A359240 -id:A359240 - cp's OEIS frontend

A078703 Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.

1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 4, 2, 1, 3, 2, 1, 4, 2, 1, 2, 1, 3, 3, 1, 2, 2, 2, 2, 2, 1, 1, 6, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 1, 3, 1, 2, 4, 1, 1, 3, 2, 2, 4, 2, 2, 2, 2, 1

Offset: 1

R. L. Coffman, K. W. McLaughlin and R. J. Dawson (robert.l.coffman(AT)uwrf.edu), Dec 19 2002

Also number of symmetric unimodal consecutive integer sequences that sum to the integer n (e.g., 4+5+6+5+4 = 24 = n). Also number of double trapezoidal arrangements of n objects, denoted SDT(n); i.e., the number of ways to arrange n objects into symmetrically-placed, congruent isosceles trapezoids adjoined at overlapping largest bases.

Also number of divisors of 4*n-1 of form 4*k+1 (or 4*k+3). - Vladeta Jovovic, Jan 05 2004. Therefore a(n) is one half of the number of divisors of A004767(n-1) (numbers 3 (mod 4)). - Wolfdieter Lang, Jul 29 2016

			SDT(34) = 4 since we have 34 or 11+12+11 or 6+7+8+7+6 or 2+3+4+5+6+5+4+3+2, Also 4*34 - 1 = 135 = (3^3)*(5^1) so that r1=3 and r2=1 (p1=3 and p2=5), resulting in SDT(34) = (3+1)*(1+1)/2 = 4.
a(4) = 2 since 4 = 2^2 - 2*0 = 4^2 - 2*6. Also A034178(4*4 - 1) = 2 since 15 = 4^2 - 1^2 = 8^2 - 7^2. - _Michael Somos_, May 11 2011
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + x^11 + ...
Number of divisors of numbers 3 (mod 4) (see the Jovovic Jan 05 2004 comment): a(16) = 3 from the 2*3 = 6 divisors [1, 3, 7, 9, 21, 63] of 63 = A004767(15), being 1, -1, -1, 1, 1, -1 (mod 4). - _Wolfdieter Lang_, Jul 29 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.

Cf. A000005, A001620, A004767, A001227, A034178, A359227, A359240.

(* This defines SDT(n): *)
SDT[n_] := Length[Cases[Range[1, n], j_ /; Cases[Range[1, j], k_ /; Plus @@ Join[Range[k, j], Range[j - 1, k, -1]] == n] != {}]] The restricted factorization technique for obtaining SDT(n) is encoded as follows: SDT[n_] := (Times @@ Cases[FactorInteger[4 n - 1], {p_, r_} -> r + 1])/2
Rest[ CoefficientList[ Series[ Sum[x^k/(1 - x^(4k - 1)), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
a[ n_] := If[ n < 1, 0, With[{m = 4 n - 1}, Sum[1 - Sign@Mod[m - k^2, 2 k], {k, Sqrt@m}]]]; (* Michael Somos, Aug 01 2016 *)
a[n_] := DivisorSigma[0, 4*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

{a(n) = if( n<1, 0, n = 4*n-1; sum(k=1, sqrtint(n), 0 == (n - k^2) % (2*k)))}; /* Michael Somos, Aug 01 2016 */

a(n) = ((r1 + 1)*(r2 + 1)*...*(rk + 1))/2, where ((p1^r1)*(p2^r2)*...*(pk^rk)) is the factorization of 4*n - 1 into (odd) primes.
G.f.: Sum_{n>0} x^n/(1-x^(4*n-1)). - Vladeta Jovovic, Jan 05 2004
a(n) = A034178(4*n - 1). - Michael Somos, May 11 2011
G.f.: Sum_{n >= 1} x^(3*n-2)/(1 - x^(4*n-3)). - Peter Bala, Jan 08 2021
From Amiram Eldar, Dec 26 2022: (Start)
a(n) = A000005(A004767(n-1))/2.
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 4*log(2))*n/4 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)
G.f.: Sum_{n >= 1} x^(n^2)/(1-x^(2*n-1)) (conjecture). - Joerg Arndt, Jan 04 2024

A359239 Number of divisors of 3n-2 of form 3k+2.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 2, 1, 2, 0, 2, 0, 4, 0, 2, 0, 2, 2, 2, 0, 3, 0, 4, 0, 2, 0, 2, 2, 4, 0, 2, 0, 4, 0, 2, 0, 4, 2, 2, 1, 2, 0, 4, 0, 4, 0, 2, 2, 2, 0, 4, 0, 6, 0, 2, 0, 2, 2, 2, 0, 4, 2, 4, 0, 3, 0, 2, 2, 4, 0, 2, 0, 6, 0, 2, 0, 4, 2, 4, 0, 2, 0, 4, 2, 4, 0, 2, 2, 2

Offset: 1

Seiichi Manyama, Dec 22 2022

Cf. A001822, A359211, A359212.

Cf. A359240, A359241.

