A001504 -id:A001504 - cp's OEIS frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316

Offset: 1

Henry Bottomley, Apr 02 2001

Triangular numbers not == 0 (mod 3). - Amarnath Murthy, Nov 13 2005
Shallow diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003
Equals the triangular numbers convolved with [1, 7, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
a(n) is congruent to 1 (mod 9) for all n. The sequence of digital roots of the a(n) is A000012(n). The sequence of units' digits of the a(n) is period 20: repeat [1, 0, 8, 5, 1, 6, 0, 3, 5, 6, 6, 5, 3, 0, 6, 1, 5, 8, 0, 1]. - Ant King, Jun 18 2012
Divide each side of any triangle ABC with area (ABC) into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_(2n) on side a, and similarly for sides b and c. If the hexagon with area (Hex(n)) delimited by AA_n, AA_(n+1), BB_n, BB_(n+1), CC_n and CC_(n+1) cevians, we have a(n+1) = (ABC)/(Hex(n)) for n >= 1, (see link with java applet). - Ignacio Larrosa Cañestro, Jan 02 2015; edited by Wolfdieter Lang, Jan 30 2015
For the case n = 1 see the link for Marion's Theorem (actually Marion Walter's Theorem, see the Cugo et al, reference). Also, the generalization considered here has been called there (Ryan) Morgan's Theorem.  - Wolfdieter Lang, Jan 30 2015
Pollock states that every number is the sum of at most 11 terms of this sequence, but note that "1, 10, 28, 35, &c." has a typo (35 should be 55). - Michel Marcus, Nov 04 2017
a(n) is also the number of (nontrivial) paths as well as the Wiener sum index of the (n-1)-alkane graph. - Eric W. Weisstein, Jul 15 2021

T. D. Noe, Table of n, a(n) for n = 1..1000
Ignacio Larrosa Cañestro, Hexágono y estrella determinados por tres pares de cevianas simétricas, (java applet).
Al Cugo et al., Marion's theorem, The Mathematics Teacher 86 (1993) p. 619.
John Elias, Illustration of Initial Terms
F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
Eric Weisstein's World of Mathematics, Alkane Graph
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Marion's Theorem
Eric Weisstein's World of Mathematics, Wiener Sum Index
Index entries for two-way infinite sequences
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A027468, A081266, A190152.

List([1..50],n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019

[(2*n-1)^2+(n-1)*n/2: n in [1..50]]; // Vincenzo Librandi, Nov 18 2015

H := n -> simplify(1/hypergeom([-3*n,3*n+3,1],[3/2,2],3/4)); A060544 := n -> H(n-1); seq(A060544(i),i=1..19); # Peter Luschny, Jan 09 2012

Take[Accumulate[Range[150]], {1, -1, 3}] (* Harvey P. Dale, Mar 11 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* Harvey P. Dale, Mar 11 2013 *)
FoldList[#1 + #2 &, 1, 9 Range @ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(3 n - 1) (3 n - 2)/2, {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[Binomial[3 n - 1, 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[PolygonalNumber[3 n - 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)

PARI
```
a(n)=(3*n-1)*(3*n-2)/2
```

[(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019

a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.
a(n) = a(n-1) + 9*(n-1) = A060543(n, 3) = A006566(n)/n.
a(n) = A025035(n)/A025035(n-1) = A027468(n-1) + 1 = A000217(3*n-2).
a(1-n) = a(n).
From Paul Barry, Mar 15 2003: (Start)
a(n) = C(n-1, 0) + 9*C(n-1, 1) + 9*C(n-1, 2); binomial transform of (1, 9, 9, 0, 0, 0, ...).
a(n) = 9*A000217(n-1) + 1.
G.f.: x*(1 + 7*x + x^2)/(1-x)^3. (End)
Narayana transform (A001263) of [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).
a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - Peter Luschny, Jan 09 2012
From Ant King, Jun 18 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 9.
a(n) = A000217(n) + 7*A000217(n-1) + A000217(n-2).
Sum_{n>=1} 1/a(n) = 2*Pi/(3*sqrt(3)) = A248897.
(End)
a(n) = (2*n-1)^2 + (n-1)*n/2. - Ivan N. Ianakiev, Nov 18 2015
a(n) = A101321(9,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: (2 + 9*x^2)*exp(x)/2 - 1. - G. C. Greubel, Mar 02 2019
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 11*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)
a(n) = A000567(n) + A005449(n-1) (see illustration in links). - John Elias, Nov 10 2020
a(n) = P(2*n,4)*P(3*n,3)/24 for n>=2, where P(s,k) = ((s - 2)*k^2 - (s - 4)*k)/2 is the k-th s-gonal number. - Lechoslaw Ratajczak, Jul 18 2021

