A001318 -id:A001318 - cp's OEIS frontend

A175009 Triangle read by rows, antidiagonals of an array formed from variants of A001318, generalized pentagonal numbers.

1, 1, 2, 1, 3, 5, 1, 4, 9, 7, 1, 5, 13, 13, 12, 1, 6, 17, 19, 23, 15, 1, 7, 21, 25, 34, 29, 22, 1, 8, 25, 31, 45, 43, 43, 26, 1, 9, 29, 37, 56, 57, 64, 51, 35, 1, 10, 33, 43, 67, 71, 85, 76, 69, 40, 1, 11, 37, 49, 78, 85, 106, 101, 103, 79, 51

Offset: 1

Gary W. Adamson, Apr 03 2010

			First few rows of the array:
  1, 2,  5,  7,  12,  15,  22,  26,  35,  40, ...
  1, 3,  9, 13,  23,  29,  43,  51,  69,  79, ...
  1, 4, 13, 19,  34,  43,  64,  76, 103, 118, ...
  1, 5, 17, 25,  45,  57,  85, 101, 137, 157, ...
  1, 6, 21, 31,  56,  71, 106, 126, 171, 196, ...
  ...
Example: row 3 is generated from 3 * (1, 3, 2, 5, 3, 7, ...) = (3, 9, 6, 15,...)
Preface with a 1 getting (1, 3, 9, 6, 15, ...) then take partial sums, = (1, 4, 13, 19, 34, 43, 64, ...).
...
First few rows of the triangle:
  1;
  1,  2
  1,  3,  5;
  1,  4,  9,  7;
  1,  5, 13, 13,  12;
  1,  6, 17, 29,  23,  15;
  1,  7, 21, 25,  34,  29,  22;
  1,  8, 25, 31,  45,  43,  43,  26;
  1,  9, 29, 37,  56,  57,  64,  51,  35;
  1, 10, 33, 43,  67,  71,  85,  76,  69,  40;
  1, 11, 37, 49,  78,  85, 106, 101, 103,  79,  51;
  1, 12, 41, 55,  89,  99, 127, 126, 137, 118, 101,  57;
  1, 13, 45, 61, 100, 113, 148, 151, 171, 157, 151, 113,  70;
  1, 14, 49, 67, 111, 127, 169, 176, 205, 196, 201, 169, 139, 77;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A175006.

Cf. A001318, A026741.

T(n,k)=if(k<=n, 1 + (n-k+1)*(binomial(k+1, 2) - 1 - binomial(k\2+1, 2)), 0) \\ Andrew Howroyd, Sep 08 2018

Let row 1 of the array = A001318 starting with offset 1: (1, 2, 5, 7, 12,...)
For rows k>1, begin with A026741 starting (1, 3, 2, 5, 3, 7, 4, 9, 5, 11,...)
= generator Q. Then k-th row = partial sums of (1,...(k * Q)).
T(n,k) = 1 + (n-k+1)*(binomial(k+1, 2) - 1 - binomial(floor(k/2)+1, 2)). - Andrew Howroyd, Sep 08 2018

a(22) corrected by Andrew Howroyd, Sep 08 2018

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 4, 1, 4, 3, 4, 2, 2, 5, 3, 5, 3, 5, 4, 5, 7, 3, 4, 4, 6, 6, 4, 6, 3, 5, 7, 6, 4, 1, 7, 7, 6, 5, 6, 9, 5, 7, 7, 8, 6, 8, 4, 6, 6, 7, 9, 4, 10, 3, 6, 9, 7, 8, 5, 9, 7, 6, 7, 5, 11, 9, 7, 3, 7, 12, 13, 7, 7, 6, 9, 11, 6, 11, 8, 7, 10, 10, 8, 8, 8, 11, 5, 8, 5, 8, 11, 10, 10, 6, 14, 10, 6, 7, 7

Offset: 1

Robert G. Wilson v, May 14 2014

It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].

Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242442.

planeFigurative[n_, r_] := pf[n, r] = (n - 2) Binomial[r, 2] + r; s = Sort@ Table[ planeFigurative[3, i] + planeFigurative[3, j] + planeFigurative[3, k], {i, 0, 14}, {j, 0, 10, 2}, {k, -8, 8}]; Table[ Count[s, n], {n, 0, 50}]

A305538 a(n) is the smallest positive generalized pentagonal number such that A001318(n) - a(n) is twice a generalized pentagonal number.

