A000466 -id:A000466 - cp's OEIS frontend

A226379 a(5n) = 2n(2n+1), a(5n+1) = (2n-3)(2n+5), a(5n+2) = (2n-1)(2n+3), a(5n+3) = (2n+2)(2n+1), a(5n+4) = (2n+1)(2*n+3).

0, -15, -3, 2, 3, 6, -7, 5, 12, 15, 20, 9, 21, 30, 35, 42, 33, 45, 56, 63, 72, 65, 77, 90, 99, 110, 105, 117, 132, 143, 156, 153, 165, 182, 195, 210, 209, 221, 240, 255, 272, 273, 285, 306, 323, 342, 345, 357, 380, 399, 420, 425, 437

Offset: 0

Paul Curtz, Jun 05 2013

The sequence is the fifth row of the following array:
0,   6,  20, 42, 72, 110, 156, 210, 272, ...   A002943
0,   3,   6, 15, 20,  35,  42,  63,  72, ...   bisections A002943, A000466
0,   2,   3,  6, 12,  15,  20,  30,  35, ...   A226023 (trisections A002943, A000466, A002439)
0,  -3,   2,  3,  6,   5,  12,  15,  20, ...   A214297 (quadrisections A078371)
0, -15,  -3,  2,  3,   6,  -7,   5,  12, ...   a(n)
0, -63, -15, -3,  2,   3,   6, -55,  -7, ...
The principle of construction is that (i) the lower left triangular portion has constant values down the diagonals (6, 3, 2, -3, -15, ...), defined from row 4 on by the negated values of A024036. (ii) The extension along the rows is defined by maintaining bisections, trisections, quadrisections etc of the form (2*n+x)*(2*n+y) with some constants x and y. In the fifth line this needs the quintisections shown in the NAME.
Each row in the array has the subsequences of the previous row plus another subsequence of the format (2*n+1)*(2*n+y) shuffled in; the first A002943, the second also A000466, the third also A002439, the fourth also A078371, and the fifth (2*n+3)*(2*n-5).
Only the first three rows are monotonically increasing everywhere.
a(n) is divisible by A226203(n).
Numerators of: 0, -15/4, -3/4, 2/9, 3/16, 6/25, -7/36, 5/36, 12/49, 15/64, 20/81, ... = a(n)/A226096(n). A permutation of A225948(n+1)/A226008(n+1).
Is the sequence increasing monotonically from 221 on?

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,2,-2,0,0,0,-1,1).

Cf. A000466, A002439, A002939, A002943, A024036, A078371, A145923.

Cf. A214297, A225948, A226008, A226023, A226096, A226203.

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) )); // G. C. Greubel, Mar 23 2024

CoefficientList[Series[x*(15 - 12*x - 5*x^2 - x^3 - 3*x^4 - 17*x^5 + 12*x^6 + 3*x^7 - x^8 + x^9)/((x^4 + x^3 + x^2 + x + 1)^2*(x - 1)^3), {x, 0, 80}], x] (* Wesley Ivan Hurt, Oct 03 2017 *)

def A226379_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) ).list()
A226379_list(50) # G. C. Greubel, Mar 23 2024

4*a(n) = A226096(n) - period 5: repeat [1, 64, 16, 1, 4].
G.f.: x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9) / ( (x^4+x^3+x^2+x+1)^2 *(x-1)^3 ). - R. J. Mathar, Jun 13 2013
a(n) = a(n-1)+2*a(n-5)-2*a(n-6)-a(n-10)+a(n-11) for n > 10. - Wesley Ivan Hurt, Oct 03 2017

A289870 a(n) = n(n + 1) for n odd, otherwise a(n) = (n - 1)(n + 1).

Original entry on oeis.org

-1, 2, 3, 12, 15, 30, 35, 56, 63, 90, 99, 132, 143, 182, 195, 240, 255, 306, 323, 380, 399, 462, 483, 552, 575, 650, 675, 756, 783, 870, 899, 992, 1023, 1122, 1155, 1260, 1295, 1406, 1443, 1560, 1599, 1722, 1763, 1892, 1935, 2070, 2115, 2256, 2303, 2450, 2499

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 14 2017

a(n) is a fifth-order linear recurrence whose main interest is that it is related to (at least) eight other sequences (see the formula section).

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

After -1, subsequence of A035106, A198442 and A214297.

Cf. A000466, A002378, A093178, A109613, A114285, A124356, A193356, A289296.

a[n_] := (n + 1)(n - 1 + Mod[n, 2]); Table[a[n], {n, 0, 50}]

a(n)=if(n%2, n, n-1)*(n+1) \\ Charles R Greathouse IV, Jul 14 2017

a(n) = (n + 1)*(n - 1 + (n mod 2)).
a(n) = n * A109613(n-1) for n>0.
a(n) = -A114285(n) * A109613(n).
a(n) = A002378(n) - A193356(n).
a(n) = A289296(-n).
a(n) = n^2 - (-1)^n * A093178(n).
a(2*k) = A000466(k).
G.f.: (1-3*x-3*x^2-3*x^3)/((-1+x)^3*(1+x)^2).

