A194275 -id:A194275 - cp's OEIS frontend

A152734 5 times pentagonal numbers: 5n(3*n-1)/2.

0, 5, 25, 60, 110, 175, 255, 350, 460, 585, 725, 880, 1050, 1235, 1435, 1650, 1880, 2125, 2385, 2660, 2950, 3255, 3575, 3910, 4260, 4625, 5005, 5400, 5810, 6235, 6675, 7130, 7600, 8085, 8585, 9100, 9630, 10175, 10735, 11310, 11900, 12505, 13125, 13760, 14410

Offset: 0

Omar E. Pol, Dec 11 2008

a(n) can be represented as a figurate number using n concentric pentagons (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 22 2011 (Start):
Illustration of initial terms as concentric pentagons (in a precise representation the pentagons should be strictly concentric):
.
.                                          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 o o o o     o
.                                 o                 o
.                                  o               o
.                                   o o o o o o o o
.
.  5             25                        60
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A033581, A085250, A152751, A194275.

Cf. sequences of the form n*(d*n+10-d)/2 indexed in A140090.

[5*n*(3*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 19 2014

A152734:=n->5*n*(3*n-1)/2: seq(A152734(n), n=0..50); # Wesley Ivan Hurt, Sep 19 2014

Table[5 n (3 n - 1)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 19 2014 *)
5*PolygonalNumber[5,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 13 2020 *)

a(n)=5*n*(3*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*A000326(n).
a(n) = a(n-1)+15*n-10 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: 5*x*(1+2*x)/(1-x)^3. a(n) = 4*A000217(n)+A051865(n). - Bruno Berselli, Feb 11 2011
E.g.f.: (5/2)*(3*x^2 + 2*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi- 6*log(2))/15. (End)

A194274 Concentric square numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, 72, 84, 97, 112, 128, 144, 161, 180, 200, 220, 241, 264, 288, 312, 337, 364, 392, 420, 449, 480, 512, 544, 577, 612, 648, 684, 721, 760, 800, 840, 881, 924, 968, 1012, 1057, 1104, 1152, 1200, 1249, 1300, 1352, 1404

Offset: 0

Omar E. Pol, Aug 20 2011

Cellular automaton on the first quadrant of the square grid. The sequence gives the number of cells "ON" in the structure after n-th stage. A098181 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194273, A194275 and A032528.
Also, union of A046092 and A077221, the bisections of this sequence.
Also row sums of an infinite square array T(n,k) in which column k lists 4*k-1 zeros followed by the numbers A008574 (see example).

			Using the numbers A008574 we can write:
0, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
0, 0, 0, 0, 0,  1,   4,  8, 12, 16, 20, ...
0, 0, 0, 0, 0,  0,   0,  0,  0,  1,  4, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, ...
...
Illustration of initial terms:
.                                         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
.
. 1    4      8        12         17           24

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A000290, A005563, A008574, A056594, A032528, A046092.

Cf. A077221, A085250, A098181, A194273, A194275.

[n le 2 select n-1 else (n-1)^2 - Self(n-2): n in [1..61]]; // G. C. Greubel, Jan 31 2024

Table[Floor[3*n/4] + Floor[(n*(n + 2) + 1)/2] - Floor[(3*n + 1)/4], {n, 0, 52}] (* Arkadiusz Wesolowski, Nov 08 2011 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==n^2-a[n-2]},a,{n,60}] (* or *) LinearRecurrence[{3,-4,4,-3,1},{0,1,4,8,12},60] (* Harvey P. Dale, Sep 11 2013 *)

prpr = 0
prev = 1
for n in range(2,777):
    print(str(prpr), end=", ")
    curr = n*n - prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 03 2012

def A194274(n): return (3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2) # Chai Wah Wu, Jul 15 2023

def A194274(n): return n if n<2 else n^2 - A194274(n-2)
[A194274(n) for n in range(41)] # G. C. Greubel, Jan 31 2024

a(n) = n^2 - a(n-2), with a(0)=0, a(1)=1. - Alex Ratushnyak, Aug 03 2012
From R. J. Mathar, Aug 22 2011: (Start)
G.f.: x*(1 + x)/((1 + x^2)*(1 - x)^3).
a(n) = (A005563(n) - A056594(n-1))/2. (End)
a(n) = a(-n-2) = (2*n*(n+2) + (1-(-1)^n)*i^(n+1))/4, where i=sqrt(-1). - Bruno Berselli, Sep 22 2011
a(n) = floor(3*n/4) + floor((n*(n+2)+1)/2) - floor((3*n+1)/4). - Arkadiusz Wesolowski, Nov 08 2011
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=4, a(3)=8, a(4)=12. - Harvey P. Dale, Sep 11 2013
E.g.f.: (exp(x)*x*(3 + x) - sin(x))/2. - Stefano Spezia, Feb 26 2023

A281026 a(n) = floor(3n(n+1)/4).

