A194274 -id:A194274 - cp's OEIS frontend

A077221 a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.

0, 1, 8, 17, 32, 49, 72, 97, 128, 161, 200, 241, 288, 337, 392, 449, 512, 577, 648, 721, 800, 881, 968, 1057, 1152, 1249, 1352, 1457, 1568, 1681, 1800, 1921, 2048, 2177, 2312, 2449, 2592, 2737, 2888, 3041, 3200, 3361, 3528, 3697, 3872, 4049, 4232

Offset: 0

Amarnath Murthy, Nov 03 2002

This sequence arises from reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the triangular numbers A000217. Cf. A139591, etc. - Omar E. Pol, May 03 2008
The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=2, thus a(k) = |(P(2,0)-(-1)^k*P(2,2*k))/2|. - Peter Luschny, Jul 12 2009
Axis perpendicular to A046092 in the square spiral whose vertices are the triangular numbers A000217. See the comment above. - Omar E. Pol, Sep 14 2011
Column 8 of A195040. - Omar E. Pol, Sep 28 2011
Concentric octagonal numbers. A139098 and A069129 interleaved. - Omar E. Pol, Sep 17 2011
Subsequence of A194274. - Bruno Berselli, Sep 22 2011
Partial sums of A047522. - Reinhard Zumkeller, Jan 07 2012
Alternating sum of the first n odd squares in decreasing order, n >= 1. Also number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton. The rules are: on the infinite square grid, start with all cells OFF, so a(0) = 0. Turn a single cell to the ON state, so a(1) = 1. At each subsequent step, the neighbor cells of each cell of the old generation are turned ON, and the cells of the old generation are turned OFF. Here "neighbor" refers to the eight adjacent cells of each ON cell. See example. - Omar E. Pol, Feb 16 2014

			From _Omar E. Pol_, Feb 16 2014: (Start)
Illustration of initial terms as a cellular automaton:
.
.                                   O O O O O O O
.                     O O O O O     O           O
.           O O O     O       O     O   O O O   O
.     O     O   O     O   O   O     O   O   O   O
.           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       8           17              32
.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, An origin of A077221 (illustration) (see Pol's comment).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to cellular automata.

Cf. A032528, A069129, A077222, A131925, A139098, A194274, A195040, A195041, A195042, A195142.

a077221 n = a077221_list !! n
a077221_list = scanl (+) 0 a047522_list
-- Reinhard Zumkeller, Jan 07 2012

[2*n^2 - (n mod 2): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011

a := n -> 2*n^2 - (n mod 2); # Peter Luschny, Jul 12 2009

a=1;lst={a};Do[b=n^2-a;AppendTo[lst,b];a=b,{n,3,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, May 18 2009 *)

a(2n) = 8*n^2, a(2n+1) = 8*n(n+1) + 1.
From Ralf Stephan, Mar 31 2003: (Start)
a(n) = 2*n^2 + 4*n + 1 [+1 if n is odd] with a(0)=1.
G.f.: x*(x^2+6*x+1)/(1-x)^3/(1+x). (End)
Row sums of triangle A131925; binomial transform of (1, 7, 2, 4, -8, 16, -32, ...). - Gary W. Adamson, Jul 29 2007
a(n) = a(-n); a(n+1) = A195605(n) - (-1)^n. - Bruno Berselli, Sep 22 2011
a(n) = 2*n^2 + ((-1)^n-1)/2. - Omar E. Pol, Sep 28 2011
Sum_{n>=1} 1/a(n) = Pi^2/48 + tan(Pi/(2*sqrt(2)))*Pi /(4*sqrt(2)). - Amiram Eldar, Jan 16 2023

Extended by Ralf Stephan, Mar 31 2003

A085250 4 times hexagonal numbers: a(n) = 4n(2*n-1).