Original entry on oeis.org

1, 3, 2, 1, 8, 2, 5, 10, 7, 5, 18, 7, 12, 20, 3, 12, 5, 1, 22, 3, 10, 22, 13, 8, 35, 11, 20, 35, 2, 18, 5, 22, 33, 1, 12, 31, 16, 8, 49, 12, 25, 47, 30, 21, 2, 26, 41, 66, 11, 37, 16, 5, 60, 10, 27, 56, 33, 21, 82, 27, 46, 78, 7, 40, 13, 47, 68, 5, 26, 62, 33

Offset: 3

Peter Luschny, Jun 04 2018

nonn

Such a number exists for n >= 3 due to a theorem of Enge, Hart and Johansson (see the link, p. 11).

			(A001318(328) - a(328))/2 = (40426 - 174)/2 = 20126 = A001318(231).

Peter Luschny, Table of n, a(n) for n = 3..1000
Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.4. Also available as arXiv:1608.06810 [math.NT].

Cf. A001318, A305539.

A339885 Triangle read by rows: T(n, m) gives the sum of the weights of weighted partitions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 0, -1, -1, 0, 1, 1, 1, 1, 0, -1, -2, -1, 0, 1, 1, 1, 1, 0, 1, -1, -2, 0, 0, 1, 1, 1, 1, 0, 0, 0, -2, -2, 0, 0, 1, 1, 1, 1

Offset: 1

Wolfdieter Lang, Feb 15 2021

The row sums are given in A341417.
One could add a row n=0 and the column (1,repeat(0)) including the empty partition with no parts, and number of parts m = 0. The weight w(0) = -1.
The weight from {-1, 0, +1} of a positive number n is w(n) = 0 if n is not an element of the generalized pentagonal numbers {Pent(k) = A001318(k)}_{k>=1}, and if n = Pent(k) then w(n) = (-1)^(ceiling(Pent(k)/2)+1). The sequence
{(n, w(n))}_{n>=1} begins: {(1,+1), (2,+1), (3,0), (4,0), (5,-1), (6,0), (7,-1), ...}. One can also use w(0) = -1. w(n) = -A010815(n), for n >= 0. For n >= 1, w(n) = A257628(n) also.
The weight of a partition is the product of the weights of its parts.
For the triangle giving the sum of the weights of weighted compositions of n with m parts from the generalized pentagonal numbers see A341418.

			The triangle T(n, m) begins:
  n\m   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ... A341417
  ----------------------------------------------------------------------------
  1:    1                                                                 1
  2:    1  1                                                              2
  3:    0  1  1                                                           2
  4:    0  1  1  1                                                        3
  5:   -1  0  1  1  1                                                     2
  6:    0 -1  1  1  1  1                                                  3
  7:   -1 -1 -1  1  1  1  1                                               1
  8:    0 -1 -1  0  1  1  1  1                                            2
  9:    0 -1 -2 -1  0  1  1  1  1                                         0
  10:   0  1 -1 -2  0  0  1  1  1  1                                      2
  11:   0  0  0 -2 -2  0  0  1  1  1  1                                   0
  12:   1  1  1  0 -2 -1  0  0  1  1  1  1                                4
  13:   0  1  1  0 -1 -2 -1  0  0  1  1  1  1                             2
  14:   0  2  2  2  0 -1 -1 -1  0  0  1  1  1  1                          7
  15:   1  0  1  2  1 -1 -1 -1 -1  0  0  1  1  1  1                       5
  16:   0  1  2  2  3  1 -1  0 -1 -1  0  0  1  1  1  1                   10
  17:   0  0  0  1  2  2  0 -1  0 -1 -1  0  0  1  1  1  1                 6
  18:   0  0  0  2  2  3  2  0  0  0 -1 -1  0  0  1  1  1  1             11
  19:   0 -1 -1 -1  1  2  2  1  0  0  0 -1 -1  0  0  1  1  1  1           5
  20:   0 -1 -1  0  1  2  3  2  1  1  0  0 -1 -1  0  0  1  1  1  1       10
  ...
n = 5: (Partition; weight w) with | separating same m numbers (in Abramowitz -Stegun order):
(5;-1) | (1,4;0), (2,3;0) | (1^2,3;0), (1,2^2;1) | (1^3,2;1) | (1^5;1), hence row n=5 is [-1, 0, 1, 1, 1] from the sum of same m weights.

Cf. A000045, A001318, A008284, -A010815, A257628, A341417, A341418.

T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j)), where p(n, m) = A008284(n, m), and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, by w(n) = -A010815(n), for n >= 1 and m = 1, 2, ..., n.