A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

Original entry on oeis.org

1, 1, 3, 1, 15, 105, 1, 35, 945, 10395, 1, 63, 3465, 135135, 2027025, 1, 99, 9009, 675675, 34459425, 654729075, 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225, 1, 195, 36465, 6235515, 916620705, 105411381075, 7905853580625, 213458046676875

Offset: 0

Peter Luschny, May 05 2023

			[0] 1;
[1] 1,   3;
[2] 1,  15,   105;
[3] 1,  35,   945,   10395;
[4] 1,  63,  3465,  135135,   2027025;
[5] 1,  99,  9009,  675675,  34459425,   654729075;
[6] 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225;

Cf. A362848 (row sums), A000466 (column 1), A101485 (main diagonal).

T := (n, k) -> 4^k * GAMMA(n + k + 1/2) / GAMMA(n - k + 1/2):
seq(seq(T(n, k), k = 0..n), n = 0..7);

T[n_,k_]:=(2*(n+k)-1)!!/(2*(n-k)-1)!!;Flatten[Table[T[n,k],{n,0,7},{k,0,n}]] (* Detlef Meya, Oct 09 2023 *)

T(n ,k ) = (2*(n + k) - 1)!!/(2*(n - k) - 1)!!; 0 <= n <= k. - Detlef Meya, Oct 09 2023

A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

Original entry on oeis.org

1, 1, -1, 1, 1, 3, 1, 3, -1, -5, 1, 5, 3, 1, 35, 1, 7, 15, -1, -5, -63, 1, 9, 35, 5, 3, 7, 231, 1, 11, 63, 35, -5, -3, -21, -429, 1, 13, 99, 105, 35, 3, 7, 33, 6435, 1, 15, 143, 231, 315, -7, -5, -9, -429, -12155, 1, 17, 195, 429, 1155, 63, 7, 5, 99, 715, 46189

Offset: 0

Stefano Spezia, Feb 12 2025

Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.

			The array of the binomial coefficients for half-integers begins as:
  1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
  1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
  1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
  1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
  1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
  1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
  1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals of the array)
Gábor Hegedüs, Sho Suda, and Ziqing Xiang, Codes with symmetric distances, arXiv:2501.11461 [math.CO], 2025. See p. 10.

Cf. A000466, A161200, A161202, A162540, A173755.
Columns k=0..1 give A000012, A060747.
Row n=1 gives A002596.
Main diagonal gives A001790.

A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=Numerator[(2n-1)!!/((2(n-k)-1)!!2^k k!)]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).
A(n,2) = A000466(n-1) for n > 0.
A(n,3) = A162540(n-3) for n > 3.
A(0,n) = (-1)^n*A001790(n).
abs(A(2,n)) = abs(A161200(n)).
abs(A(3,n)) = abs(A161202(n)).

A384488 Numbers k having a divisor d such that d - k/d is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 26, 28, 30, 32, 35, 36, 38, 40, 42, 44, 48, 50, 54, 60, 62, 63, 66, 68, 70, 72, 74, 78, 80, 84, 86, 88, 90, 92, 96, 98, 99, 102, 104, 108, 110, 114, 120, 122, 126, 128, 130, 132, 138, 140, 143, 144, 146, 150, 152, 154, 158, 162, 164, 168, 170, 174, 176, 180

Offset: 1

Juri-Stepan Gerasimov, May 30 2025

nonn

Presumably, all odd terms are in A000466.

			a(6) = 12 is a term because 12 = 1*12 with 12 - 1 = 11 prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000466, A005408, A355643. Includes A005563 and 2 * A052147.

[k: k in [1..180] | not #[d: d in Divisors (k) | IsPrime(d-(k div d))] eq 0];

filter:= k -> ormap(d -> d^2 > k and isprime(d - k/d), numtheory:-divisors(k)):
select(filter, [$1..200]); # Robert Israel, Jun 30 2025

A384488Q[k_] := AnyTrue[Divisors[k], PrimeQ[# - k/#] &];
Select[Range[200], A384488Q] (* Paolo Xausa, Jun 30 2025 *)

isok(k) = fordiv(k, d, if (isprime(d - k/d), return(1))); \\ Michel Marcus, Jun 01 2025

A268484 a(n) = (n + 1)(4n^2 + 14*n + 9)/3.

Original entry on oeis.org

3, 18, 53, 116, 215, 358, 553, 808, 1131, 1530, 2013, 2588, 3263, 4046, 4945, 5968, 7123, 8418, 9861, 11460, 13223, 15158, 17273, 19576, 22075, 24778, 27693, 30828, 34191, 37790, 41633, 45728, 50083, 54706, 59605, 64788, 70263, 76038, 82121, 88520, 95243

Offset: 0

Ilya Gutkovskiy, Feb 12 2016

			a(0) = 1*3 = 3;
a(1) = 1*3 + 3*5 = 18;
a(2) = 1*3 + 3*5 + 5*7 = 53;
a(3) = 1*3 + 3*5 + 5*7 + 7*9 = 116, etc.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000466, A005408, A007290, A135036.