Original entry on oeis.org

0, 4, 24, 60, 112, 180, 264, 364, 480, 612, 760, 924, 1104, 1300, 1512, 1740, 1984, 2244, 2520, 2812, 3120, 3444, 3784, 4140, 4512, 4900, 5304, 5724, 6160, 6612, 7080, 7564, 8064, 8580, 9112, 9660, 10224, 10804, 11400, 12012, 12640, 13284

Offset: 0

Gary W. Adamson, Jun 23 2003

a(n) also can represented as n concentric squares (see example). - Omar E. Pol, Aug 21 2011

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric squares:
.
.                           o o o o o o o o o o
.                           o                 o
.            o o o o o o    o   o o o o o o   o
.            o         o    o   o         o   o
.     o o    o   o o   o    o   o   o o   o   o
.     o o    o   o o   o    o   o   o o   o   o
.            o         o    o   o         o   o
.            o o o o o o    o   o o o o o o   o
.                           o                 o
.                           o o o o o o o o o o
.
.      4          24                 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. A067239, A000384, A033581, A046092, A152734, A152751, A194274.

Table[8*n^2 - 4*n, {n, 0, 50}] (* G. C. Greubel, Jul 14 2017 *)
4 PolygonalNumber[6,Range[0,50]] (* Harvey P. Dale, Oct 19 2022 *)

a(n)=4*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A067239(n)/2, for n>0.
Sum_{n>0} 1/a(n) = log(2)/2.
a(n) = A000384(n)*4. - Omar E. Pol, Dec 11 2008
a(n) = 16*n + a(n-1) - 12 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010
G.f.: 4*x*(1 + 3*x)/(1 - 3*x + 3*x^2 - x^3). - Colin Barker, Jan 04 2012
E.g.f.: 4*x*(2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A046092(2n-1), for n > 0. - Bruce J. Nicholson, Sep 04 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 - log(2)/4. - Amiram Eldar, Mar 17 2022

Edited by Don Reble, Nov 13 2005

Added zero, better definition, corrected offset and edited original formula. - Omar E. Pol, Dec 11 2008

A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

Original entry on oeis.org

1, 4, 1, 9, 4, 1, 16, 9, 4, 1, 25, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 49, 36, 25, 16, 9, 4, 1, 64, 49, 36, 25, 16, 9, 4, 1, 81, 64, 49, 36, 25, 16, 9, 4, 1, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1

Offset: 1

Henry Bottomley, Jun 26 2000

			From _Omar E. Pol_, Jan 26 2014: (Start)
Triangle begins:
    1;
    4,  1;
    9,  4,  1;
   16,  9,  4,  1;
   25, 16,  9,  4,  1;
   36, 25, 16,  9,  4,  1;
   49, 36, 25, 16,  9,  4,  1;
   64, 49, 36, 25, 16,  9,  4,  1;
   81, 64, 49, 36, 25, 16,  9,  4,  1;
  100, 81, 64, 49, 36, 25, 16,  9,  4,  1;
  ...
For n = 7 the row sum is 49 + 36 + 25 + 16 + 9 + 4 + 1 = A000330(7) = 140.
The alternating row sum is 49 - 36 + 25 - 16 + 9 - 4 + 1 = A000217(7) = 28.
(End)

Robert Israel, Rows n = 1..141, flattened

Cf. A000290, A000292, A004736, A129371, A194274.

Cf. A000217 (alternating row sums), A000330 (row sums).

[(n-k)^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Jan 31 2024

for n from 1 to 10 do
  seq((n-k)^2, k=0..n-1)
od; # Robert Israel, Jan 18 2018

Table[Range[n,1,-1]^2,{n,20}]//Flatten (* Harvey P. Dale, Apr 17 2020 *)

flatten([[(n-k)^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Jan 31 2024

a(n) = A004736(n)^2.
Sum_{k=0..n-1} T(n, k) = A000330(n) (row sums). - Michel Marcus, Dec 31 2012
G.f. as triangle: x*(1+x)/((1-x*y)*(1-x)^3). - Robert Israel, Jan 18 2018
Sum_{k=0..n-1} (-1)^k*T(n, k) = A000217(n) (alternating row sums). - Omar E. Pol, Jan 24 2014
From G. C. Greubel, Jan 31 2024: (Start)
T(2*n-1, n-1) = A000290(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000292(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A194274(n).
Sum_{k=0..floor(n/2)} T(n, k) = A129371(n). (End)

A187093 a(0)=0, a(1)=a(2)=1; thereafter, a(n+1) = n^2 - a(n-1).