A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 5, 15, 30, 45, 56, 65, 85, 115, 150, 171, 175, 185, 205, 260, 300, 325, 340, 350, 415, 440, 485, 500, 505, 580, 581, 650, 645, 675, 815, 815, 910, 845, 865, 985, 951, 1130, 1030, 1060, 1155, 1150, 1370, 1265, 1410, 1495, 1420, 1545, 1460, 1600, 1675, 1690

Offset: 0

Judson Neer, Feb 14 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001318, A008443, A014110, A080995.

a[n_] := SeriesCoefficient[(QPochhammer[-q, q^3]* QPochhammer[-q^2, q^3]*QPochhammer[q^3, q^3])^5, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jun 12 2017 *)

G.f.: f(x, x^2)^5 where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Jun 08 2012

A154565 One-half of averages of twin prime pairs of A001318.

Original entry on oeis.org

2, 15, 51, 210, 330, 651, 1365, 1650, 1926, 3480, 5430, 5985, 6501, 11310, 16485, 16590, 21660, 25026, 27270, 28635, 35190, 38001, 39285, 46905, 48690, 58905, 64170, 90651, 109485, 143376, 148995, 151845, 190995, 311676, 316251, 332526

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 12 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A154563, A074378.

q=3;lst={};s=0;Do[s+=n/q;If[Floor[s]==s,If[PrimeQ[2*s-1]&&PrimeQ[2*s+1],AppendTo[lst,s]]],{n,0,8!}];lst

A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 3, 6, 7, 6, 6, 7, 12, 12, 12, 9, 6, 12, 12, 18, 13, 12, 18, 12, 18, 12, 13, 18, 12, 24, 12, 12, 24, 21, 30, 12, 18, 18, 12, 24, 18, 19, 18, 24, 24, 18, 24, 36, 24, 18, 19, 18, 24, 24, 30, 18, 12, 36, 30, 24, 21, 18, 36, 24, 36, 24, 12, 36, 36, 36, 18, 25, 30, 24, 24, 24, 30, 24, 36, 30, 24

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

Conjecture: every number is the sum of at most k - 4 generalized k-gonal numbers (for k >= 8).

In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m-2)^3 can be written as the sum of five m-gonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper Zhi-Wei Sun proved that each natural number is the sum of four octagonal numbers. - Zhi-Wei Sun, Oct 03 2020

			a(6) = 7 because we have [5, 1, 0], [5, 0, 1], [2, 2, 2], [1, 5, 0], [1, 0, 5], [0, 5, 1] and [0, 1, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A001318, A002175, A008443, A080995, A093518, A093519, A255350, A255934, A256132, A256171, A280718.

N:= 100;
bds:= [fsolve(k*(3*k-1)/2 = N)];
G:= add(x^(k*(3*k-1)/2),k=floor(min(bds))..ceil(max(bds)))^3:
seq(coeff(G,x,n),n=0..N); # Robert Israel, Aug 16 2017

nmax = 75; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, -nmax, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Sum[x^((6 k^2 + 6 k + (-1)^(k + 1) (2 k + 1) + 1)/16), {k, 0, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[EllipticTheta[4, 0, x^3]^3/QPochhammer[x, x^2]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.
G.f.: (Sum_{k>=0} x^A001318(k))^3.
G.f.: Product_{n >= 1} ( (1 - q^(3*n))/(1 - q^n + q^(2*n)) )^3. - Peter Bala, Jan 04 2025

A306592 Number of ways to write n as u^2 + 2v^2 + 3x^2 + 4y^2 + 5z^2, where u,v,x,y,z are generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 4, 5, 7, 6, 4, 5, 7, 6, 6, 5, 8, 8, 6, 6, 10, 10, 5, 8, 9, 8, 8, 5, 8, 9, 5, 5, 10, 7, 4, 7, 6, 6, 6, 3, 5, 7, 3, 2, 7, 6, 4, 6, 4, 4, 9, 4, 5, 8, 6, 5, 6, 7, 4, 7, 2, 4, 8, 5, 3, 6, 7, 5, 6, 7, 6, 9, 3, 5, 9, 5, 4, 8, 6, 7, 7, 5, 5, 10, 4, 3, 6, 4, 4, 5, 2, 3, 6, 4, 5, 10, 5

Offset: 0

Zhi-Wei Sun, Feb 26 2019

nonn

Conjecture: a(n) > 0 for any nonnegative integer n.
I'd like to call this the 1-2-3-4-5 conjecture. I have verified it for all n = 0..4*10^5.
It seems that a(n) = 1 only for n = 0,1,2.

