A017569 -id:A017569 - cp's OEIS frontend

A017573 a(n) = (12n+4)^5.

1024, 1048576, 17210368, 102400000, 380204032, 1073741824, 2535525376, 5277319168, 10000000000, 17623416832, 29316250624, 46525874176, 71008211968, 104857600000, 150536645632, 210906087424, 289254654976, 389328928768, 515363200000, 672109330432, 864866612224

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000584, A013663, A017569.

[(12*n+4)^5: n in [0..20]]; // Vincenzo Librandi, Feb 27 2014

A017573:=n->(12*n + 4)^5; seq(A017573(n), n=0..20); # Wesley Ivan Hurt, Feb 25 2014

Table[(12 n + 4)^5, {n, 0, 20}] (* Wesley Ivan Hurt, Feb 25 2014 *)
(12*Range[0,20]+4)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1024,1048576,17210368,102400000,380204032,1073741824},20] (* Harvey P. Dale, Apr 24 2017 *)

a(n) = (12n + 4)^5 = 1024(3n + 1)^5 = A000584(A017569(n)). - Wesley Ivan Hurt, Feb 25 2014

Sum_{n>=0} 1/a(n) = Pi^5/(373248*sqrt(3)) + 121*zeta(5)/248832. - Amiram Eldar, Jul 14 2024

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 4, 12, 14, 10, 5, 16, 21, 20, 13, 6, 20, 28, 30, 26, 16, 7, 24, 35, 40, 39, 32, 19, 8, 28, 42, 50, 52, 48, 38, 22, 9, 32, 49, 60, 65, 64, 57, 44, 25, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28, 11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, 12, 44, 70, 90, 104, 112, 114

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

The triangle is generated from the product A*B
of the infinite lower triangular matrices A =
1 0 0 0...
1 1 0 0...
1 1 1 0...
1 1 1 1...
... and B =
1 0 0 0...
1 4 0 0...
1 4 7 0...
1 4 7 10...
...
Row sums give pentagonal pyramidal numbers A002411 T(n+0,0)= 1*n=A000027(n) T(n+0,1)= 4*n=A008586(n) T(n+1,2)= 7*n=A008589(n) T(n+2,3)=10*n=A008592(n) ...
so for example T(n+1,n-0)=6*n+2=A016933(n) T(n+1,n-1)=9*n+3=A017197(n) T(n+2,n-1)=12*n+4=A017569(n)
T(n,0)*T(n,1) = A033996(n) (8 times triangular numbers)
T(n,n)*T(n,0) = A000567(n+1) (Octagonal numbers) etc.

			Triangle begins:
1,
2,  4,
3,  8,  7,
4,  12, 14, 10,
5,  16, 21, 20, 13,
6,  20, 28, 30, 26, 16,
7,  24, 35, 40, 39, 32, 19,
8,  28, 42, 50, 52, 48, 38, 22,
9,  32, 49, 60, 65, 64, 57, 44, 25,
10, 36, 56, 70, 78, 80, 76, 66, 50, 28,
11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, etc.
[_Bruno Berselli_, Feb 10 2014]

Cf. A095871 (product B*A), A002411.

t[n_, k_] := If[n < k, 0, (3*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 21 2005 *)

T(n,k)=if(k>n,0,(n-k+1)*(3*k+1)) for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A278814 a(n) = ceiling(sqrt(3n+1)).

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

Mohammad K. Azarian, Nov 28 2016

Cf. A002162, A016777, A016789, A016933, A017569, A032765, A058183, A131033, A007494, A051536, A007559.

PROG(y := [], n := 100, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(1 + 3·n)), y), n := n - 1))

seq(ceil(sqrt(3*k+1)), k=0..100); # Robert Israel, Nov 28 2016

Mathematica
```
Table[Ceiling[Sqrt[3n+1]],{n,0,100}]
```

a(n)=sqrtint(3*n)+1 \\ Charles R Greathouse IV, Nov 29 2016

from math import isqrt
def A278814(n): return 1+isqrt(3*n) # Chai Wah Wu, Jul 28 2022

a(n) = ceiling(sqrt(3n+1)).
From Robert Israel, Nov 28 2016: (Start)
G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(3*k^2)+x^(3*k^2+2*k+1)+x^(3*k^2+4*k+2)).
a(n+1) = a(n)+1 if n is in A032765, otherwise a(n+1) = a(n). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Jun 18 2025

A017575 a(n) = (12n+4)^7.

