A001082 -id:A001082 - cp's OEIS frontend

A086500 Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.

1, 9, 18, 50, 75, 147, 196, 324, 405, 605, 726, 1014, 1183, 1575, 1800, 2312, 2601, 3249, 3610, 4410, 4851, 5819, 6348, 7500, 8125, 9477, 10206, 11774, 12615, 14415, 15376, 17424, 18513, 20825, 22050, 24642, 26011, 28899, 30420, 33620, 35301

Offset: 1

Amarnath Murthy, Jul 28 2003

The number of terms in the groups is given by A063196. i.e., 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, ...

Also the arithmetic mean of the n-th group is T(n), the n-th triangular number.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A001082, A022998, A063196, A181900.

a086500 n = a086500_list !! (n-1)
a086500_list = scanl1 (+) $ tail a181900_list
-- Reinhard Zumkeller, Mar 31 2012

Table[n*(n + 1)*(2*n + 1 + (-1)^n)/4, {n, 1, 40}] (* Amiram Eldar, Feb 22 2022 *)

Vec(x*(x^4+8*x^3+6*x^2+8*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 20 2014

a(n) = n*(n+1)*(2*n+1+(-1)^n)/4. - Wesley Ivan Hurt, Sep 19 2014
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>7. - Colin Barker, Sep 19 2014
G.f.: x*(x^4+8*x^3+6*x^2+8*x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 19 2014
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 4. (End)

More terms from Ray Chandler, Sep 17 2003

A104567 Triangle read by rows: T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i).

Original entry on oeis.org

1, 2, 2, 3, 4, 1, 4, 6, 2, 2, 5, 8, 3, 4, 1, 6, 10, 4, 6, 2, 2, 7, 12, 5, 8, 3, 4, 1, 8, 14, 6, 10, 4, 6, 2, 2, 9, 16, 7, 12, 5, 8, 3, 4, 1, 10, 18, 8, 14, 6, 10, 4, 6, 2, 2, 11, 20, 9, 16, 7, 12, 5, 8, 3, 4, 1, 12, 22, 10, 18, 8, 14, 6, 10, 4, 6, 2, 2, 13, 24, 11, 20, 9, 16, 7, 12, 5, 8, 3, 4, 1, 14

Offset: 1

Gary W. Adamson, Mar 16 2005

T(i,j) is the (i,j)-entry (1<=j<=i) of the product R*H of the infinite lower triangular matrices R = [1; 1,1; 1,1,1; 1,1,1,1; ...] and H = [1; 1,2; 1,2,1; 1 2,1,2; ...]. Row sums yield A006578. H*R yields A104566. - Emeric Deutsch, Mar 24 2005

			The first few rows are:
  1;
  2, 2;
  3, 4, 1;
  4, 6, 2, 2;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001082, A006578, A104566, A104568.

Cf. A006578, A104566.

T:=proc(i,j) if j>i then 0 elif j mod 2 = 1 then i-j+1 elif j mod 2 = 0 then 2*(i-j+1) else fi end: for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 24 2005

Table[If[OddQ[j],i-j+1,2(i-j+1)],{i,20},{j,i}]//Flatten (* Harvey P. Dale, Sep 03 2018 *)

T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i). - Emeric Deutsch, Mar 24 2005

More terms from Emeric Deutsch, Mar 24 2005

A118282 Conjectured largest number that is not the sum of three generalized n-gonal numbers, or -1 if there is no largest number.

Original entry on oeis.org

0, -1, 0, 0, 307, -1, 2027, 5200, 18180, -1, 10795, -1, 87740, -1, 75150, 212048, 122818, -1, 146970, 199153, 585513

Offset: 3

T. D. Noe, Apr 21 2006

sign

Extensive calculations show that if a(n)>=0, then every number greater than a(n) can be represented as the sum of three generalized n-gonal numbers. a(n)=0 for n=3 and 6 because generalized triangular and generalized hexagonal numbers are the same a triangular numbers and every number can be written as the sum of three triangular numbers. When n is a multiple of 4, there is an infinite set of numbers not representable. For n=14, there appears to be a sparse, but infinite, set of numbers not representable. See A118283 for the number of numbers not representable.

Cf. A001318 (generalized pentagonal numbers), A085787 (generalized heptagonal numbers), A001082 (generalized octagonal numbers), A118277 (generalized 9-gonal numbers), A118278-A118285.

