %% 10; k = 1; while (k != as.integer(substring(A121805$a[i-1]+LSD*10+k,1,1))){k = k+1; if(k>9) break} A121805$a[i]=A121805$a[i-1]+LSD*10+k} # Simon Demers, Oct 19 2017

More terms from Zak Seidov, Dec 11 2006
Edited by N. J. A. Sloane, Sep 17 2023
Changed name from "commas sequence" to "comma sequence". - N. J. A. Sloane, Dec 20 2023

A014601 Numbers congruent to 0 or 3 mod 4.

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124

Offset: 0

Eric Rains (rains(AT)caltech.edu)

Discriminants of orders in imaginary quadratic fields (negated). [Comment corrected by Christopher E. Thompson, Dec 11 2016]
Numbers such that Langford-Skolem problem has a solution - see A014552.
Complement of A042963. - Reinhard Zumkeller, Oct 04 2004
Also called skew amenable numbers; a number k is skew amenable if there exist a set {a(i)} of integers satisfying the relations k = Sum_{i=1..k} a(i) = -Product_{i=1..k} a(i). Thus we have 8 = 1 + 1 + 1 + 1 + 1 + 1 - 2 + 4 = -(1*1*1*1*1*1*(-2)*4). - Lekraj Beedassy, Jan 07 2005
Possible nonpositive discriminants of quadratic equation a*x^2 + b*x + c or discriminants of binary quadratic forms a*x^2 + b*x*y + c*y^2. - Artur Jasinski, Apr 28 2008
Also, disregarding the 0 term, positive integers m such that, equivalently,
  (i) +-1 +-2 +-... +-m is even for all choices of signs,
  (ii) +-1 +-2 +-... +-m = 0 for some choices of signs,
  (iii) for all -m <= k <= m, k = +-1 +-2 +-... +-(k-1) +-(k+1) +-(k+2) +-... +-m for at least one choice of signs. - Rick L. Shepherd, Oct 29 2008
A145768(a(n)) is even. - Reinhard Zumkeller, Jun 05 2012
Multiples of 4 interleaved with 1 less than multiples of 4. - Wesley Ivan Hurt, Nov 08 2013
((2*k+0) + (2*k+1) + ... + (2*k+m-1) + (2*k+m)) is even if and only if m = a(n) for some n where k is any nonnegative integer. - Gionata Neri, Jul 24 2015
Numbers whose binary reflected Gray code (A014550) ends with 0. - Amiram Eldar, May 17 2021

			G.f. = 3*x + 4*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 12*x^6 + 15*x^7 + 16*x^8 + ...

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 108.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. F. Barger, Solution to problem 10454: Amenable Numbers, Amer. Math. Monthly, Vol. 105, No. 4 (April 1998), p. 368.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Heiko Harborth, Solution of Steinhaus's problem with plus and minus signs, Journal of Combinatorial Theory, Series A, Volume 12, Issue 2 (March 1972), Pages 253-259.
Mickaël Launay, Les routes de Numland, L’énigme maths du "Monde" n°2. In French.
Michael Penn, The most beautiful arrangement of numbers, YouTube video, 2024.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004676, A014550, A042948, A079896.

Cf. A274406. - Bruno Berselli, Jun 26 2016

a014601 n = a014601_list !! n
a014601_list = [x | x <- [0..], mod x 4 `elem` [0, 3]]
-- Reinhard Zumkeller, Jun 05 2012

[n: n in [0..200]|n mod 4 in {0,3}]; // Vincenzo Librandi, Dec 24 2010

A014601:=n->3*n-2*floor(n/2); seq(A014601(k), k=0..100); # Wesley Ivan Hurt, Nov 08 2013

aa = {}; Do[Do[Do[d = b^2 - 4 a c; If[d <= 0, AppendTo[aa, -d]], {a, 0, 50}], {b, 0, 50}], {c, 0, 50}]; Union[aa] (* Artur Jasinski, Apr 28 2008 *)
Select[Range[0, 124], Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] (* Ant King, Nov 18 2010 *)
CoefficientList[Series[2 x/(1 - x)^2 + (1/(1 - x) + 1/(1 + x)) x/2, {x, 0, 100}], x] (* Vincenzo Librandi, May 18 2014 *)
a[ n_] := 2 n + Mod[n, 2]; (* Michael Somos, Jul 24 2015 *)