Table[Count[Divisors[3 n-2],?(IntegerQ[(#-2)/3]&)],{n,100}] (* _Harvey P. Dale, Apr 23 2023 *)
a[n_] := DivisorSum[3*n-2, 1 &, Mod[#, 3] == 2 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(3*n-2, d, d%3==2);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(3*k-1)))))

a(n) = A001822(3*n-2).

G.f.: Sum_{k>0} x^(2*k)/(1 - x^(3*k-1)).

A359241 Number of divisors of 5n-4 of form 5k+4.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 22 2022

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

Cf. A359239, A359240.

Cf. A001899.

a[n_] := DivisorSum[5*n-4, 1 &, Mod[#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(5*n-4, d, d%5==4);
```

my(N=100, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^(5*k-1)))))

a(n) = A001899(5*n-4).

G.f.: Sum_{k>0} x^(4*k)/(1 - x^(5*k-1)).

A359327 Number of divisors of 6n-5 of form 6k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A319995, A359305, A359324, A359325, A359326.

Cf. A359239, A359240, A359241.

a[n_] := DivisorSum[6*n-5, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(6*n-5, d, d%6==5);
```

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/(1-x^(6*k-1)))))

a(n) = A319995(6*n-5).

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(6*k-1)).

A359290 Number of divisors of 4n-2 of form 4k+3.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 24 2022

nonn

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

Cf. A001842, A078703, A359240.

Table[Count[Divisors[4 n-2],?(IntegerQ[(#-3)/4]&)],{n,100}] (* _Harvey P. Dale, May 09 2023 *)
a[n_] := DivisorSum[4*n-2, 1 &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(4*n-2, d, d%4==3);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(4*k-1)))))

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)/(1-x^(4*k-2)))))

a(n) = A001842(4*n-2).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(4*k-1)).
G.f.: Sum_{k>0} x^(3*k-1)/(1 - x^(4*k-2)).

A364084 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(4k-1)).

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 0, 4, 0, 0, 5, 2, 0, 6, 0, 0, 7, 0, 5, 8, 0, 0, 9, 0, 0, 16, 0, 0, 11, 3, 0, 12, 7, 0, 13, 0, 0, 14, 0, 8, 22, 0, 0, 16, 0, 0, 26, 0, 0, 18, 0, 8, 19, 10, 0, 24, 0, 0, 21, 0, 11, 22, 9, 0, 23, 0, 0, 36, 0, 0, 34, 0, 0, 36, 13, 0, 27, 0, 0, 28, 0, 14, 29, 0, 11, 40, 0, 0, 46, 5, 0, 32, 0, 0

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A359240.

a[n_] := DivisorSum[4*n - 3, # + 1 &, Mod[#, 4] == 3 &]/4; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)

a(n) = sumdiv(4*n-3, d, (d%4==3)*(d+1))/4;

a(n) = (1/4) * Sum_{d | 4*n-3, d==3 (mod 4)} (d+1).

G.f.: Sum_{k>0} x^(3*k) / (1 - x^(4*k-1))^2.

A363407 Sum of divisors of 4n-3 of form 4k+3.

Original entry on oeis.org

0, 0, 3, 0, 0, 10, 0, 0, 14, 0, 0, 18, 7, 0, 22, 0, 0, 26, 0, 18, 30, 0, 0, 34, 0, 0, 60, 0, 0, 42, 11, 0, 46, 26, 0, 50, 0, 0, 54, 0, 30, 84, 0, 0, 62, 0, 0, 100, 0, 0, 70, 0, 30, 74, 38, 0, 93, 0, 0, 82, 0, 42, 86, 34, 0, 90, 0, 0, 140, 0, 0, 132, 0, 0, 140, 50, 0, 106, 0, 0, 110, 0, 54, 114, 0, 42, 156

Offset: 1

Seiichi Manyama, Jul 08 2023

nonn

Cf. A050452, A359240, A364084.

a[n_] := DivisorSum[4*n - 3, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)

PARI
```
a(n) = sumdiv(4*n-3, d, (d%4==3)*d);
```

a(n) = A050452(4*n-3).

G.f.: Sum_{k>0} (4*k-1) * x^(3*k) / (1 - x^(4*k-1)).

cp's OEIS Frontend

A078703 Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.

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A359239 Number of divisors of 3*n-2 of form 3*k+2.

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A359241 Number of divisors of 5*n-4 of form 5*k+4.

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A359327 Number of divisors of 6*n-5 of form 6*k+5.

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A359290 Number of divisors of 4*n-2 of form 4*k+3.

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A364084 Expansion of Sum_{k>0} k * x^(3*k) / (1 - x^(4*k-1)).

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A363407 Sum of divisors of 4*n-3 of form 4*k+3.

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A359239 Number of divisors of 3n-2 of form 3k+2.

A359241 Number of divisors of 5n-4 of form 5k+4.

A359327 Number of divisors of 6n-5 of form 6k+5.

A359290 Number of divisors of 4n-2 of form 4k+3.

A364084 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(4k-1)).

A363407 Sum of divisors of 4n-3 of form 4k+3.