A121628 Nonnegative k such that 3*k + 1 is a perfect cube.

Original entry on oeis.org

0, 21, 114, 333, 732, 1365, 2286, 3549, 5208, 7317, 9930, 13101, 16884, 21333, 26502, 32445, 39216, 46869, 55458, 65037, 75660, 87381, 100254, 114333, 129672, 146325, 164346, 183789, 204708, 227157, 251190, 276861, 304224, 333333, 364242, 397005

Offset: 1

Zak Seidov, Aug 12 2006

Intersection of this sequence and A001082 is {0, 21, 1365, 87381,...} all of the form (2^(6*m)-1)/3.

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

Cf. A001082: 3*m + 1 is a perfect square.

Cf. A287335 (see Crossrefs).

[3*n*(1+3*n+3*n^2): n in [1..40]]; // Vincenzo Librandi, Apr 11 2012

CoefficientList[Series[3 (7 + 10 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 11 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,21,114,333},40] (* Harvey P. Dale, Mar 08 2018 *)

a(n) = 3*(n - 1)*(3*n^2 - 3*n + 1) with n>0. Corresponding cubes are 3*a(n) + 1 = (3*n - 2)^3.

G.f.: 3*x^2*(7 + 10*x + x^2)/(1-x)^4. - Colin Barker, Apr 11 2012

0 added and b-file updated by Bruno Berselli, May 23 2017

A135556 Squares of numbers not divisible by 3: a(n) = A001651(n)^2.

Original entry on oeis.org

1, 4, 16, 25, 49, 64, 100, 121, 169, 196, 256, 289, 361, 400, 484, 529, 625, 676, 784, 841, 961, 1024, 1156, 1225, 1369, 1444, 1600, 1681, 1849, 1936, 2116, 2209, 2401, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3721, 3844, 4096, 4225, 4489, 4624, 4900

Offset: 1

Artur Jasinski, Nov 25 2007

From Fermat's Little Theorem all these numbers are congruent to 1 mod 3.
From Peter Bala, Jan 26 2025: (Start)
The sequence terms are the exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(3*n))*(1 - q^(18*n))^2/( (1 - q^(6*n))*(1 - q^(9*n)) ) = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + + - - ....
Also, the exponents of q in the expansion of q*Product_{n >= 1} (1 - q^(6*n))^5/(1 - q^(3*n))^2 = q + 2*q^4 - 4*q^16 - 5*q^25 + 7*q^49 + 8*q^64 -10*q^100 - 11*q^121 + + - - ....  See Lemke Oliver, Theorem 1.2. (End)

Colin Barker, Table of n, a(n) for n = 1..1000
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651, A001082, A214549.

Partial sums of A298028.

LinearRecurrence[{1,2,-2,-1,1}, {1,4,16,25,49}, 25] (* or *) Table[(18*n^2-6*(-1)^n*n-18*n+3*(-1)^n+5)/8, {n,1,25}] (* G. C. Greubel, Oct 19 2016 *)
Flatten[Partition[Range[70],2,3,{1,1},{}]]^2 (* Harvey P. Dale, Jun 19 2018 *)

isok(n) = issquare(n) && (n % 3 == 1); \\ Michel Marcus, Nov 02 2013

Vec(-x*(1+3*x+10*x^2+3*x^3+x^4) / ( (1+x)^2*(x-1)^3 ) + O(x^100)) \\ Colin Barker, Jan 26 2016

G.f.: -x*(1+3*x+10*x^2+3*x^3+x^4) / ((1+x)^2*(x-1)^3). - R. J. Mathar, Feb 16 2011
From Colin Barker, Jan 26 2016: (Start)
a(n) = (18*n^2-6*(-1)^n*n-18*n+3*(-1)^n+5)/8.
a(n) = (9*n^2-12*n+4)/4 for n even.
a(n) = (9*n^2-6*n+1)/4 for n odd. (End)
E.g.f.: (1/8)*( (3 + 6*x)*exp(-x) - 8 + (5 + 18*x^2)*exp(x)). - G. C. Greubel, Oct 19 2016
Sum_{n>=1} 1/a(n) = 4*Pi^2/27 (A214549). - Amiram Eldar, Dec 19 2020