{a(n) = 2*n + n%2}; /* Michael Somos, Dec 27 2010 */

a(n) = (n + 1)*2 + 1 - n mod 2. - Reinhard Zumkeller, Apr 21 2003
A014494(n) = A000217(a(n)). - Reinhard Zumkeller, Oct 04 2004
a(n) = Sum_{k=1..n} (2 - (-1)^k). - William A. Tedeschi, Mar 20 2008
A139131(a(n)) = A078636(a(n)). - Reinhard Zumkeller, Apr 10 2008
From R. J. Mathar, Sep 25 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2.
G.f.: x*(3+x)/((1+x)*(x-1)^2). (End)
a(n) = 2*n + (n mod 2). - Paolo Valzasina (p.valzasina(AT)gmail.com), Nov 24 2009
a(n) = (4*n - (-1)^n + 1)/2. - Bruno Berselli, Oct 06 2010
a(n) = 4*n - a(n-1) - 1 (with a(0) = 0). - Vincenzo Librandi, Dec 24 2010
a(n) = -A042948(-n) for all n in Z. - Michael Somos, Dec 27 2010
G.f.: 2*x / (1 - x)^2 + (1 / (1 - x) + 1 / (1 + x)) * x/2. - Michael Somos, Dec 27 2010
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0) = 3 and b(k) = 2^(k+1) for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((4/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
a(n) = 3n - 2*floor(n/2). - Wesley Ivan Hurt, Nov 08 2013
a(n) = A042948(n+1) - 1 for all n in Z. - Michael Somos, Jul 24 2015
a(n) + a(n+1) = A004767(n) for all n in Z. - Michael Somos, Jul 24 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)/4 - Pi/8. - Amiram Eldar, Dec 05 2021
E.g.f.: ((4*x + 1)*exp(x) - exp(-x))/2. - David Lovler, Aug 04 2022

A133872 Period 4: repeat [1, 1, 0, 0].

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

Partial sums of A056594.
Let i=sqrt(-1) and S(n) = Sum_{k=0..n-1} exp(2*Pi*i*k^2/n) for n>=1 the famous Gauss sum. Then S(n) = (a(n)+a(n+1)*i)*sqrt(n). - Franz Vrabec, Nov 08 2007
a(A042948(n)) = 1; a(A042964(n)) = 0. - Reinhard Zumkeller, Oct 03 2008
a(n) is also the real part of partial sum of powers of the complex unit i. - Enrique Pérez Herrero, Aug 16 2009
Periodic sequences having a period of 2k and composed of k ones followed by k zeros have a closed formula of floor(((n+k) mod 2k)/k). Listed sequences of this form are: k=1..A000035(n+1), k=2..A133872(n), k=3..A088911, k=4..A131078(n), k=5..A112713(n-1). - Gary Detlefs, May 17 2011
0.repeat(0,0,1,1) is 1/5 in base 2, due to 1/5 = (3/16)/(1-1/16). For the general case see 1/A062158(n) in base n >= 2. Here n = 2. - Wolfdieter Lang, Jun 20 2014
a(n) (for n>=1) is the determinant of the n X n Toeplitz matrix M satisfying: M(i,j)=1 if -1<=j-i<=2 and 0 otherwise. - Dmitry Efimov, Jun 23 2015
a(n) (for n>=1) is the difference between numbers of even and odd permutations p of 1,2,...,n such that -1 <= p(i)-i <= 2 for i=1,2,...,n. - Dmitry Efimov, Jan 08 2016
The binomial transform is 1, 2, 3, 4, 6, 12,... (see A038504). - R. J. Mathar, Feb 25 2023

			G.f. = 1 + x + x^4 + x^5 + x^8 + x^9 + x^12 + x^13 + x^16 + x^17 + x^20 + ...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Psychedelic Geometry Blogspot, Curious Series-001 [_Enrique Pérez Herrero_, Aug 16 2009]
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A056594, A133620-A133625, A133630, A038509, A133634-A133636, A021913, A000217, A133882, A133880, A133890, A133900, A133910, A000035, A088911, A131078, A112713.

[ (1 + (-1)^Floor(n/2))/2 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

&cat[[1,1,0,0]^^25]; // Vincenzo Librandi, Jan 09 2016

A133872:=n->(1+(-1)^((2n-1+(-1)^n)/4))/2;
seq(A133872(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[(1 + (-1)^((2 n - 1 + (-1)^n)/4))/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)
PadRight[{},120,{1,1,0,0}] (* Harvey P. Dale, Jan 26 2014 *)

a(n)=n%4<2 \\ Jaume Oliver Lafont, Mar 17 2009

Vec((1+x)/(1-x^4) + O(x^100)) \\ Altug Alkan, Jan 08 2015

def A133872(n): return int(not n&2) # Chai Wah Wu, Jan 31 2023

maxn <- 63 # by choice
a <- c(1,0,0)
for(n in 4:maxn) a[n] <- a[n-1] - a[n-2] + a[n-3]
(a <- a(1,a))
# Yosu Yurramendi, Oct 25 2020

