A204542 -id:A204542 - cp's OEIS frontend

A204221 Integers of the form (N^2 - 1) / 15.

0, 1, 8, 13, 17, 24, 45, 56, 64, 77, 112, 129, 141, 160, 209, 232, 248, 273, 336, 365, 385, 416, 493, 528, 552, 589, 680, 721, 749, 792, 897, 944, 976, 1025, 1144, 1197, 1233, 1288, 1421, 1480, 1520, 1581, 1728, 1793, 1837, 1904, 2065, 2136, 2184, 2257, 2432

Offset: 0

Michael Somos, Jan 13 2012

Equivalently, numbers in increasing order of the form m(15m+2) or m(15m+8)+1, where m = 0,-1,1,-2,2,-3,3,... [Bruno Berselli, Nov 27 2012]

The sequence terms occur as exponents in the expansion of the identity Product_{n >= 0} (1 - x^(20*n+1))*(1 - x^(20*n+19))*(1 - x^(20*n+8))*(1 - x^(20*n+12))*(1 - x^(20*n+9))*(1 - x^(20*n+11))*(1 - x^(10*n+10)) = Sum_{n >= 0} x^(n^2+n)*Product_{k >= 2*n+1} 1 - x^k = 1 - x - x^8 + x^13 + x^17 - - + + .... See Andrews et al., p. 591, Exercise 6(c). - Peter Bala, Feb 22 2021.

George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Cambridge University Press, 1999.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
John Greene and James A. Sellers, Extending recent parity results of Nyirenda and Mugwangwavari for partitions with initial repetitions, Integers (2025), Vol. 25, Art. No. A32. See p. 8.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A204220, A204542 (square roots of 15*a(n)+1), A379210.

Cf. similar sequences listed in A219257.

[n: n in [0..2500] | IsSquare(15*n+1)]; // Bruno Berselli, Nov 23 2012

/* By comment: */ s:=[0, 1] cat &cat[[t*(15*t+2), t*(15*t+8)+1]: t in [-n,n], n in [1..13]]; Sort(s); // Bruno Berselli, Nov 27 2012

A204221 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(15*n+1), integer) then print(n);
fi; od; end:
A204221(2500); # Peter Bala, Dec 18 2024

Select[Range[0, 2500], IntegerQ[Sqrt[15 # + 1]] &] (* Bruno Berselli, Nov 23 2012 *)

{a(n) = (15*n^2 + n*[8, 2, 28, 22][n%4 + 1] + 12) \ 16}

|A204220(n)| is the characteristic function of the numbers in this sequence.
a(-1 - n) = a(n).
G.f. x*(x^2-x+1)*(x^4+8*x^3+12*x^2+8*x+1) / ( (1+x)^2*(1+x^2)^2*(1-x)^3 ). - R. J. Mathar, Jan 28 2012
a(n) = (30*n-10*i^(n(n-1))+3*(-1)^n+7)*(30*n-10*i^(n(n-1))+3*(-1)^n+23)/960, where i=sqrt(-1). - Bruno Berselli, Nov 28 2012
Sum_{n>=1} 1/a(n) = 15/4 - cot(2*Pi/15)*Pi/2 - Pi/(2*sqrt(3)) + sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
From Peter Bala, Dec 17 2024: (Start)
a(n) is quasi-polynomial in n: for n >= 0,
a(4*n+1) = 15*n^2 + 8*n + 1; a(4*n+2) = 15*n^2 + 22*n + 8;
a(4*n+3) = 15*n^2 + 28*n + 13; a(4*n+4) = 15*n^2 + 32*n + 17.
For 1 <= k <= 4, a(4*n+k) = (N_k(n)^2 - 1)/15, where N_1(n) = 15*n + 4, N_2(n) = 15*n + 11, N_3(n) = 15*n + 14 and N_4(n) = 15*n + 16. (End)

A379210 List of integers of the form (N^2 - 4)/15.