Original entry on oeis.org

0, 1, 1, 3, 8, 13, 17, 23, 32, 41, 49, 59, 72, 85, 97, 111, 128, 145, 161, 179, 200, 221, 241, 263, 288, 313, 337, 363, 392, 421, 449, 479, 512, 545, 577, 611, 648, 685, 721, 759, 800, 841, 881, 923, 968, 1013, 1057, 1103, 1152, 1201, 1249, 1299, 1352, 1405, 1457

Offset: 0

Benjamin Coinsin, Mar 04 2011

nonn

The original definition was equivalent to: Let S(n) = sum_{i=0..n} a(i), then n^2+a(n)-S(n+1) = S(n-2). This in turn simplifies to the present definition.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A194274, A081654.

A000034 := proc(n) op(1+(n mod 2),[1,2]) ; end proc:
A187093 := proc(n) (n^2-1+(-1)^floor(n/2)*A000034(n))/2 ;end proc: # R. J. Mathar

LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 1, 3, 8}, 60] (* Jean-François Alcover, Mar 30 2020 *)
Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[n+1]==n^2-a[n-1]},a,{n,60}]] (* Harvey P. Dale, Jan 05 2023 *)

a(n) = (n^2-1+(-1)^(n\2)*(1 + (n % 2)))/2; \\ Michel Marcus, Sep 11 2016

print(0, end=',')       # a(-1)=0
prpr = prev = 1         # a(0)=a(1)=1
for n in range(2, 77):
    print(prpr, end=',')
    curr = n*n - prpr   # a(n) = n^2 - a(n-2)
    prpr = prev
    prev = curr
# from Alex Ratushnyak, Aug 05 2012

a(n) = (n^2 - 1 + (-1)^floor(n/2) * A000034(n))/2.
G.f.: x*(-1+2*x+x^3-4*x^2) / ( (x^2+1)*(x-1)^3 ).
a(2^(n+1)) = A081654(n). - Anton Zakharov, Sep 13 2016

Edited by N. J. A. Sloane, Mar 09 2011

More terms from Alex Ratushnyak, Aug 05 2012

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)).

A215097 a(n) = n^3 - a(n-2) for n >= 2 and a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 8, 26, 56, 99, 160, 244, 352, 485, 648, 846, 1080, 1351, 1664, 2024, 2432, 2889, 3400, 3970, 4600, 5291, 6048, 6876, 7776, 8749, 9800, 10934, 12152, 13455, 14848, 16336, 17920, 19601, 21384, 23274, 25272, 27379, 29600, 31940, 34400, 36981, 39688, 42526

Offset: 0

Alex Ratushnyak, Aug 03 2012

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

Cf. A000217 (n^2 - a(n-1)).
Cf. A125577 (n^2 - a(n-1) with a(0)=1).
Cf. A011934 (n^3 - a(n-1)).
Cf. A153026 (n^3 - a(n-1) with a(1)=0).
Cf. A194274 (n^2 - a(n-2)).
Cf. A187093 (n^2 - a(n-2) with a(0)=a(1)=1, a(-1)=0).
Cf. A107386 ((n-2)^2 - a(n-1) with a(0)=0, a(1)=a(2)=1, a(3)=2).
Cf. A206481 ((n-1)^3 - a(n-2)).

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == n^3 - a[n - 2]}, a[n], {n, 0, 43}] (* Bruno Berselli, Aug 07 2012 *)

prpr = 0
prev = 1
for n in range(2,77):
    print(prpr, end=',')
    curr = n*n*n - prpr
    prpr = prev
    prev = curr

G.f.: (x+4*x^2+x^3)/((-1+x)^4*(1+x^2)). - David Scambler, Aug 06 2012

a(n) = (n*(n^2-3)-(1-(-1)^n)*i^(n+1))/2, where i=sqrt(-1). - Bruno Berselli, Aug 07 2012

A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 18, 24, 25, 29, 34, 38, 39, 41, 44, 48, 53, 55, 56, 58, 71, 73, 78, 84, 85, 89, 94, 102, 103, 109, 120, 124, 131, 133, 138, 144, 145, 149, 162, 164, 169, 173, 178, 180, 181, 187, 192, 196, 197, 201

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

For n>0 and (n mod 4)<2, a(n) is odd.
Same definition, but k+a(n-2) is a
Fibonacci number: A006498 except first two terms,
Lucas number: A000045 except first two terms,
Pell number: A089928(n-1),
Jacobsthal number: A215095,
factorial: A215096,
square: A194274,
cube: A215097,
triangular number: A011848(n+2),
oblong number: A215098.
Example of a related sequence definition: a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a cube.