a(n) = (1 + floor(n/2)) mod 2.
a(n) = A004526(A000035(n+2)).
a(n) = 1 + floor(n/2) - 2*floor((n+2)/4).
a(n) = (((n+2) mod 4) - (n mod 2))/2.
a(n) = ((n + 2 - (n mod 2))/2) mod 2.
a(n) = ((2*n + 3 + (-1)^n)/4) mod 2.
a(n) = (1 + (-1)^((2*n - 1 + (-1)^n)/4))/2.
a(n) = binomial(n+2, n) mod 2 = binomial(n+2, 2) mod 2.
a(n) = A000217(n+1) mod 2.
G.f.: (1+x)/(1-x^4) = 1/((1-x)(1+x^2)).
a(n) = 1/2 + (1/2)*cos(Pi*n/2) + (1/2)*sin(Pi*n/2). a(n) = A021913(n+2). - R. J. Mathar, Nov 15 2007
From Jaume Oliver Lafont, Dec 05 2008: (Start)
a(n) = 1/2 + sin((2n+1)Pi/4)/sqrt(2).
a(n) = 1/2 + cos((2n-1)Pi/4)/sqrt(2). (End)
a(n) = Re(Sum_{k=0..n} i^k), where i=sqrt(-1) and Re is the real part of a complex number. a(n) = (1/2)*((Sum_{k=0..n} i^k) + Sum_{k=0..n} i^-k) = Re((1/2)*(1 + i)*(1 - i^(n+1))). - Enrique Pérez Herrero, Aug 16 2009
a(n) = (1 + i^(n*(n-1)))/2, where i=sqrt(-1). - Bruno Berselli, May 18 2011
a(n) = (Sum_{k=1..n} k^j) mod 2, for any j. - Gary Detlefs, Dec 28 2011
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Jean-Christophe Hervé, May 01 2013
a(n) = 1 - floor(n/2) + 2*floor(n/4) = 1 - A004526(n) + A122461(n). - Wesley Ivan Hurt, Dec 06 2013
a(n) = (1 + (-1)^floor(n/2))/2. - Wesley Ivan Hurt, Apr 17 2014
a(n) = A054925(n+2) - A011848(n+2). - Wesley Ivan Hurt, Jun 09 2014
Euler transform of length 4 sequence [1, -1, 0, 1]. - Michael Somos, Sep 26 2014
a(n) = a(1-n) for all n in Z. - Michael Somos, Sep 26 2014
From Ilya Gutkovskiy, Jul 09 2016: (Start)
Inverse binomial transform of A038504(n+1).
E.g.f.: (exp(x) + sin(x) + cos(x))/2. (End)
a(n) = (1 + (-1)^(n*(n-1)/2))/2. - Guenther Schrack, Apr 04 2019

Definition rewritten by N. J. A. Sloane, Apr 30 2009

A042963 Numbers congruent to 1 or 2 mod 4.

Original entry on oeis.org

1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 25, 26, 29, 30, 33, 34, 37, 38, 41, 42, 45, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66, 69, 70, 73, 74, 77, 78, 81, 82, 85, 86, 89, 90, 93, 94, 97, 98, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 121, 122, 125, 126, 129, 130, 133, 134, 137, 138

Offset: 1

N. J. A. Sloane

Complement of A014601. - Reinhard Zumkeller, Oct 04 2004
Let S(x) = (1, 2, 2, 2, ...). Then A042963 = ((S(x))^2 + S(x^2))/2 = ((1, 4, 8, 12, 16, 20, ...) + (1, 0, 2, 0, 2, 0, 2, ...))/2 = (1, 2, 5, 6, 9, 10, ...). - Gary W. Adamson, Jan 03 2011
(a(n)*(a(n) + 1 + 4*k))/2 is odd, for k >= 0. - Gionata Neri, Jul 19 2015
Equivalent to the following variation on Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is not a square; this sequence, which we'll call sequence S, is produced by the following algorithm. At the start, S is initially empty. At stage n, starting at n = 1, the algorithm checks whether there exists a number m already in the sequence, such that mn+1 is a perfect square. If such a number m is found, then n is not added to the sequence; otherwise, n is added. Then n is incremented to n + 1, and we repeat the procedure. Proof by Clark R. Lyons: We prove by strong induction that n is in the sequence S if and only if n == 1 (mod 4) or n == 2 (mod 4). Suppose now that this holds for all k < n. In case 1, either n == 1 (mod 4) or n == 2 (mod 4), and we wish to show that n does indeed enter the sequence S. That is, we wish to show that there does not exist m < n, already in the sequence at this point such that mn+1 is a square. By the inductive hypothesis m == 1 (mod 4) or m == 2 (mod 4). This means that both m and n are one of 1, 2, 5, or 6 mod 8. Using a multiplication table mod 8, we see that this implies mn+1 is congruent to one of 2, 3, 5, 6, or 7 mod 8. But we also see that mod 8, a perfect square is congruent to 0, 1, or 4. Thus mn+1 is not a perfect square, so n is added to the sequence. In case 2, n == 0 (mod 4) or n == 3 (mod 4), and we wish to show that n is not added to the sequence. That is, we wish to show that there exists m < n already in the sequence such that mn+1 is a perfect square. For this we let m = n - 2, which is positive since n >= 3. By the inductive hypothesis, since m == 1 (mod 4) or m == 2 (mod 4) and m < n, m is already in the sequence. And we have m*n + 1 = (n - 2)*n + 1 = n^2 - 2*n + 1 = (n - 1)^2, so mn+1 is indeed a perfect square, and so n is not added to the sequence. Thus n is added to the sequence if and only if n == 1 (mod 4) or n == 2 (mod 4). This completes the proof. - Robert C. Lyons, Jun 30 2016
Also the number of maximal cliques in the (n + 1) X (n + 1) black bishop graph. - Eric W. Weisstein, Dec 01 2017
Lexicographically earliest sequence of distinct positive integers such that the average of any two or more consecutive terms is never an integer. (For opposite property see A005408.) - Ivan Neretin, Dec 21 2017
Numbers whose binary reflected Gray code (A014550) ends with 1. - Amiram Eldar, May 17 2021
Also: append its negated last bit to n-1. - M. F. Hasler, Oct 17 2022

