A020757 -id:A020757 - cp's OEIS frontend

A322431 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^10 is zero.

6, 13, 17, 27, 28, 34, 36, 39, 41, 48, 55, 59, 61, 62, 72, 74, 76, 82, 83, 90, 93, 94, 97, 104, 105, 111, 112, 116, 121, 125, 127, 128, 131, 132, 138, 139, 146, 149, 151, 152, 153, 160, 168, 169, 174, 181, 182, 183, 188, 193, 195, 197, 202, 204, 207, 209, 211, 214, 215

Offset: 1

Seiichi Manyama, Dec 07 2018

nonn

Indices of zero entries in A010818.

Also: numbers k such that 24k + 10 cannot be written as (12m+3)^2 + (4n+1)^2 with integers m, n. In this case, 12k + 5 is never prime. - M. F. Hasler, Jun 30 2025

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

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), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), this sequence (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

my(x='x+O('x^300)); Vec(select(x->(x==0), Vec(eta(x)^10 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A322432 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^14 is zero.

Original entry on oeis.org

4, 9, 15, 19, 24, 26, 29, 32, 34, 37, 44, 48, 49, 54, 55, 59, 66, 69, 74, 78, 79, 81, 83, 84, 92, 94, 99, 100, 101, 103, 104, 109, 113, 114, 117, 119, 124, 125, 129, 134, 136, 142, 144, 147, 149, 151, 154, 158, 159, 160, 169, 170, 171, 174, 179, 180, 184, 185, 193, 194

Offset: 1

Seiichi Manyama, Dec 07 2018

nonn

Indices of zero entries in A010821.

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

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), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), A322431 (m=10), this sequence (m=14), A322043 (m=15), A322433 (m=26).

my(x='x+O('x^300)); Vec(select(x->(x==0), Vec(eta(x)^14 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

Original entry on oeis.org

6, 16, 21, 31, 36, 42, 46, 51, 55, 56, 66, 72, 76, 81, 91, 94, 106, 111, 120, 121, 123, 126, 133, 136, 141, 146, 156, 157, 171, 172, 174, 181, 186, 191, 196, 198, 210, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306

Offset: 1

Greg Huber, May 13 2006

A052343(a(n)) > 1; gives A020756 together with A119345. - Reinhard Zumkeller, May 15 2006

			a(1) = 6 = 0 + 6 = 3 +3.
a(2) = 16 = 1 + 15 = 6 + 10.
a(3) = 21 = 0 + 21 = 6 + 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020757, A000217.

Cf. A052343.

a118139 n = a118139_list !! (n-1)
a118139_list = filter ((> 1) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

Sort[Transpose[Select[Tally[Total/@(Union[Sort/@Tuples[Accumulate[ Range[ 0,30]],2]])],#[[2]]>1&]][[1]]] (* Harvey P. Dale, Jul 21 2015 *)

More terms from Reinhard Zumkeller, May 15 2006

A203861 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n))^3 where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -3, -9, 20, 45, 0, -151, -231, 0, 140, 1107, 2052, 49, -1305, 0, -15004, -28260, 0, 17710, 0, 81, 324040, 589953, 0, -375570, -1089, 0, -124124, -10659705, -19764180, -121, 12605358, 0, 0, 4158315, 0, 567552368, 1052295189, -780030, -669901660, 0, 0, -221399431, -85965, 0

Offset: 0

Paul D. Hanna, Jan 07 2012

sign

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

			G.f.: A(x) = 1 - 3*x - 9*x^2 + 20*x^3 + 45*x^4 - 151*x^6 - 231*x^7 +...
-log(A(x))/3 = 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)^3 * (1-3*x^2+x^4)^3 * (1-4*x^3-x^6)^3 * (1-7*x^4+x^8)^3 * (1-11*x^5-x^10)^3 * (1-18*x^6+x^12)^3 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...]
which are numbers that are not the sum of two triangular numbers.

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

Cf. A203860, A203850, A156234, A020757.

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

{a(n)=polcoeff(exp(sum(k=1, n, -3*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))^3, n)}

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A000204(n) * x^n/n ).