Original entry on oeis.org

16384, 268435456, 13492928512, 163840000000, 1028071702528, 4398046511104, 14645194571776, 40867559636992, 100000000000000, 221068140740608, 450766669594624, 860542568759296, 1555363874947072, 2684354560000000, 4453476124377088, 7140436495826944, 11112006825558016

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015, A013665, A016783, A017569.

a[n_] := (12*n + 4)^7; Array[a, 20, 0] (* Amiram Eldar, Jul 14 2024 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A001015(A017569(n)) = A017569(n)^7.
a(n) = 16384 * A016783(n).
Sum_{n>=0} 1/a(n) = 7*Pi^7/(403107840*sqrt(3)) + 1093*zeta(7)/35831808. (End)

More terms from Amiram Eldar, Jul 14 2024

A017577 a(n) = (12n+4)^9.

Original entry on oeis.org

262144, 68719476736, 10578455953408, 262144000000000, 2779905883635712, 18014398509481984, 84590643846578176, 316478381828866048, 1000000000000000000, 2773078757450186752, 6930988311686938624, 15916595351771938816, 34068690316840665088, 68719476736000000000

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001017, A013667, A016785, A017569.

(12*Range[0,30]+4)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{262144,68719476736,10578455953408,262144000000000,2779905883635712,18014398509481984,84590643846578176,316478381828866048,1000000000000000000,2773078757450186752},40] (* Harvey P. Dale, Sep 07 2018 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A001017(A017569(n)) = A017569(n)^9.
a(n) = 262144 * A016785(n).
Sum_{n>=0} 1/a(n) = 809*Pi^9/(7313988648960*sqrt(3)) + 9841*zeta(9)/5159780352. (End)

More terms from Amiram Eldar, Jul 14 2024

A017579 a(n) = (12n+4)^11.

Original entry on oeis.org

4194304, 17592186044416, 8293509467471872, 419430400000000000, 7516865509350965248, 73786976294838206464, 488595558857835544576, 2450808588882738675712, 10000000000000000000000, 34785499933455142617088, 106570876280498368282624, 294393347626373780340736

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A008455, A013669, A016787, A017569.

(12*Range[0,20]+4)^11 (* Harvey P. Dale, Jan 07 2015 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A008455(A017569(n)) = A017569(n)^11.
a(n) = 4194304 * A016787(n).
Sum_{n>=0} 1/a(n) = 1847*Pi^11/(2633035913625600*sqrt(3)) + 88573*zeta(11)/743008370688. (End)

More terms from Amiram Eldar, Jul 14 2024

A360927 Expansion of the g.f. x(1 + 3x + 4x^2 + 4x^3)/((1 - x)^2*(1 + x)).

Original entry on oeis.org

0, 1, 4, 9, 16, 21, 28, 33, 40, 45, 52, 57, 64, 69, 76, 81, 88, 93, 100, 105, 112, 117, 124, 129, 136, 141, 148, 153, 160, 165, 172, 177, 184, 189, 196, 201, 208, 213, 220, 225, 232, 237, 244, 249, 256, 261, 268, 273, 280, 285, 292, 297, 304, 309, 316, 321, 328

Offset: 0

Stefano Spezia, Feb 25 2023

The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the perimeter and the two diagonals of the square.

			Illustrations for n = 1..8:
      o          o o          o o o
                 o o          o o o
                              o o o
  a(1) = 1    a(2) = 4      a(3) = 9
   o o o o    o o o o o    o o o o o o
   o o o o    o o   o o    o o     o o
   o o o o    o   o   o    o   o o   o
   o o o o    o o   o o    o   o o   o
              o o o o o    o o     o o
                           o o o o o o
  a(4) = 16   a(5) = 21     a(6) = 28
   o o o o o o o       o o o o o o o o
   o o       o o       o o         o o
   o   o   o   o       o   o     o   o
   o     o     o       o     o o     o
   o   o   o   o       o     o o     o
   o o       o o       o   o     o   o
   o o o o o o o       o o         o o
                       o o o o o o o o
     a(7) = 33            a(8) = 40

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

Cf. A194274, A016777, A017569, A017629.

LinearRecurrence[{1,1,-1},{0,1,4,9,16},57]

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.
a(0) = 0, a(1) = 1, a(n) = 6*n - 8 for n even, and a(n) = 6*n - 9 for n odd.
E.g.f.: 4*(x + 2) + (6*x - 8)*cosh(x) + (6*x - 9)*sinh(x).
a(2*n) = A017569(n-1) = 4*A016777(n-1).
a(2*n+1) = A017629(n-1).

cp's OEIS Frontend

A017573 a(n) = (12n+4)^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A101468 Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A278814 a(n) = ceiling(sqrt(3n+1)).

Views

Author

Keywords

Crossrefs

Programs

Derive

Maple

Mathematica

PARI

Python

Formula

A017575 a(n) = (12n+4)^7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A017577 a(n) = (12n+4)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A017579 a(n) = (12n+4)^11.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A360927 Expansion of the g.f. x*(1 + 3*x + 4*x^2 + 4*x^3)/((1 - x)^2*(1 + x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).

A360927 Expansion of the g.f. x(1 + 3x + 4x^2 + 4x^3)/((1 - x)^2*(1 + x)).