Original entry on oeis.org

0, 1, 4, 9, 15, 22, 31, 42, 54, 67, 82, 99, 117, 136, 157, 180, 204, 229, 256, 285, 315, 346, 379, 414, 450, 487, 526, 567, 609, 652, 697, 744, 792, 841, 892, 945, 999, 1054, 1111, 1170, 1230, 1291, 1354, 1419, 1485, 1552, 1621, 1692, 1764, 1837, 1912, 1989, 2067, 2146

Offset: 0

Bruno Berselli, Jan 13 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of the initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Subsequence of A214068.
Partial sums of A047273.
Cf. A011865, A045943, A274757 (subsequence).
Cf. sequences with formula floor(k*n*(n+1)/4): A011848 (k=1), A000217 (k=2), this sequence (k=3), A002378 (k=4).
Cf. sequences with formula floor(k*n*(n+1)/(k+1)): A000217 (k=1), A143978 (k=2), this sequence (k=3), A281151 (k=4), A194275 (k=5).

Magma
```
[3*n*(n+1) div 4: n in [0..60]];
```

A281026:=n->floor(3*n*(n+1)/4): seq(A281026(n), n=0..100); # Wesley Ivan Hurt, Jan 13 2017

Table[Floor[3 n (n + 1)/4], {n, 0, 60}]
LinearRecurrence[{3,-4,4,-3,1},{0,1,4,9,15},60] (* Harvey P. Dale, Jun 04 2023 *)

makelist(floor(3*n*(n+1)/4), n, 0, 60);

PARI
```
vector(60, n, n--; floor(3*n*(n+1)/4))
```
Python
```
[int(3*n*(n+1)/4) for n in range(60)]
```

[floor(3*n*(n+1)/4) for n in range(60)]

O.g.f.: x*(1 + x + x^2)/((1 + x^2)*(1 - x)^3).
E.g.f.: -(1 - 6*x - 3*x^2)*exp(x)/4 - (1 + i)*(i - exp(2*i*x))*exp(-i*x)/8, where i=sqrt(-1).
a(n) = a(-n-1) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) = a(n-4) + 6*n - 9.
a(n) = 3*n*(n+1)/4 + (i^(n*(n+1)) - 1)/4. Therefore:
a(4*k+r) = 12*k^2 + 3*(2*r+1)*k + r^2, where 0 <= r <= 3.
a(n) = n^2 - floor((n-1)*(n-2)/4).
a(n) = A011865(3*n+2).

A194273 Concentric triangular numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55, 63, 72, 81, 90, 99, 109, 120, 132, 144, 156, 168, 181, 195, 210, 225, 240, 255, 271, 288, 306, 324, 342, 360, 379, 399, 420, 441, 462, 483, 505, 528, 552, 576, 600, 624, 649, 675, 702, 729, 756, 783, 811

Offset: 0

Omar E. Pol, Aug 20 2011

This can be interpreted as a cellular automaton on the infinite hexagonal net. The sequence gives the number of cells "ON" in the structure after n-th stage. A194272 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194274, A194275 and A032528.

Also, row sums of an infinite square array T(n,k) in which column k lists 6*k-1 zeros followed by the numbers A008486 (see example).

			Using the numbers A008486 we can write:
0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
0, 0, 0, 0, 0,  0,  0,  1,  3,  6,  9, 12, 15, 18,...
0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,...
And so on.
=========================================================
The sums of the columns give this sequence:
0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55,...
...
Illustration of initial terms:
.                                              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
.
. 1    3      6        9          12           15
.
.                                           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
.
.       19               24                 30

Index entries for sequences related to cellular automata

Cf. A000217, A008486, A032528, A069131, A152751, A194272, A194274, A194275.

