A042963 -id:A042963 - cp's OEIS frontend

A193868 Even central polygonal numbers.

2, 4, 16, 22, 46, 56, 92, 106, 154, 172, 232, 254, 326, 352, 436, 466, 562, 596, 704, 742, 862, 904, 1036, 1082, 1226, 1276, 1432, 1486, 1654, 1712, 1892, 1954, 2146, 2212, 2416, 2486, 2702, 2776, 3004, 3082, 3322, 3404, 3656, 3742, 4006, 4096, 4372

Offset: 1

Omar E. Pol, Aug 15 2011

Odd triangular numbers plus 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000124, A014493, A174114, A193867.

[1+((2*n-1)*(2*n-1-(-1)^n)/2): n in [1..50]]; // Vincenzo Librandi, Aug 18 2011

Table[(3 + (-1)^n - 2 (2 + (-1)^n) n + 4 n^2)/2, {n, 50}] (* or *)
Select[PolygonalNumber@ Range@ 100, OddQ] + 1 (* Version 10.4, or *)
Table[If[EvenQ@ n, 2 n^2 - 3 n + 2, 2 n^2 - n + 1], {n, 50}] (* or *)
Rest@ CoefficientList[Series[-2 x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{2,4,16,22,46},50] (* Harvey P. Dale, Sep 13 2020 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)/2+1 \\ Charles R Greathouse IV, Jun 11 2015

Vec(2*x*(1+x+4*x^2+x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = A000124(A042963(n-1)).
a(n) = 1 + A014493(n).
a(n) = 2*A174114(n).
G.f.: -2*x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (3+(-1)^n-2*(2+(-1)^n)*n+4*n^2)/2.
a(n) = 2*n^2-3*n+2 for n even.
a(n) = 2*n^2-n+1 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 11 2025

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

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

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 5, 15, 12, 1, 5, 21, 45, 29, 1, 5, 21, 77, 129, 70, 1, 5, 21, 89, 265, 361, 169, 1, 5, 21, 89, 353, 865, 991, 408, 1, 5, 21, 89, 377, 1325, 2717, 2681, 985, 1, 5, 21, 89, 377, 1549, 4733, 8281, 7169, 2378, 1, 5, 21, 89, 377, 1597, 6125

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: F(2+3k), where F=A000045 (Fibonacci numbers)
Coefficient of x^n in u(n,x): 1,2,5,12,.... A000129(n)
Row sums:  1,3,11,33,101,303,911,2733,..... A081250
Alternating row sums: 1,-1,1,-1,1,-1,,..... A033999
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...5...5
1...5...15...12
1...5...21...45...29
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.

Cf. A209766, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209766 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 13, 17, 1, 3, 13, 43, 41, 1, 3, 13, 55, 133, 99, 1, 3, 13, 55, 209, 391, 239, 1, 3, 13, 55, 233, 739, 1113, 577, 1, 3, 13, 55, 233, 939, 2469, 3095, 1393, 1, 3, 13, 55, 233, 987, 3589, 7903, 8457, 3363, 1, 3, 13, 55, 233, 987, 4085

Offset: 1

Clark Kimberling, Mar 14 2012

Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)
Coefficient of x^n in u(n,x): A001333(n)
Row sums:  1,4,11,34,101,304,... A060925.
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...3
1...3...7
1...3...13...17
1...3...13...43...41
First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^2.

Cf. A209665, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209766 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

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

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 13, 14, 14, 16, 18, 20, 23, 24, 27, 30, 31, 34, 37, 41, 46, 49, 53, 58, 62, 67, 73, 80, 88, 94, 101, 109, 117, 127, 136, 147, 161, 172, 184, 198, 211, 228, 245, 262, 284, 304, 324, 347, 370, 397, 425, 454, 488

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 1 or 2 mod 4.

			a(11) = 3 because we have [10, 1], [9, 2] and [6, 5].

Index entries for sequences related to partitions

Cf. A035365, A042963, A070048, A108932, A169975, A301504, A301505, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 1)) (1 + x^(4 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x, x^4] QPochhammer[-x^2, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042963(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 2, 2, 2, 4, 5, 5, 2, 2, 5, 5, 6, 5, 5, 3, 5, 5, 6, 7, 6, 5, 4, 4, 5, 6, 7, 8, 6, 5, 5, 4, 5, 5, 6, 8, 9, 9, 5, 5, 5, 5, 5, 5, 9, 9, 10, 9, 10, 4, 5, 5, 5, 4, 10, 9, 10, 11, 10, 10, 11, 4, 5, 5, 4, 11, 10, 10, 11, 12, 10, 10, 10, 11, 5, 6, 5, 11, 10, 10, 10, 12

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.