A204383 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n))^3 where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -6, -9, 70, 90, 0, -1411, -1722, 0, 490, 60534, 75222, 49, -21510, 0, -6067754, -7542180, 0, 2156110, 0, 81, 1420032740, 1764323886, 0, -504516870, -8118, 0, -50196874, -783087782910, -973096740630, -121, 278263575996, 0, 0, 27685627830, 0, 1024173639305948

Offset: 0

Paul D. Hanna, Jan 14 2012

sign

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

			G.f.: A(x) = 1 - 6*x - 9*x^2 + 70*x^3 + 90*x^4 - 1411*x^6 - 1722*x^7 +...
-log(A(x))/3 = 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)^3 * (1-6*x^2+x^4)^3 * (1-14*x^3-x^6)^3 * (1-34*x^4+x^8)^3 * (1-82*x^5-x^10)^3 * (1-198*x^6+x^12)^3 *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...].

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

Cf. A203861, A204382, A204384, A020757.

/* 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, -3*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))^3, n)}

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A002203(n) * x^n/n ).

A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

Original entry on oeis.org

23, 33, 47, 62, 63, 86, 118, 134, 138, 143, 158, 167, 188, 195, 203, 204, 209, 223, 230, 243, 248, 275, 283, 294, 318, 323, 348, 368, 383, 385, 395, 398, 408, 411, 413, 418, 419, 426, 437, 440, 448, 454, 467, 473, 476, 489, 492, 503, 508, 518, 523, 558, 563, 566, 572, 608

Offset: 1

Alex Ratushnyak, Nov 03 2015

nonn

Intersection of A014134, A020757, A022544.

			Since 22 = 16+6, because 16 is a square and 6 is a triangular number, 22 is not a term.
23 is a term because there is no representation as S+T or S1+S2 or T1+T2, where S, S1, S2 are squares, and T, T1, T2 are triangular numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A014134, A020757, A022544, A264118.

N:= 1000: # for terms <= N
S:= [seq(i^2,i=0..floor(sqrt(N)))]: nS:= nops(S):
T:= [seq(i*(i+1)/2, i=0..floor(sqrt(2*N)))]: nT:= nops(T):
sort(convert({$1..N} minus {seq(seq(S[i]+S[j], j=1..i),i=1..nS),
seq(seq(S[i]+T[j],i=1..nS),j=1..nT),
seq(seq(T[i]+T[j],j=1..i),i=1..nT)}, list)); # Robert Israel, May 19 2020

mx = 610; Complement[ Range@ mx, Union@ Flatten@ Table[{i^2 + j^2, i(i + 1)/2 + j^2, i(i + 1)/2 + j(j + 1)/2}, {i, 0, Sqrt[2 mx]}, {j, 0, Sqrt[2 mx]}]] (* Robert G. Wilson v, Nov 29 2015 *)

A357505 Numbers that are not sum of two distinct triangular numbers.

Original entry on oeis.org

0, 2, 5, 8, 12, 14, 17, 19, 20, 23, 26, 30, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 90, 95, 96, 98, 103, 104, 107, 109, 110, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 132, 134, 138, 140, 143, 145, 147

Offset: 1

Stefano Spezia, Oct 01 2022

This sequence differs from A020757 in including the terms that are twice a triangular number and that cannot be expressed as a sum of two distinct triangular numbers: 0, 2, 12, 20, 30, 90, 110, 132, ... = 2*A357529.

Cf. A000217, A020757 (subsequence), A357504 (complement).

Cf. A357529.

TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=0, k<=148, k++, ok=1; For[h=0, A000217[h]A000217[h]] , ok=0]]; If[ok==1, AppendTo[a, k]]]; a (* Stefano Spezia, Nov 06 2022 *)

A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

Original entry on oeis.org

0, 1, 6, 10, 15, 45, 55, 66, 91, 120, 136, 231, 276, 300, 406, 435, 496, 561, 595, 630, 741, 780, 820, 861, 1081, 1225, 1326, 1431, 1830, 2016, 2080, 2145, 2211, 2415, 2485, 2701, 2850, 3240, 3321, 3486, 3655, 3916, 4005, 4465, 4560, 4950, 5050, 5356, 5460, 5565

Offset: 1

Stefano Spezia, Oct 02 2022

Subset of even terms of A357505, divided by 2. - Michel Marcus, Nov 05 2022

Cf. A000217 (supersequence), A002378.
Half of the complement of A357504 in A020756.
Half of the complement of A020757 in A357505.
Subsequence of A008851.

TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=0, k<=105, k++, ok=1; For[h=0, h<2k, h++, If[TriangularQ[2*A000217[k] - A000217[h]] && k!=h, ok=0]]; If[ok==1, AppendTo[a,k(k+1)/2]]]; a (* Stefano Spezia, Nov 05 2022 *)

A253195 Numbers congruent to 5 or 8 mod 9.

