A093519 -id:A093519 - cp's OEIS frontend

A213250 Numbers n such that the coefficient of x^n in the expansion of Product_{k>=1} (1-x^k)^2 is zero.

7, 11, 12, 17, 18, 21, 22, 25, 32, 37, 39, 41, 42, 43, 46, 47, 49, 54, 57, 58, 60, 62, 65, 67, 68, 72, 74, 75, 76, 81, 82, 87, 88, 90, 92, 95, 97, 98, 99, 106, 107, 109, 111, 112, 113, 116, 117, 120, 122, 123, 125, 126, 128, 130, 132, 136, 137

Offset: 1

William J. Keith, Jun 07 2012

Indices of zero entries in A002107.
Asymptotic density is 1.
Contains A093519, numbers with no representation as sum of two or fewer pentagonal numbers.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002107, A093519.

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), this sequence (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322043 (m=15).

# DedekindEta is defined in A000594.
function A213250List(upto)
    eta = DedekindEta(upto, 2)
    [n - 1 for (n, z) in enumerate(eta) if z == 0] end
println(A213250List(140))  # Peter Luschny, Jul 19 2022

LongPoly = Series[Product[1 - q^n, {n, 1, 300}]^2, {q, 0, 300}]; ZeroTable = {}; For[i = 1, i < 301, i++, If[Coefficient[LongPoly, q^i] == 0, AppendTo[ZeroTable, i]]]; ZeroTable

x='x+O('x^200);
v=Vec(eta(x)^2 - 1);
for(k=1,#v,if(v[k]==0,print1(k,", ")));
/* Joerg Arndt, Jun 07 2012 */

A203860 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n)) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -1, -4, -1, 1, 11, 7, 25, 18, -11, -1, 0, -325, -199, 122, -1364, -843, 550, 0, 11, 123, 0, 39650, 24476, -15126, 0, 271443, 164194, -103682, -1364, -1, -24476, 0, -9349, -123, -20633239, -12752043, 7881225, -843, 0, -226965629, -141422125, 88114450, 0, 1

Offset: 0

Paul D. Hanna, Jan 07 2012

sign

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

			G.f.: A(x) = 1 - x - 4*x^2 - x^3 + x^4 + 11*x^5 + 7*x^6 + 25*x^7 +...
-log(A(x)) = x + 3*3*x^2/2 + 4*4*x^3/3 + 7*7*x^4/4 + 6*11*x^5/5 + 12*18*x^6/6 +...+ sigma(n)*A000204(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10) * (1-18*x^6+x^12) *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...]
which are numbers that are not the sum of two generalized pentagonal numbers.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A203861, A203850, A156234, A093519.

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

{a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*Lucas(k)*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} -sigma(n) * A000204(n) * x^n/n ).

A093518 Number of ways of representing n as the sum of two (not necessarily distinct) generalized pentagonal numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 2, 1, 2, 1, 1, 2, 0, 1, 1, 0, 2, 1, 2, 0, 1, 3, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 2, 2, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 0, 2, 0, 0, 1, 2, 1, 2, 1, 0, 2, 0, 3, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 0, 1, 0, 1, 4, 1, 1, 0, 1, 2, 0, 2, 1, 1, 2, 1

Offset: 0

Jon Perry, Mar 29 2004

nonn

			a(7)=2 as we have 7+0 = 5+2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001318 (generalized pentagonal numbers), A093519 (where a(n)=0).

GP:= [0,seq(op([m*(3*m-1)/2, m*(3*m+1)/2]),m=1..11)]:
V:= Array(0..200):
for i from 1 to 23 do
  for j from 1 to i do
    r:= GP[i]+GP[j];
    if r > 200 then break fi;
    V[r]:= V[r]+1
od od:
convert(V,list); # Robert Israel, Feb 26 2025

{ v=vector(101); v[1]=0; for (i=1,50, v[2*i]=i*(3*i-1)/2; v[2*i+1]=i*(3*i+1)/2); x=vector(500); for (a=1,50, for (b=a,50, if (v[a]+v[b]<500, x[v[a]+v[b]+1]++))); x }

Definition clarified by Robert Israel, Feb 26 2025

A204382 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n)) where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -2, -7, -2, 1, 82, 34, 464, 198, -82, -1, 0, -39208, -16238, 6725, -551614, -228486, 95120, 0, 82, 6726, 0, 263673800, 109216786, -45239073, 0, 8957108166, 3706940654, -1536796802, -551614, -1, -109216786, 0, -18738638, -6726, -24954506565518, -10336495061766

Offset: 0

Paul D. Hanna, Jan 14 2012

sign

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

			G.f.: A(x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4 + 82*x^5 + 34*x^6 + 464*x^7 +...
-log(A(x)) = 1*2*x + 3*6*x^2/2 + 4*14*x^3/3 + 7*34*x^4/4 + 6*82*x^5/5 + 12*198*x^6/6 +...+ sigma(n)*A002203(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...].
which are numbers that are not the sum of two generalized pentagonal numbers.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A203860, A204383, A204384, A203534, A093519, A002203.

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} -sigma(n) * A002203(n) * x^n/n ).