David Lovler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Black Bishop Graph.
Eric Weisstein's World of Mathematics, Maximal Clique.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A153284 (first differences), A014848 (partial sums).
Cf. A014550, A046712 (subsequence).
Union of A016813 and A016825.

a042963 n = a042963_list !! (n-1)
a042963_list = [x | x <- [0..], mod x 4 `elem` [1,2]]
-- Reinhard Zumkeller, Feb 14 2012

[ n : n in [1..165] | n mod 4 eq 1 or n mod 4 eq 2 ]; // Vincenzo Librandi, Jan 25 2011

A046923:=n->(n mod 2) + 2n - 2; seq(A046923(n), n=1..100); # Wesley Ivan Hurt, Oct 10 2013

Select[Range[109], Or[Mod[#, 4] == 1, Mod[#, 4] == 2] &] (* Ant King, Nov 17 2010 *)
Table[(4 n - 3 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{1, 1, -1}, {1, 2, 5}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + x + 2 x^2)/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=2*n-1-(n-1)%2 \\ Jianing Song, Oct 06 2018; adapted to offset by Michel Marcus, Sep 09 2022

apply( A042963(n)=n*2-2+n%2, [1..99]) \\ M. F. Hasler, Oct 17 2022

a(n) = 1 + A042948(n-1). [Corrected by Jianing Song, Oct 06 2018]
From Michael Somos, Jan 12 2000: (Start)
G.f.: x*(1 + x + 2*x^2)/((1 - x)^2*(1 + x)).
a(n) = a(n-1) + 2 + (-1)^n, a(0) = 1. (End) [This uses offset 0. - Jianing Song, Oct 06 2018]
A014493(n) = A000217(a(n)). - Reinhard Zumkeller, Oct 04 2004, Feb 14 2012
a(n) = Sum_{k=0..n} (A001045(k) mod 4). - Paul Barry, Mar 12 2004
A145768(a(n)) is odd. - Reinhard Zumkeller, Jun 05 2012
a(n) = A005843(n-1) + A059841(n-1). - Philippe Deléham, Mar 31 2009 [Corrected by Jianing Song, Oct 06 2018]
a(n) = 4*n - a(n-1) - 5 for n > 1. [Corrected by Jerzy R Borysowicz, Jun 09 2023]
From Ant King, Nov 17 2010: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = (4*n - 3 - (-1)^n)/2. (End)
a(n) = (n mod 2) + 2*n - 2. - Wesley Ivan Hurt, Oct 10 2013
A163575(a(n)) = n - 1. - Reinhard Zumkeller, Jul 22 2014
E.g.f.: 2 + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - Ilya Gutkovskiy, Jun 30 2016
E.g.f.: 2 + (2*x - 1)*exp(x) - cosh(x). - David Lovler, Jul 19 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Dec 05 2021

Offset corrected by Reinhard Zumkeller, Feb 14 2012

More terms by David Lovler, Jul 19 2022

cp's OEIS Frontend

A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.

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A329649 Let D = A042948(n) be the n-th positive integer congruent to 0 or 1 mod 4, then a(n) = b(D) := Sum_{i=1..D} Kronecker(D,i)*i^2, where Kronecker(D,i) is the Kronecker symbol.

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A104571 Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.

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A367269 Triangle T(n, k) read by rows and based on A042948 yields a permutation of the natural numbers.

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A002620 Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).

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A007696 Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

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