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A255934 Number of ways to write n as the sum of four unordered generalized octagonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 3, 2, 3, 2, 1, 3, 3, 3, 3, 2, 4, 2, 1, 3, 2, 3, 4, 3, 4, 2, 2, 4, 4, 3, 4, 3, 6, 5, 2, 4, 3, 4, 5, 4, 6, 4, 1, 4, 5, 4, 5, 5, 7, 4, 1, 5, 5, 5, 6, 5, 8, 5, 3, 4, 6, 6, 6, 6, 7, 6, 3, 6, 6, 5, 6, 6, 10, 7, 1, 5, 8, 7, 7, 7, 8, 5, 3, 6, 7, 6, 8, 7, 10, 8, 3

Offset: 0

Zhi-Wei Sun, Mar 11 2015

nonn

I have proved that a(n) > 0 for all n, i.e., any nonnegative integer can be expressed as the sum of four generalized octagonal numbers. I can show that a(n) = 1 if 3*n+4 is among 7, 13, 19, 31, 43, 2^(2k), 5*2^(2k+1), 11*2^(2k+1), 23*2^(2k+1) (k = 0,1,2,...), and conjecture the converse.

I also conjecture that each nonnegative integer can be written as the sum of two heptagonal numbers, a second heptagonal number and a generalized heptagonal number.

			a(60) = 1 since 60 = 1*(3*1-2) + (-1)*(3*(-1)-2) + 3*(3*3-2) + (-3)*(3*(-3)-2).
a(1876) = 1 since 1876 = (-5)*(3*(-5)-2) + (-5)*(3*(-5)-2) + 11*(3*11-2) + (-21)*(3*(-21)-2).
a(15700) = 1 since 15700 = 11*(3*11-2) + (-21)*(3*(-21)-2) + 43*(3*43-2) + (-53)*(3*(-53)-2).
a(21844) = 1 since 21844 = 43*(3*43-2) + 43*(3*43-2) + 43*(3*43-2) + 43*(3*43-2).
a(30036) = 1 since 30036 = (-21)*(3*(-21)-2) + (-21)*(3*(-21)-2) + 43*(3*43-2) + (-85)*(3*(-85)-2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv:1503.03743 [math.NT], 2015.

Cf. A000566, A001082, A085787, A147875, A255350.

T[n_]:=Union[Table[x(3x-2),{x,-Floor[(Sqrt[3n+1]-1)/3],Floor[(Sqrt[3n+1]+1)/3]}]]
Do[r=0;Do[If[n-Part[T[n],x]-Part[T[n],y]-Part[T[n],z]

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 4*x^2 + 3*x^3 + 8*x^4 + 5*x^5 + 12*x^6 + 7*x^7 + 16*x^8 + ...

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

Cf. A001082, A005408, A007310, A008574, A040001, A215947.

CF. A257083 (partial sums), A246695.

import Data.List (transpose)
a257088 n = a257088_list !! n
a257088_list = concat $ transpose [a008574_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 2 n];
a[ n_] := SeriesCoefficient[ (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

