A016885 -id:A016885 - cp's OEIS frontend

A001509 a(n) = (5n + 1)(5n + 2)(5*n + 3).

6, 336, 1716, 4896, 10626, 19656, 32736, 50616, 74046, 103776, 140556, 185136, 238266, 300696, 373176, 456456, 551286, 658416, 778596, 912576, 1061106, 1224936, 1404816, 1601496, 1815726, 2048256, 2299836, 2571216, 2863146, 3176376, 3511656, 3869736, 4251366

Offset: 0

N. J. A. Sloane, Dec 11 1996

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001622, A016861, A016873, A016885.

A001509:=n->(5*n+1)*(5*n+2)*(5*n+3); seq(A001509(n), n=0..50); # Wesley Ivan Hurt, May 07 2014

Table[(5*n + 1)*(5*n + 2)*(5*n + 3), {n, 0, 100}] (* Harvey P. Dale, Apr 21 2011 *)

a(n) = A016861(n) * A016873(n) * A016885(n). - Wesley Ivan Hurt, May 07 2014
G.f.: 6*(1 + 52*x + 68*x^2 + 4*x^3)/(1 - x)^4. - Stefano Spezia, Jan 03 2023
Sum_{n>=0} 1/a(n) = sqrt(2*(25-11*sqrt(5))/5)*Pi/20 + log(phi)/(2*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Jan 26 2023
From Elmo R. Oliveira, Sep 07 2025: (Start)
E.g.f.: exp(x)*(6 + 330*x + 525*x^2 + 125*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A013828 a(n) = 3^(5*n + 3).

Original entry on oeis.org

27, 6561, 1594323, 387420489, 94143178827, 22876792454961, 5559060566555523, 1350851717672992089, 328256967394537077627, 79766443076872509863361, 19383245667680019896796723, 4710128697246244834921603689

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (243).

Cf. A000244 (3^n), A016885 (5*n+3).

[3^(5*n+3): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

3^(5*Range[0,20]+3) (* or *) NestList[243#&,27,20] (* Harvey P. Dale, Jun 07 2016 *)

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 243*a(n-1), n > 0; a(0)=27.
G.f.: 27/(1-243*x).
a(n) = 3*A013827(n) = 9*A013826(n) = A013829(n)/3. (End)

A013836 a(n) = 5^(5*n + 3).

Original entry on oeis.org

125, 390625, 1220703125, 3814697265625, 11920928955078125, 37252902984619140625, 116415321826934814453125, 363797880709171295166015625, 1136868377216160297393798828125, 3552713678800500929355621337890625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3125).

Cf. A000351 (5^n), A016885 (5*n+3).

[5^(5*n+3): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

NestList[3125#&,125, 20] (* Harvey P. Dale, Jan 14 2025 *)

a(n)=5^(5*n+3) \\ Edward Jiang, Sep 06 2014

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 3125*a(n-1), n > 0; a(0)=125.
G.f.: 125/(1-3125*x). (End)

A013852 a(n) = 9^(5*n + 3).

Original entry on oeis.org

729, 43046721, 2541865828329, 150094635296999121, 8862938119652501095929, 523347633027360537213511521, 30903154382632612361920641803529, 1824800363140073127359051977856583921, 107752636643058178097424660240453423951129

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (59049).

Cf. A001019, A016885.

[9^(5*n+3): n in [0..15]]; // Vincenzo Librandi, Jul 08 2011

A013852:=n->9^(5*n + 3); seq(A013852(n), n=0..15); # Wesley Ivan Hurt, Jan 28 2014

Table[9^(5n + 3), {n, 0, 15}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n) = 9^A016885(n) = A001019(A016885(n)). - Wesley Ivan Hurt, Jan 28 2014

A016887 a(n) = (5*n+3)^3.

Original entry on oeis.org

27, 512, 2197, 5832, 12167, 21952, 35937, 54872, 79507, 110592, 148877, 195112, 250047, 314432, 389017, 474552, 571787, 681472, 804357, 941192, 1092727, 1259712, 1442897, 1643032, 1860867, 2097152, 2352637, 2628072, 2924207, 3241792, 3581577, 3944312

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A016885.