Additional description from Terrel Trotter, Jr., Apr 06 2002

Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015

A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

Original entry on oeis.org

2, 2, 12, 20, 10, 42, 56, 24, 90, 110, 44, 156, 182, 70, 240, 272, 102, 342, 380, 140, 462, 506, 184, 600, 650, 234, 756, 812, 290, 930, 992, 352, 1122, 1190, 420, 1332, 1406, 494, 1560, 1640, 574, 1806, 1892, 660, 2070, 2162, 752, 2352, 2450, 850, 2652, 2756, 954, 2970, 3080, 1064, 3306, 3422, 1180, 3660

Offset: 0

Paul Curtz, Sep 02 2011

The autosequence of first kind from (-1)^n/(n+1) is A189733.
For the second kind (the second increasing diagonal is (-1)^n/(n+1), half of the main one):
      2,     1,     0,  -1/2,  -1/3,   1/6,   1/2,  5/12,
     -1,    -1,  -1/2,   1/6,   1/2,   1/3, -1/12, -7/20,
      0,   1/2,   2/3,   1/3,  -1/6, -5/12, -4/15,  1/12,
    1/2,   1/6,  -1/3,  -1/2,  -1/4,  3/20,  7/20, 13/60,
   -1/3,  -1/2,  -1/6,   1/4,   2/5,   1/5, -2/15, -3/10,
   -1/6,   1/3,  5/12,  3/20,  -1/5,  -1/3,  -1/6,  5/42,
    1/2,  1/12, -4/15, -7/20, -2/15,   1/6,   2/7,   1/7,
  -5/12, -7/20, -1/12, 13/60,  3/10,  5/42,  -1/7,  -1/4.
Main diagonal: (period 2:repeat 2, -1)/A026741(n+1).
Second (increasing) diagonal: (-1)^n / (n+1).
Third (increasing) diagonal: (-1)^(n+1)*A026741(n) / A045896(n).
Fourth (increasing) diagonal: (-1)^(n+1)*A146535(n)/ a(n).

OEIS Wiki, Autosequence.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A001504 = 2*A060544, A049450 = 2*A000326, A045945 = 6*A005449.

Cf. A026741, A045896, A051176, A146535, A189733, A306368.

c = Table[1/9 (7 n + 7 n^2 + 2 n Cos[2 n *Pi/3] + 2 n^2 Cos[2 n *Pi/3] + 2 Sqrt[3] n Sin[2 n *Pi/3] + 2 Sqrt[3] n^2 Sin[2 n *Pi/3]), {n, 1, 50}] (* Roger Bagula, Mar 25 2012 *)
a[n_] := (n+1) * Numerator[(n+2)/3]; Array[a, 60, 0] (* Amiram Eldar, Sep 17 2023 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{2,2,12,20,10,42,56,24,90},60] (* Harvey P. Dale, May 15 2025 *)

a(3*n) = (3*n+1)*(3*n+2), a(3*n+1) = (n+1)*(3*n+2), a(3*n+2) = 3*(n+1)*(3*n+4).
G.f.: 2*(1+x+6*x^2+7*x^3+2*x^4+3*x^5+x^6)/(1-x^3)^3. - Jean-François Alcover, Nov 11 2016
a(n+2) = 2 * A306368(n) for n >= 0. - Joerg Arndt, Aug 25 2023
a(n) = (n+1) * A051176(n+2) for n >= 0. - Paul Curtz, Sep 13 2023
Sum_{n>=0} 1/a(n) = 1 + log(3) - Pi/(3*sqrt(3)). - Amiram Eldar, Sep 17 2023