Table[(n + 1) ((4 n^2 + 14 n + 9)/3), {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {3, 18, 53, 116}, 40]

a(n)=(n+1)*(4*n^2+14*n+9)/3 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: (3 + 6*x - x^2)/(x - 1)^4.
a(n) = Sum_{k = 0..n} (2*k + 1)*(2*k + 3) = Sum_{k = 0..n} A005408(k)*A005408(k + 1).
Sum_{n>=0} 1/a(n) = 0.4315109123788144393864...

A269342 a(n) = (n + 1)(2n + 1)(4n + 9)/3.

Original entry on oeis.org

3, 26, 85, 196, 375, 638, 1001, 1480, 2091, 2850, 3773, 4876, 6175, 7686, 9425, 11408, 13651, 16170, 18981, 22100, 25543, 29326, 33465, 37976, 42875, 48178, 53901, 60060, 66671, 73750, 81313, 89376, 97955, 107066, 116725, 126948, 137751, 149150, 161161

Offset: 0

Ilya Gutkovskiy, Feb 24 2016

			a(0) = 0*2 + 1*3 = 3;
a(1) = 0*2 + 1*3 + 2*4 + 3*5 = 26;
a(2) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 = 85;
a(3) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 = 196;
a(4) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 + 8*10 + 9*11 = 375, etc.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000466, A005563, A033996, A195241.

[(n+1)*(2*n+1)*(4*n+9)/3: n in [0..50]]; // Vincenzo Librandi, Feb 25 2016

Table[(n + 1) (2 n + 1) (4 n + 9)/3, {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {3, 26, 85, 196}, 39]
Table[Sum[8 k^2 + 12 k + 3, {k, 0, n}], {n, 0, 38}]

Vec((3 + 14*x - x^2)/(1 - x)^4 + O(x^50)) \\ Michel Marcus, Feb 25 2016

G.f.: (3 + 14*x - x^2)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = Sum_{k=0..n} (8*k^2 + 12*k + 3).
Sum_{n>=0} 1/a(n) = 3*(80*log(2) + 5*Pi - 48)/175 = 0.397024075075621559...

A365983 Even numbers k such that k^2 - 1 is a powerful number.

Original entry on oeis.org

26, 70226, 130576328, 189750626, 512706121226, 13837575261124, 99612037019890, 1385331749802026

Offset: 1

Jud McCranie, Sep 24 2023

This sequence is a subsequence of A060860 (the even terms) and a supersequence of A094835. All the terms of A094835 are in this sequence, but 130576328 is not in A094835. A094835 also shows that this sequence is infinite.
Terms A076445(n)+1 are terms of this sequence because A076445(n) and A076445(n)+2 are powerful and (A076445(n)+1)^2-1 = A076445(n) * (A076445(n)+2), which is also powerful.
a(n) - 1 is an odd powerful number (A062739). - Amiram Eldar, Feb 23 2024

			26^2 - 1 = 675 = 3^3 * 5^2 is powerful.
130576328^2 - 1 = 17050177433963583 = 3^2 * 7^3 * 13^2 * 293^2 * 617^2, whose exponents are all greater than 1, so it is powerful.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, entries 70226 and 485.

Index entries for sequences related to powerful numbers.

Cf. A000466, A001694, A062739, A094835, A060860, A076445.

seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3), 2}, {i, 1, Sqrt[max/j^3], 2}]]], i}, i = Position[Differences[p], 2] // Flatten; Sqrt[p[[i]]*(p[[i]] + 2) + 1]]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)

isok(k) = !(k%2) && ispowerful(k^2-1); \\ Michel Marcus, Sep 25 2023

a(5)-a(8) from Amiram Eldar, Feb 23 2024

cp's OEIS Frontend

A226379 a(5n) = 2*n*(2*n+1), a(5n+1) = (2*n-3)*(2*n+5), a(5n+2) = (2*n-1)*(2*n+3), a(5n+3) = (2*n+2)*(2*n+1), a(5n+4) = (2*n+1)*(2*n+3).

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A289870 a(n) = n*(n + 1) for n odd, otherwise a(n) = (n - 1)*(n + 1).

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A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

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A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

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A384488 Numbers k having a divisor d such that d - k/d is prime.

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A268484 a(n) = (n + 1)*(4*n^2 + 14*n + 9)/3.

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A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3.

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A226379 a(5n) = 2n(2n+1), a(5n+1) = (2n-3)(2n+5), a(5n+2) = (2n-1)(2n+3), a(5n+3) = (2n+2)(2n+1), a(5n+4) = (2n+1)(2*n+3).

A289870 a(n) = n(n + 1) for n odd, otherwise a(n) = (n - 1)(n + 1).

A268484 a(n) = (n + 1)(4n^2 + 14*n + 9)/3.

A269342 a(n) = (n + 1)(2n + 1)(4n + 9)/3.