[(5*n+3)^3: n in [0..40]]; // Vincenzo Librandi, May 26 2016

(5 Range[0, 30] + 3)^3 (* or *) LinearRecurrence[{4, -6, 4, -1}, {27, 512, 2197, 5832}, 30] (* Harvey P. Dale, Nov 26 2014 *)
Table[(5 n + 3)^3, {n, 0, 40}] (* Vincenzo Librandi, May 26 2016 *)

From Ilya Gutkovskiy, May 26 2016: (Start)
O.g.f.: (27 + 404*x + 311*x^2 + 8*x^3)/(1 - x)^4.
E.g.f.: (27 + 485*x + 600*x^2 + 125*x^3)*exp(x). (End)
a(n) = A000578(A016885(n)). - Michel Marcus, May 26 2016

A017125 Table read by antidiagonals of Fibonacci-type sequences.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 5, 5, 4, 4, 1, 5, 8, 8, 7, 5, 5, 1, 6, 13, 13, 11, 9, 6, 6, 1, 7, 21, 21, 18, 14, 11, 7, 7, 1, 8, 34, 34, 29, 23, 17, 13, 8, 8, 1, 9, 55, 55, 47, 37, 28, 20, 15, 9, 9, 1, 10, 89, 89, 76, 60, 45, 33, 23, 17, 10, 10, 1, 11, 144, 144, 123, 97, 73

Offset: 0

Henry Bottomley, Jul 31 2000

Rows are (essentially) A000045, A000045, A000032, A000285, A022095, A022096, A022097, etc. Columns are (essentially) A001477, A000012, A000027, A005408, A016789, A016885, etc. One of the diagonals is A007502.

Antidiagonal sums are in A019274.

T(n, k) = T(n, k-1)+T(n, k-2) [with T(n, 0) = n and T(n, 1) = 1] = 2*T(n-1, k)-T(n-2, k) = Fib(k)+n*Fib(k-1) = (s^k*(1+2n/s)-t^k*(1+2n/t))/(2^k*sqrt(5)) where s = (1+sqrt(5))/2 and t = (1-sqrt(5))/2 = 1-s.

G.f. for n-th row: (n+x-nx)/(1-x-x^2).

A093303 a(n) = a(n-1)*(2n-1) + 2n with a(0)=0.

Original entry on oeis.org

0, 2, 10, 56, 400, 3610, 39722, 516400, 7746016, 131682290, 2501963530, 52541234152, 1208448385520, 30211209638026, 815702660226730, 23655377146575200, 733316691543831232, 24199450820946430690, 846980778733125074186

Offset: 0

Emrehan Halici (emrehan(AT)halici.com.tr), Apr 24 2004

Obviously, a(n) is always an even number. a(2) and a(6) are even semiprimes. - Altug Alkan, Dec 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005843.

Flatten[{0,Table[n!*Binomial[2*n-1,n]/2^(n-1)*Sum[2^k*k/(k!*Binomial[2*k-1,k]), {k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

a(n) = if(n==0, 0, n!*binomial(2*n-1,n)/2^(n-1) * sum(k=1, n, 2^k*k/(k!*binomial(2*k-1,k)))) \\ Altug Alkan, Dec 07 2015

a(n) = if(n==0, 0, a(n-1)*(2*n-1) + 2*n); \\ Altug Alkan, Dec 07 2015

a(n) = n!*C(2*n-1,n)/2^(n-1) * Sum_{k=1..n} 2^k*k/(k!*C(2*k-1,k)), for n>0. - Vaclav Kotesovec, Oct 28 2012
From Altug Alkan, Dec 07 2015: (Start)
a(A047212(k)) mod 10 = 0.
a(A016861(k)) mod 10 = 2.
a(A016885(k)) mod 10 = 6. (End)
a(n) ~ (sqrt(2) + 2*sqrt(Pi)*exp(1/2)*erf(1/sqrt(2))) * 2^n * n^n / exp(n). - Vaclav Kotesovec, Dec 18 2015

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

A187322 a(n) = floor(n/2) + floor(3*n/4).

Original entry on oeis.org

0, 0, 2, 3, 5, 5, 7, 8, 10, 10, 12, 13, 15, 15, 17, 18, 20, 20, 22, 23, 25, 25, 27, 28, 30, 30, 32, 33, 35, 35, 37, 38, 40, 40, 42, 43, 45, 45, 47, 48, 50, 50, 52, 53, 55, 55, 57, 58, 60, 60, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 75, 77, 78, 80, 80, 82, 83, 85, 85, 87, 88, 90, 90, 92, 93, 95, 95, 97

Offset: 0

Clark Kimberling, Mar 08 2011

List of quadruples [5*k, 5*k, 5*k+2, 5*k+3]. - Luce ETIENNE, Aug 14 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008587, A016873, A016885, A004526, A057353.