G.f.: x/(1-3*x+3*x^2-3*x^4+3*x^5-x^6) = x/((1-x)^3*(1+x)*(1-x+x^2)).

A330451 a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.

Original entry on oeis.org

0, 3, 13, 30, 53, 83, 120, 163, 213, 270, 333, 403, 480, 563, 653, 750, 853, 963, 1080, 1203, 1333, 1470, 1613, 1763, 1920, 2083, 2253, 2430, 2613, 2803, 3000, 3203, 3413, 3630, 3853, 4083, 4320, 4563, 4813, 5070

Offset: 0

Paul Curtz, Mar 01 2020

Main N-S vertical in the pentagonal spiral for A002264:
                      16
                  16  10  10
              16   9   5   5  10
          15   9   4   1   2   5  11
      15   9   4   1   0   0   2   6  11
        15   8   4   1   0   2   6  11
          14   8   3   3   3   6  12
            14   8   7   7   7  12
              14  13  13  13  12
The main S-N vertical is A194275.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A002264, abs(A084103(n+3)), A194275, A244636.

Cf. A049347.

Table[2/9(-1+15n^2+Cos[2n*Pi/3]),{n,0,39}] (* Stefano Spezia, Mar 02 2020 *)

concat(0, Vec(x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Mar 02 2020

def A330451(n): return 10*n**2//3 # Chai Wah Wu, Aug 12 2025

G.f.: x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Mar 02 2020
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = (2/9)*(-1 + 15*n^2 + cos(2*n*Pi/3)). - Stefano Spezia, Mar 02 2020
a(3*n) = 30*n^2.
a(n) = floor(10*n^2/3). - Chai Wah Wu, Aug 12 2025

A281151 a(n) = floor(4n(n+1)/5).

Original entry on oeis.org

0, 1, 4, 9, 16, 24, 33, 44, 57, 72, 88, 105, 124, 145, 168, 192, 217, 244, 273, 304, 336, 369, 404, 441, 480, 520, 561, 604, 649, 696, 744, 793, 844, 897, 952, 1008, 1065, 1124, 1185, 1248, 1312, 1377, 1444, 1513, 1584, 1656, 1729, 1804, 1881, 1960, 2040, 2121, 2204, 2289

Offset: 0

Bruno Berselli, Jan 16 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Subsequence of A047462.
Partial sums of A047486.
Cf. A184005: n^2 - floor((n-2)^2/4).
Cf. sequences with formula floor(k*n*(n+1)/(k+1)): A000217 (k=1), A143978 (k=2), A281026 (k=3), this sequence (k=4), A194275 (k=5).

Magma
```
[4*n*(n+1) div 5: n in [0..60]];
```
Mathematica
```
Table[Floor[4 n (n + 1)/5], {n, 0, 60}]
```

makelist(floor(4*n*(n+1)/5), n, 0, 60);

PARI
```
vector(60, n, n--; floor(4*n*(n+1)/5))
```
Python
```
[int(4*n*(n+1)/5) for n in range(60)]
```

[floor(4*n*(n+1)/5) for n in range(60)]

O.g.f.: x*(1 + x^2)*(1 + x)^2/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n-1) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) = a(n-5) + 8*(n-2).
a(5*k+r) = 20*k^2 + 4*(2*r+1)*k + r^2, where 0 <= r <= 4. Example: for r=3, a(5*k+3) = (2*k+1)*(10*k+9), which gives: 9, 57, 145, 273, 441, 649 etc. Also, a(n) belongs to A047462, in fact: for r = 0 or 4, a(n) == 0 (mod 8); for r = 1 or 3, a(n) == 1 (mod 8); for r = 2, a(n) == 4 (mod 8).
a(n) = a(-n) + A047462(n).
a(n) = n^2 - floor((n-2)^2/5).

cp's OEIS Frontend

A152734 5 times pentagonal numbers: 5*n*(3*n-1)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A194274 Concentric square numbers (see Comments lines for definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Python

SageMath

Formula

A281026 a(n) = floor(3*n*(n+1)/4).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula

A194273 Concentric triangular numbers (see Comments lines for definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A330451 a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A281151 a(n) = floor(4*n*(n+1)/5).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

A152734 5 times pentagonal numbers: 5n(3*n-1)/2.

A281026 a(n) = floor(3n(n+1)/4).

A281151 a(n) = floor(4n(n+1)/5).