A355717 Smallest number of generalized pentagonal numbers which sum to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 1, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 1, 2, 2, 3, 2, 2, 3, 2, 1

Offset: 0

Bernard Schott, Jul 15 2022

nonn

From Euler's Pentagonal Number Theorem, every number is expressible as the sum of at most three generalized pentagonal numbers (A001318) (see Richard K. Guy reference).
Corresponding sums of only pentagonal numbers of positive rank are A100878(n). Those numbers are a subset of the generalized pentagonals so that a(n) <= A100878(n).
More specifically, by the definition given in the name, we understand the following: Given n >= 0 we seek a multiset S such that (1) S is a multiset of GPN = {0, 1, 2, 5, ...} = A001318; (2) Sum(S) = n; (3) if T is a multiset of GPN and Sum(T) = n then card(T) >= card(S). Additionally one might require that the set is not empty. If a multiset satisfies these three conditions, then a(n) = card(S). Note that no actual summation has to be performed to decide the value of a(n); only membership in GPN needs to be tested, as shown in the Maple and Python program. - Peter Luschny, Jul 18 2022

			Let GPN = {0, 1, 2, 5, ...} be the generalized pentagonal numbers.
a(0) = 0 since {} is a multiset of GPN, Sum {} = 0, and card({}) = 0.
a(1) = 1 since {1} is a multiset of GPN, Sum {1} = 1, and card({1}) = 1.
a(3) = 2 since {1, 2} is a multiset of GPN, Sum {1, 2} = 3, and card({1, 2}) = 2.
a(11) = 3 since {2, 2, 7} is a multiset of GPN, Sum {2, 2, 7} = 11, card({2, 2, 7}) = 3, and no other multiset S of GPN with Sum(S) = 11 has less members.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem.

Cf. A001318, A093519 (indices of 3's).

Cf. A100878.

A355717_list := proc(upto) local P, Q, k, q, isgpn; P := []; Q := [0];
isgpn := k -> ormap(n -> 0 = 8*k - (n + irem(n,2)) * (3*n + 2 - irem(n,2)), [$0..k]);
for k from 1 to upto do
    q := 3;
    if isgpn(k) then
        P := [op(P), k]; q := 1;
        elif ormap(p -> member(k - p, P), P) then q := 2 fi:
        Q := [op(Q), q];
od: Q end:
print(A355717_list(100));  # Peter Luschny, Jul 18 2022

def A355717_list(ln: int) -> list[int]:
    P: list[int] = []
    Q: list[int] = [0]
    def is_gpn(k: int) -> bool:
        return any(8 * k == ((n + n % 2) * (3 * n + 2 - n % 2)) for n in range(k + 1))
    for k in range(1, ln):
        q = 3
        if is_gpn(k):
            P.append(k)
            q = 1
        elif any([(k - p) in P for p in P]):
            q = 2
        Q.append(q)
    return Q
print(A355717_list(100))  # Peter Luschny, Jul 18 2022

a(n) <= 3.

a(A001318(n)) = 1.

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

A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 12, 15, 21, 22, 25, 26, 35, 39, 40, 49, 51, 57, 67, 70, 77, 87, 92, 100, 117, 120, 123, 126, 145, 153, 155, 173, 176, 182, 186, 187, 205, 210, 214, 222, 228, 241, 247, 251, 260, 283, 287, 301, 319, 330, 345, 376, 382, 392, 425, 435, 442, 448

Offset: 0

Peter Luschny, Jul 17 2022

nonn

The sequence is defined inductively. Starting from the empty sequence, the terms are added one after the other. A term is added if it is a generalized pentagonal number or if it cannot be represented as the sum of two preceding terms. Note that these exceptions form a proper subsequence of A093519.

Thus any positive number can be expressed as the sum of at most two positive terms by Euler's Pentagonal Number Theorem. Every pentagonal number and every generalized pentagonal number is in this sequence.

			32 = 7 + 25; 195 = 22 + 173.

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Burkard Polster (Mathloger), The hardest 'What comes next?' (Euler's pentagonal formula), YouTube video, 2020.

Cf. A000326, A001318, A093519, A100878, A355717, A176747 (same construction with triangular numbers).

A355774_list := proc(upto) local P, k, issum, isgpn; P := [];
isgpn := k -> ormap(n -> 0 = 8*k-(n+irem(n,2))*(3*n+2-irem(n,2)), [$0..k]);
issum := k -> ormap(p -> member(k - p, P), P);
for k from 0 to upto do
    if isgpn(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A355774_list(448));

isgpn[k_] := AnyTrue[Range[0, k], 0 == 8*k-(#+Mod[#,2])*(3*#+2-Mod[#,2])&];
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
P = {};
For[k = 0, k <= 448, k++, If[isgpn[k] || !issum[k], AppendTo[P, k]]];
P (* Jean-François Alcover, Mar 07 2024, after Peter Luschny *)

def A355774_list(upto: int) -> list[int]:
    P: list[int] = []
    for k in range(upto + 1):
        if any(
            k == ((n + n % 2) * (3 * n + 2 - n % 2)) >> 3
            for n in range(k + 1)
        ) or not any([(k - p) in P for p in P]):
            P.append(k)
    return P
print(A355774_list(448))

cp's OEIS Frontend

A213250 Numbers n such that the coefficient of x^n in the expansion of Product_{k>=1} (1-x^k)^2 is zero.

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A203860 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n).

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A093518 Number of ways of representing n as the sum of two (not necessarily distinct) generalized pentagonal numbers.

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Maple

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Extensions

A204382 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203(n) is the companion Pell numbers.

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A355717 Smallest number of generalized pentagonal numbers which sum to n.

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Maple

Python

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

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Maple

Mathematica

Formula

A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

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Examples

A203860 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n)) where Lucas(n) = A000204(n).

A204382 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n)) where A002203(n) is the companion Pell numbers.