PARI
```
{a(n) = if( n<1, n==0, n%2, n, 2*n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4) + x * O(x^n), n))};

Euler transform of length 4 sequence [ 1, 3, -1, -1].
a(n) is multiplicative with a(2^e) = 2^(e+1) if e>0, otherwise a(p^e) = p^e.
G.f.: (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
G.f.: (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^2)^3).
MOBIUS transform of A215947 is [1, 4, 3, 8, 5, ...].
a(n) = n * A040001(n) if n>0.
a(n) + a(n-1) = A007310(n) if n>0.
a(n) = A001082(n+1) - A001082(n) if n>0.
Binomial transform with a(0)=0 is A128543 if n>0.
a(2*n) = A008574(n). a(2*n + 1) = A005408(n).
a(n) = A022998(n) if n>0. - R. J. Mathar, Apr 19 2015
From Amiram Eldar, Jan 28 2025: (Start)
Dirichlet g.f.: (1+2^(1-s)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4) * n^2. (End)
a(n) = gcd(n^n, 2*n). - Mia Boudreau, Jun 27 2025

A271675 Numbers m such that 3*m + 4 is a square.

Original entry on oeis.org

0, 4, 7, 15, 20, 32, 39, 55, 64, 84, 95, 119, 132, 160, 175, 207, 224, 260, 279, 319, 340, 384, 407, 455, 480, 532, 559, 615, 644, 704, 735, 799, 832, 900, 935, 1007, 1044, 1120, 1159, 1239, 1280, 1364, 1407, 1495, 1540, 1632, 1679, 1775, 1824, 1924, 1975, 2079, 2132, 2240, 2295, 2407

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2016

7 is the unique prime in this sequence. If m is in this sequence, then 3*m + 4 = k^2 for k is nonzero integer, that is, m = (k^2 - 4)/3 = (k-2)*(k+2)/3. So m can be only prime if one of divisors is prime and another one is 1. Otherwise there should be more than 1 prime divisors, that is n must be composite. - Altug Alkan, Apr 12 2016
From Ray Chandler, Apr 12 2016: (Start)
Square roots of resulting squares gives A001651 (with a different starting point).
Sequence is the union of (positive terms) in A140676 and A270710. (End)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} q^n*(1 - q)*(1 - q^3)*...*(1 - q^(2*n+1)) = 1 - q^4 - q^7 + q^15 + q^20 - q^32 - q^50 + + - - .... - Peter Bala, Dec 19 2024

			a(4) = 32 because 3*32 + 4 = 100 = 10*10.

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

Cf. A001651, A140676, A270710.

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), this sequence (k=4), A100536 (k=6).

Magma
```
[n: n in [0..4000] | IsSquare(3*n+4)];
```

Select[Range[0,2500], IntegerQ@ Sqrt[3 # + 4] &] (* Michael De Vlieger, Apr 12 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,7,15,20},60] (* Harvey P. Dale, Dec 09 2016 *)

from gmpy2 import is_square
for n in range(0,10**5):
    if(is_square(3*n+4)):print(n)
# Soumil Mandal, Apr 12 2016

O.g.f.: x^2*(4 + 3*x - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: 1 + (1 - 2*x)*exp(-x)/8 - 3*(3 - 4*x - 2*x^2)*exp(x)/8.
a(n) = A001082(n+1) - 1 = (6*n*(n+1) + (2*n + 1)*(-1)^n - 1)/8 - 1. Therefore: a(2*k+1) = k*(3*k+4), a(2*k) = (k+1)*(3*k-1).
Sum_{n>=2} 1/a(n) = 19/16 - Pi/(4*sqrt(3)). - Amiram Eldar, Jul 26 2024

Edited and extended by Bruno Berselli, Apr 12 2016

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).

Original entry on oeis.org

1, 1, 5, 13, 34, 87, 212, 504, 1167, 2665, 5933, 13042, 28191, 60148, 126688, 263821, 543414, 1108272, 2239182, 4484482, 8907530, 17555485, 34345465, 66724969, 128772908, 246951514, 470738283, 892159198, 1681544803, 3152656375, 5880839454, 10916463171, 20169007200, 37095527149

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the generalized octagonal numbers (A001082).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A001082, A028377, A294838, A294839, A294840.

nmax = 33; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 2)) (1 + x^(2 k))^(k (3 k + 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A001082(k+1).

a(n) ~ exp(Pi/3 * (7/5)^(1/4) * 2^(3/4) * n^(3/4) + 9*Zeta(3) / (2*Pi^2) * sqrt(5*n/14) - (405*Zeta(3)^2 / (56*Pi^5) + Pi/48) * (10*n/7)^(1/4) + (6075*Zeta(3)^2 / (196*Pi^8) + 15/(224*Pi^2)) * Zeta(3)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

cp's OEIS Frontend

A086500 Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.

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A104567 Triangle read by rows: T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i).

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A118282 Conjectured largest number that is not the sum of three generalized n-gonal numbers, or -1 if there is no largest number.

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A121628 Nonnegative k such that 3*k + 1 is a perfect cube.

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A135556 Squares of numbers not divisible by 3: a(n) = A001651(n)^2.

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A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

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A255934 Number of ways to write n as the sum of four unordered generalized octagonal numbers.

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A257088 a(2*n) = 4*n if n>0, a(2*n + 1) = 2*n + 1, a(0) = 1.

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A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

A294841 Expansion of Product_{k>=1} (1 + x^(2k-1))^(k(3k-2))(1 + x^(2k))^(k(3*k+2)).