			Let f(x) = x*(3x-1)/2. Then
a(2) = 1 with 2 = f(0)^2 + 2*f(1)^2 + 3*f(0)^2 + 4*f(0)^2 + 5*f(0)^2,
a(415) = 2 with 415 = f(-1)^2 + 2*f(3)^2 + 3*f(1)^2 + 4*f(2)^2 + 5*f(-1)^2 = f(3)^2 + 2*f(0)^2 + 3*f(2)^2 + 4*f(-2)^2 + 5*f(0)^2,
a(427) = 2 with 427 = f(-3)^2 + 2*f(2)^2 + 3*f(-2)^2 + 4*f(0)^2 + 5*f(1)^2 = f(-3)^2 + 2*f(1)^2 + 3*f(2)^2 + 4*f(0)^2 + 5*f(2)^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A001318.

f[x_]:=f[x]=(x(3x-1)/2)^2;
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=SQ[n]&&SQ[24Sqrt[n]+1];
tab={};Do[r=0;Do[If[PQ[n-5f[x]-4f[y]-3f[z]-2f[w]],r=r+1],{x,-Floor[(Sqrt[24Sqrt[n/5]+1]-1)/6],(Sqrt[24Sqrt[n/5]+1]+1)/6},{y,-Floor[(Sqrt[24Sqrt[(n-5f[x])/4]+1]-1)/6],(Sqrt[24Sqrt[(n-5f[x])/4]+1]+1)/6},{z,-Floor[(Sqrt[24Sqrt[(n-5f[x]-4f[y])/3]+1]-1)/6],(Sqrt[24Sqrt[(n-5f[x]-4f[y])/3]+1]+1)/6},{w,-Floor[(Sqrt[24Sqrt[(n-5f[x]-4f[y]-3f[z])/2]+1]-1)/6],(Sqrt[24Sqrt[(n-5f[x]-4f[y]-3f[z])/2]+1]+1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

A070334 Erroneous version of A001318.

Original entry on oeis.org

0, 1, 3, 5, 7, 12, 15

Offset: 0

dead

A. Weil, Two lectures on number theory, past and present, L'Enseign. Math., XX (1974), 87-110 see p. 98; Oeuvres III, 279-302.

A143445 Triangle read by rows, A051731 * (A001318 * 0^(n-k)); 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 0, 5, 1, 2, 0, 7, 1, 0, 0, 0, 12, 1, 2, 5, 0, 0, 15, 1, 0, 0, 0, 0, 0, 22, 1, 2, 0, 7, 0, 0, 0, 26, 1, 0, 5, 0, 0, 0, 0, 0, 35, 1, 2, 0, 0, 12, 0, 0, 0, 0, 40, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 1, 2, 5, 7, 0, 15, 0, 0, 0, 0, 0, 57

Offset: 1

Gary W. Adamson, Aug 15 2008

Row sums = A079248: (1, 3, 6, 10, 13, 23,...)

			First few rows of the triangle =
1;
1, 2;
1, 0, 5;
1, 2, 0, 7;
1, 0, 0, 0, 12;
1, 2, 5, 0, 0, 15;
...

Cf. A051731, A001318, A079248.

Triangle read by rows, A051731 * (A001318 * 0^(n-k)); 1<=k<=n. A051731 = the inverse Mobius transform and (A001318 * 0^(n-k)) = an infinite lower triangular matrix with A001318 (1, 2, 5, 7, 12, 15, 22,...) in the main diagonal and the rest zeros.

cp's OEIS Frontend

A175009 Triangle read by rows, antidiagonals of an array formed from variants of A001318, generalized pentagonal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A305538 a(n) is the smallest positive generalized pentagonal number such that A001318(n) - a(n) is twice a generalized pentagonal number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A339885 Triangle read by rows: T(n, m) gives the sum of the weights of weighted partitions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A154565 One-half of averages of twin prime pairs of A001318.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A306592 Number of ways to write n as u^2 + 2*v^2 + 3*x^2 + 4*y^2 + 5*z^2, where u,v,x,y,z are generalized pentagonal numbers (A001318).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A070334 Erroneous version of A001318.

Views

Author

Keywords

A306592 Number of ways to write n as u^2 + 2v^2 + 3x^2 + 4y^2 + 5z^2, where u,v,x,y,z are generalized pentagonal numbers (A001318).