Iain Fox, Table of n, a(n) for n = 0..10000

Cf. A062042: a(1) = 2, a(n) = least k>a(n-1) such that k+a(n-1) is a prime.

Cf. A073627, A073628.

first(n) = my(res = vector(n, i, i-1), k); for(x=3, n, k=res[x-1]+1; while(!isprime(k+res[x-2]), k++); res[x]=k); res \\ Iain Fox, Apr 22 2019 (corrected by Iain Fox, Apr 25 2019)

from sympy import prime
prpr = 0
prev = 1
for n in range(77):
    print(prpr, end=', ')
    b = c = 0
    while c<=prev:
        c = prime(b+1) - prpr
        b+=1
    prpr = prev
    prev = c

A215095 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a Jacobsthal number.

Original entry on oeis.org

0, 1, 3, 4, 8, 17, 35, 68, 136, 273, 547, 1092, 2184, 4369, 8739, 17476, 34952, 69905, 139811, 279620, 559240, 1118481, 2236963, 4473924, 8947848, 17895697, 35791395, 71582788, 143165576, 286331153, 572662307, 1145324612, 2290649224, 4581298449, 9162596899

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

Same definition, but k+a(n-2) is a
Fibonacci number: A006498 except first two terms,
Lucas number: A000045 except first two terms,
Pell number: A089928(n-1),
factorial: A215096,
square: A194274,
cube: A215097,
triangular number: A011848(n+2),
oblong number: A215098,
prime number: A215099.
Example of a related sequence definition: a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a cube.

Cf. A001045, A006498, A089928, A000045.

prpr = 0
prev = 1
jac = [0]*10000
for n in range(10000):
    jac[n] = prpr
    curr = prpr*2 + prev
    prpr = prev
    prev = curr
prpr, prev = 0, 1
for n in range(1, 44):
    print(prpr, end=', ')
    b = c = 0
    while c<=prev:
        c = jac[b] - prpr
        b+=1
    prpr = prev
    prev = c

Conjecture: G.f. (x+2*x^2)/(1-x-x^2-x^3-2*x^4). - David Scambler, Aug 06 2012

A363518 Concentric square numbers on the faces of an n X n X n cube.

Original entry on oeis.org

1, 8, 20, 32, 50, 80, 116, 152, 194, 248, 308, 368, 434, 512, 596, 680, 770, 872, 980, 1088, 1202, 1328, 1460, 1592, 1730, 1880, 2036, 2192, 2354, 2528, 2708, 2888, 3074, 3272, 3476, 3680, 3890, 4112, 4340, 4568, 4802, 5048, 5300, 5552, 5810, 6080, 6356, 6632, 6914, 7208, 7508, 7808

Offset: 1

Nicolay Avilov, Jun 07 2023

a(n) is the number of colored cubes in the outer layer of a cube made up of n^3 unit cubes. The cubes are painted in such a way that concentric square numbers are obtained on each face of the n X n X n cube.

			a(3) = 6*8 - 12*1 - 2*8 = 20;
a(5) = 6*17 - 12*3 - 2*8 = 50.

Nicolay Avilov, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A194274.

Join[{1},LinearRecurrence[{3,-4,4,-3,1},{8,20,32,50,80},51]] (* Stefano Spezia, Jun 08 2023 *)

def A363518(n): return 6*((3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2))-12*n+8 if n>1 else 1 # Chai Wah Wu, Jul 15 2023

a(n) = 6*A194274 - 12*n + 8, where n>1.
From Stefano Spezia, Jun 08 2023: (Start)
G.f.: (1 + 5*x + 5*x^4 + x^5)/((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3)- 3*a(n-4) + a(n-5) for n > 6. (End)

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

A077221 a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.

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A085250 4 times hexagonal numbers: a(n) = 4*n*(2*n-1).

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A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

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A187093 a(0)=0, a(1)=a(2)=1; thereafter, a(n+1) = n^2 - a(n-1).

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

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A215097 a(n) = n^3 - a(n-2) for n >= 2 and a(0)=0, a(1)=1.

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A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

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A085250 4 times hexagonal numbers: a(n) = 4n(2*n-1).

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