A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

Original entry on oeis.org

1, 2, 2, 6, 9, 3, 24, 64, 20, 4, 120, 625, 216, 36, 5, 720, 7776, 3136, 512, 56, 6, 5040, 117649, 59049, 10000, 1000, 81, 7, 40320, 2097152, 1331000, 248832, 24336, 1728, 110, 8, 362880, 43046721, 35831808, 7529536, 759375, 50625, 2744, 144, 9, 3628800, 1000000000, 1097199376, 268435456, 28652616, 1889568, 93636, 4096, 182, 10

Offset: 1

Chai Wah Wu, Feb 23 2020

A dual sequence to A260355. See arXiv link for sets of permutations that achieve the value of T(n,k). The minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] is equal to n!*k^n.

			T(n,k)
   k    1    2     3      4      5      6      7      8      9     10     11     12
  ---------------------------------------------------------------------------------
n  1|   1    2     3      4      5      6      7      8      9     10     11     12
   2|   2    9    20     36     56     81    110    144    182    225    272    324
   3|   6   64   216    512   1000   1728   2744   4096   5832   8000  10648  13824
   4|  24  625  3136  10000  24336  50625  93636 160000 256036 390625 571536 810000

Chai Wah Wu, Table of n, a(n) for n = 1..70
Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A000027, A000142, A000169, A001504, A016743, A016766, A061711, A061718, A260355.

from itertools import permutations, combinations_with_replacement
def A331988(n,k): # compute T(n,k)
    if k == 1:
        count = 1
        for i in range(1,n):
            count *= i+1
        return count
    ntuple, count = tuple(range(1,n+1)), 0
    for s in combinations_with_replacement(permutations(ntuple,n),k-2):
        t = list(ntuple)
        for d in s:
            for i in range(n):
                t[i] += d[i]
        t.sort()
        w = 1
        for i in range(n):
            w *= (n-i)+t[i]
        if w > count:
            count = w
    return count

T(n,n) = (n*(n+1)/2)^n = A061718(n).
T(n,k) <= (k(n+1)/2)^n.
T(1,k) = k = A000027(k).
T(n,1) = n! = A000142(n).
T(2,2m) = 9m^2 = A016766(m).
T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
T(n,2) = (n+1)^n = A000169(n+1).
T(3,k) = 8k^3 = A016743(k) for k > 1.
If n divides k then T(n,k) = (k*(n+1)/2)^n.
If k is even then T(n,k) = (k*(n+1)/2)^n.
If n is odd and k >= n-1 then T(n,k) = (k*(n+1)/2)^n.
If n is even and k is odd such that k >= n-1, then T(n,k) = ((k^2*(n+1)^2-1)/4)^(n/2).

A323640 Numbers m having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 20, 56, 70, 110, 182, 272, 286, 308, 506, 646, 650, 812, 884, 992, 1150, 1406, 1672, 1748, 1798, 1892, 2162, 2756, 2990, 3422, 3526, 3782, 4030, 4466, 4556, 4606, 4930, 5402, 5510, 5704, 6032, 6068, 6806, 7198, 7310, 7378, 7832, 7904, 8084, 8170, 8246, 8584, 8710

Offset: 1

David A. Corneth, Aug 31 2019

nonn

Primitive terms of A094519.
From Bernard Schott, Aug 31 2019: (Start)
Some subsequences (this list is not exhaustive):
1) Oblong numbers of the form (3*k+1)*(3*k+2). These are in A001504 and the pair (x,y) = (1,3*k+1). Only 6 is oblong and not of this form. The first few terms are 20, 56, 110, 182, 272, ...
2) Numbers of the form 2*p*q  where (p, q) is a twin prime pair. These terms are precisely A071142 \ {30} and the pair (x,y) = (2,p). The first few terms are 70, 286, 646, ...
3) Numbers of the form 2^2 * p * q where p and q = p+4 are primes and p > 3. These primes p are in A023200 \ {3} and the pair (x,y) = (4,p). The first few terms are 308, 884, ...
4) More generally, numbers of the form 2^k * p * q where p and q = p+2^k are primes and the pair (x,y) = (2^k,p). For k = 3, the smallest such term is 1672 with p = 11. (End)