Original entry on oeis.org

5, 8, 14, 17, 23, 26, 32, 35, 41, 44, 50, 53, 59, 62, 68, 71, 77, 80, 86, 89, 95, 98, 104, 107, 113, 116, 122, 125, 131, 134, 140, 143, 149, 152, 158, 161, 167, 170, 176, 179, 185, 188, 194, 197, 203, 206, 212, 215, 221, 224, 230, 233, 239, 242, 248, 251

Offset: 1

Arkadiusz Wesolowski, Mar 24 2015

These numbers cannot be written as the sum of two triangular numbers.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Subsequence of A014132 and of A020757.

[n: n in [0..251] | n mod 9 in {5, 8}];

LinearRecurrence[{1, 1, -1}, {5, 8, 14}, 56]
Select[Range[300],MemberQ[{5,8},Mod[#,9]]&] (* Harvey P. Dale, Mar 17 2020 *)

Vec(x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2) + O(x^80)) \\ Michel Marcus, Mar 25 2015

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.
G.f.: x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-2) + 9.
a(n) = 9*n - a(n-1) - 5.
a(n) = 4*n + 2*ceiling(n/2) - floor(n/2) - 1.
a(n) = (9*n - (3/2)*(1 + (- 1)^n) + 1)/2.
E.g.f.: 1 + ((18*x - 1)*exp(x) - 3*exp(-x))/4. - David Lovler, Sep 06 2022

A375249 Integers which cannot be partitioned into the sum of a hexagonal number plus a pentagonal number, nor a hexagonal number plus a square, nor a pentagonal number plus a square.

Original entry on oeis.org

3, 8, 34, 43, 56, 59, 62, 68, 72, 73, 83, 89, 90, 97, 104, 110, 111, 114, 131, 138, 139, 148, 152, 163, 164, 167, 168, 186, 193, 200, 203, 205, 207, 222, 227, 228, 229, 233, 244, 249, 250, 252, 258, 269, 273, 279, 299, 300, 305, 306, 308, 309, 318, 319, 321, 333, 343, 344, 356, 364, 365

Offset: 1

Robert G. Wilson v, Aug 07 2024

Inspired by A160324 and analogous to A020757.
C. F. Gauss proved that all positive integers can be expressed as the sum of three triangular numbers. However, Zhi-Wei Sun (2009) has shown that there are 95 candidates for universal triples. This sequence looks at the {p4, p5, p6} triples and ask which integers require all three members to satisfy the sum.
Obviously, a(n) cannot be either a square, a pentagonal number, nor a hexagonal number.
There are more terms less than some integer in A020757 than in this sequence. In a sense, a square plus a pentagonal plus a hexagonal number is more efficient than the sum of three triangular numbers.
a(n) =~ 5.14 n^(.97).

			7 is not in the sequence since the third hexagonal number 6 plus the second square or pentagonal number sum to 7;
8 is in the sequence because s = {0, 1, 4}, p = {0, 1, 5}, and h = {0, 1, 6} with no two sets having members which sum to 8.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000290, A000326, A000384, A020757, A160324.

planeFiguratePi[r_,n_] := Floor[((r -4) +Sqrt[(r -4)^2 + 8n (r -2)])/(2 (r - 2))];
h = Table[PolygonalNumber[6, n], {n, 0, planeFiguratePi[6, 500]}];
p = Table[PolygonalNumber[5, n], {n, 0, planeFiguratePi[5, 500]}];
s = Table[PolygonalNumber[4, n], {n, 0, planeFiguratePi[4, 500]}];
Complement[ Range@ 500, Flatten[{Outer[Plus, h, p], Outer[Plus, h, s], Outer[Plus, p, s]} ]]

cp's OEIS Frontend

A322431 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^10 is zero.

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A322432 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^14 is zero.

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A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

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Haskell

Mathematica

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A203861 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^3 where Lucas(n) = A000204(n).

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A204383 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 where A002203(n) is the companion Pell numbers.

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A264101 Numbers that can't be represented as the sum of two squares, two triangular numbers, or a square and a triangular number.

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Maple

Mathematica

A357505 Numbers that are not sum of two distinct triangular numbers.

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Mathematica

A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

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A203861 G.f.: Product_{n>=1} (1 - Lucas(n)x^n + (-1)^nx^(2*n))^3 where Lucas(n) = A000204(n).

A204383 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n))^3 where A002203(n) is the companion Pell numbers.