Table[Floor[n/2]+Floor[3n/4], {n,0,120}]
LinearRecurrence[{1,0,0,1,-1},{0,0,2,3,5},80] (* Harvey P. Dale, Dec 05 2018 *)

a(n) = n\2 + 3*n\4; \\ Altug Alkan, Aug 14 2017

concat(vector(2), Vec(x^2*(2 + x + 2*x^2) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^100))) \\ Colin Barker, Aug 14 2017

def A187322(n): return (n>>1)+(3*n>>2) # Chai Wah Wu, Jan 31 2023

a(n) = A004526(n) + A057353(n). - Michel Marcus, Aug 14 2017
a(n) = (10*n-5+3*cos(n*Pi)+2*(cos(n*Pi/2)-sin(n*Pi/2)))/8. - Luce ETIENNE, Aug 14 2017
From Colin Barker, Aug 14 2017: (Start)
G.f.: x^2*(2 + x + 2*x^2) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
(End)

A290781 a(n) = 20*n + 15.

Original entry on oeis.org

15, 35, 55, 75, 95, 115, 135, 155, 175, 195, 215, 235, 255, 275, 295, 315, 335, 355, 375, 395, 415, 435, 455, 475, 495, 515, 535, 555, 575, 595, 615, 635, 655, 675, 695, 715, 735, 755, 775, 795, 815, 835, 855, 875, 895, 915, 935, 955, 975, 995, 1015, 1035

Offset: 0

Arkadiusz Wesolowski, Aug 10 2017

Bisection of A017329.

None of the numbers in this sequence is a Fermat pseudoprime to base 2.

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

Cf. A001567, A004767, A008587, A016885, A017329.

Magma
```
[n: n in [15..1035 by 20]];
```
Mathematica
```
Range[15, 1035, 20]
```

G.f.: 5*(3 + x)/(1 - x)^2.
a(n) = A004767(A016885(n)) = A004767(A004767(n) + n). - Torlach Rush, Oct 10 2019
E.g.f.: 5*exp(x)*(3 + 4*x). - Stefano Spezia, Oct 12 2019
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 5*A004767(n) = A017329(2*n+1) = A008587(4*n+3).
a(n) = 2*a(n-1) - a(n-2). (End)

A276489 a(n) = 25^(n+1)*Gamma(n+8/5)/Gamma(3/5).

Original entry on oeis.org

15, 600, 39000, 3510000, 403650000, 56511000000, 9324315000000, 1771619850000000, 380898267750000000, 91415584260000000000, 24225129828900000000000, 7025287650381000000000000, 2212965609870015000000000000, 752408307355805100000000000000, 274629032184868861500000000000000

Offset: 0

Ilya Gutkovskiy, Sep 05 2016

			a(0) = (1+2+3+4+5) = 15;
a(1) = (1+2+3+4+5)*(6+7+8+9+10) = 600;
a(2) = (1+2+3+4+5)*(6+7+8+9+10)*(11+12+13+14+15) = 39000, etc.

Cf. A016885, A268730.

FullSimplify[Table[25^(n + 1) (Gamma[n + 8/5]/Gamma[3/5]), {n, 0, 14}]]

a(n) = prod(k=0, n, 5*(5*k + 3)); \\ Michel Marcus, Sep 06 2016

E.g.f.: 15/(1 - 25*x)^(8/5).
D-finite with recurrence: a(n) = 5*(5*n + 3)*a(n - 1), a(0)=15.
a(n) = Product_{k=0..n} 5*(5*k + 3).
a(n) = Product_{k=0..n} 5*A016885(k).
a(n) ~ sqrt(2*Pi)*25^(n+1)*n^(n+11/10)/(Gamma(3/5)*exp(n)).
Sum_{n>=0} 1/a(n) = exp(1/25)*(Gamma(3/5) - Gamma(3/5, 1/25))/5^(4/5)
= 0.06835926175445652444604..., where Gamma(a, x) is the incomplete Gamma function.

cp's OEIS Frontend

A001509 a(n) = (5*n + 1)*(5*n + 2)*(5*n + 3).

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A013828 a(n) = 3^(5*n + 3).

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A013836 a(n) = 5^(5*n + 3).

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A013852 a(n) = 9^(5*n + 3).

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A016887 a(n) = (5*n+3)^3.

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Magma

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A017125 Table read by antidiagonals of Fibonacci-type sequences.

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A093303 a(n) = a(n-1)*(2n-1) + 2n with a(0)=0.

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A187322 a(n) = floor(n/2) + floor(3*n/4).

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A001509 a(n) = (5n + 1)(5n + 2)(5*n + 3).