			56 is in the sequence as 1, 7 and 1 + 7 = 8 are divisors of 56 and no divisor of 56 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11106 (terms <= 5*10^7)

Cf. A094519.

filter:= proc(n) local D,i,j,nD;
  D:= numtheory:-divisors(n);
  nD:= nops(D);
  for i from 1 to nD-1 do
    for j from i+1 to nD do
      if (n/(D[i]+D[j]))::integer then return true fi
  od od;
  false
end proc:
N:= 10000: # for terms <= N
C:= Vector(N):
R:= NULL:
for i from 1 to N do
  if C[i]=0 and filter(i) then
    R:= R, i;
    C[[seq(i*j,j=2..N/i)]]:= 1
  fi
od:
R; # Robert Israel, Sep 02 2019

upto(n) = {my(charprim = vector(n, i, 1), res = List()); for(i = 1, n, if(charprim[i] == 1, if(isA094519(i), listput(res, i); for(k = 2, n \ i, charprim[i*k] = 0 ) , charprim[i] = 0; ) ) ); res }
isA094519(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 }

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

4, 6, 2, 0, 9, 8, 1, 2, 0, 3, 7, 3, 2, 9, 6, 8, 7, 2, 9, 4, 4, 8, 2, 1, 4, 1, 4, 3, 0, 5, 4, 5, 1, 0, 4, 5, 3, 8, 3, 6, 6, 6, 7, 5, 6, 2, 4, 0, 1, 7, 0, 1, 6, 9, 4, 1, 3, 7, 8, 6, 6, 7, 2, 9, 9, 5, 5, 9, 5, 7, 4, 7, 9, 7, 9, 7, 9, 6, 4, 7, 7, 0, 7, 0, 5, 7, 5, 5, 5, 1, 3, 3, 0, 9, 4, 5, 7, 9, 1, 6

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A269232 a(n) = (n + 1)(6n^2 + 15*n + 4)/2.

Original entry on oeis.org

2, 25, 87, 206, 400, 687, 1085, 1612, 2286, 3125, 4147, 5370, 6812, 8491, 10425, 12632, 15130, 17937, 21071, 24550, 28392, 32615, 37237, 42276, 47750, 53677, 60075, 66962, 74356, 82275, 90737, 99760, 109362, 119561, 130375, 141822, 153920, 166687, 180141

Offset: 0

Ilya Gutkovskiy, Feb 20 2016

			a(0) = 0 + 1*2 = 2;
a(1) = 0 + 1*2 + 3 + 4*5 = 25;
a(2) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
a(3) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
a(4) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + 13*14 = 400, etc.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001477, A001504, A008585.

[(n+1)*(6*n^2+15*n+4)/2: n in [0..40]]; // Vincenzo Librandi, Feb 22 2016

Table[(n + 1) ((6 n^2 + 15 n + 4)/2), {n, 0, 38}]
Table[Sum[3 k + (3 k + 1) (3 k + 2), {k, 0, n}], {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {2, 25, 87, 206}, 39]

Vec((2 + 17*x - x^2)/(x - 1)^4 + O(x^50)) \\ Michel Marcus, Feb 22 2016

G.f.: (2 + 17*x - x^2)/(x - 1)^4.
a(n) = Sum_{k=0..n} (3*k + (3*k+1)*(3*k+2)) = Sum_{k=0..n} (A008585(k) + A001504(k)).
Sum_{n>=0} 1/a(n) = 0.56407113696623548787861365289...

cp's OEIS Frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

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A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

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A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

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A323640 Numbers m having at least one pair (x,y) of divisors with x

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A387235 Decimal expansion of 2*log(2)/3.

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A269232 a(n) = (n + 1)*(6*n^2 + 15*n + 4)/2.

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A269232 a(n) = (n + 1)(6n^2 + 15*n + 4)/2.