The underlying idea is to break in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

			Array T(n, k) begins:
  n\k|    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+-----------------------------------------------------------------
    0|    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|    1   1   2   2   5   5   6   6   9   9  10  10  13  13  14  14
    2|    2   2   2   2   5   5   5   5  10  10  10  10  13  13  13  13
    3|    3   2   2   3   4   5   5   4  11  10  10  11  12  13  13  12
    4|    4   5   5   4   4   5   5   4  11  10  10  11  11  10  10  11
    5|    5   5   5   5   5   5   5   5  10  10  10  10  10  10  10  10
    6|    6   6   5   5   5   5   6   6   9   9  10  10  10  10   9   9
    7|    7   6   5   4   4   5   6   7   8   9  10  11  11  10   9   8
    8|    8   9  10  11  11  10   9   8   8   9  10  11  11  10   9   8
    9|    9   9  10  10  10  10   9   9   9   9  10  10  10  10   9   9
   10|   10  10  10  10  10  10  10  10  10  10  10  10  10  10  10  10
   11|   11  10  10  11  11  10  10  11  11  10  10  11  11  10  10  11
   12|   12  13  13  12  11  10  10  11  11  10  10  11  12  13  13  12
   13|   13  13  13  13  10  10  10  10  10  10  10  10  13  13  13  13
   14|   14  14  13  13  10  10   9   9   9   9  10  10  13  13  14  14
   15|   15  14  13  12  11  10   9   8   8   9  10  11  12  13  14  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A005811, A042963, A070939, A101211, A227736, A341840, A341841.

T(n,k) = { my (r=[], v=0); while (n||k, my (w=min(valuation(n+n%2,2), valuation(k+k%2,2))); r=concat(w,r); n\=2^w; k\=2^w); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }

T(n, k) = T(k, n)
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
A070939(T(n, k)) = max(A070939(n), A070939(k)).
A003188(T(n, k)) = A003188(n) OR A003188(k) (where OR denotes the bitwise OR operator).
T(n, 1) = A042963(ceiling((n+1)/2)).

A350396 Numbers m such that there are two or more terms with the largest coefficient in Product_{k=1..m} (1 + x^k).

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 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, 141, 142, 145, 146, 149, 150, 153, 154, 157, 158, 161, 162, 165, 166, 169, 170, 173, 174, 177, 178

Offset: 1

Max Alekseyev, Dec 28 2021

nonn

Numbers m such that A350393(m) < A350394(m).

Apparently, a(n) = A042963(n-2) for n >= 7. - Hugo Pfoertner, Dec 30 2021

Complement of A350395.
Cf. A025591 (largest coefficient), A350393, A350394.
Cf. A042963.

A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 4, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
  1;
  1, 2;
  3, 2, 3;
  3, 4, 3, 4;
  5, 4, 5, 4, 5;
  5, 6, 5, 6, 5, 6;
  7, 6, 7, 6, 7, 6, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.

Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

[n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024

Table[n - Mod[n+k,2], {n,16}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[n - ((n+k)%2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.
Sum_{k=1..n} T(n, k) = A014848(n) (row sums).
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = n - (1 - (-1)^(n+k))/2 = n - (n+k mod 2).
T(n, 1) = A109613(n+1).
T(n, n) = A000027(n).
T(2*n-1, n) = A042963(n).
T(3*n-1, n) = A016777(n+1).
T(4*n-3, n) = A047461(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 14 2024

A237989 Numbers m such that the numbers of primes, even positive integers and odd positive integers less than or equal to m are all odd.

Original entry on oeis.org

2, 6, 18, 26, 34, 42, 50, 70, 74, 78, 86, 98, 106, 110, 130, 138, 150, 158, 162, 170, 198, 214, 218, 222, 234, 238, 242, 246, 250, 258, 262, 270, 278, 286, 290, 310, 314, 334, 354, 358, 370, 382, 390, 394, 402, 406, 442, 450, 454, 462, 470, 474, 478, 490, 502

Offset: 1

Ivan N. Ianakiev, Feb 16 2014

			A cubic die whose faces are marked with the numbers from 1 to 6 has odd number of sides marked with prime numbers (2, 3 and 5), even integers (2, 4 and 6) and odd integers (1, 3 and 5). Therefore, 6 is in the sequence.

Ivan N. Ianakiev, Table of n, a(n) for n = 1..10000

Cf. A042963, A042964, A057812, A237990.

Select[Range[2, 1000, 4], OddQ[PrimePi[#]] &] (* Paolo Xausa, Jun 24 2024 *)

isok(n) = (primepi(n) % 2) && ((n % 4) == 2); \\ Michel Marcus, Mar 12 2014

Intersection of A042963 (odd number of odd numbers), A042964 (odd number of even numbers), A057812 (odd number of primes). - Michel Marcus, Feb 26 2014

cp's OEIS Frontend

A193868 Even central polygonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

Mathematica

PARI

Formula

A209765 Triangle of coefficients of polynomials u(n,x) jointly generated with A209766; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

PARI

Formula

A301507 Expansion of Product_{k>=0} (1 + x^(4*k+1))*(1 + x^(4*k+2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A350396 Numbers m such that there are two or more terms with the largest coefficient in Product_{k=1..m} (1 + x^k).

A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).