Original entry on oeis.org

0, 3, 4, 11, 19, 32, 35, 52, 68, 91, 96, 123, 147, 180, 187, 224, 256, 299, 308, 355, 395, 448, 459, 516, 564, 627, 640, 707, 763, 836, 851, 928, 992, 1075, 1092, 1179, 1251, 1344, 1363, 1460, 1540, 1643, 1664, 1771, 1859, 1972, 1995, 2112, 2208, 2331, 2356, 2483

Offset: 1

Peter Bala, Dec 18 2024

Compare with A204221.
The sequence terms occur as exponents in the expansion of Sum_{n >= 1} x^(n*(n-1)) * Product_{k >= 2*n} 1 - x^k = 1 - x^3 - x^4 + x^11 + x^19 - x^32 - x^35 + + - - ....
|A379212(n)| is the characteristic function of the numbers in this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A204220, A204221, A204542, A379211 (square roots of 15*a(n) + 4), A379212.

A379210 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(15*n+4), integer) then print(n);
fi; od; end:
A379210(2500);

LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0, 3, 4, 11, 19, 32, 35, 52, 68},52] (* James C. McMahon, Dec 24 2024 *)

a(n) = (1/32)*(30*n^2 - 30*n + 1) + (-1)^(n)*(5/32)*(2*n - 1) - (-1)^(n*(n+1)/2)*(1/16)*(6*n - 3 + (-1)^n).
a(n) is quasi-periodic in n: for n >= 0,
a(4*n+1) = 15*n^2 - 26*n + 11; a(4*n+2) = 15*n^2 - 16*n + 4;
a(4*n+3) = 15*n^2 - 14*n + 3; a(4*n+4) = 15*n^2 - 4*n.
a(1-n) = a(n).
15*a(n) + 4 = A379211(n)^2.
G.f: x*(3*x^6 + x^5 + 7*x^4 + 8*x^3 + 7*x^2 + x + 3)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
E.g.f.: ((15*x^2 + 35*x - 2)*cosh(x) + 2*(cos(x) + 3*x*cos(x) + 2*sin(x) - 3*x*sin(x)) + (15*x^2 + 25*x + 3)*sinh(x))/16. - Stefano Spezia, Dec 23 2024

A379211 List of positive integers that are congruent to {2, 7, 8, 13} mod 15.

Original entry on oeis.org

2, 7, 8, 13, 17, 22, 23, 28, 32, 37, 38, 43, 47, 52, 53, 58, 62, 67, 68, 73, 77, 82, 83, 88, 92, 97, 98, 103, 107, 112, 113, 118, 122, 127, 128, 133, 137, 142, 143, 148, 152, 157, 158, 163, 167, 172, 173, 178, 182, 187, 188, 193, 197, 202, 203, 208, 212, 217, 218, 223, 227, 232, 233, 238, 242, 247, 248, 253, 257, 262

Offset: 1

Peter Bala, Dec 18 2024

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

Cf. A001622, A072703, A151972, A204542, A315211, A379210.

a := proc(n) option remember;
      `if`(n < 5, [0, 2, 7, 8, 13][n+1], 15 + a(n-4))
     end:
seq(a(n), n = 1..70);

LinearRecurrence[{1, 0, 0, 1, -1}, {2, 7, 8, 13, 17}, 70] (* Amiram Eldar, Dec 24 2024 *)

a(n) = 15 + a(n-4); a(n) = - a(1-n).
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 6.
G.f.: x*(x^2 + 3*x + 1)*(2*x^2 - x + 2)/((1 + x)*(1 - x)^2*(1 + x^2)).
a(n)^2 = 15 * A379210(n) + 4.
For n >= 2, a(n-1) + a(n+1) = A072703(n).
It appears that a(n) + a(n+1) = (3/2) * A315211(n).
E.g.f.: (8 - 3*cos(x) + 5*(3*x - 1)*cosh(x) + 3*sin(x) + 5*(3*x - 2)*sinh(x))/4. - Stefano Spezia, Dec 23 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(5*sqrt(3)*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 24 2024

A257645 a(n) = 15*n + 14.

Original entry on oeis.org

14, 29, 44, 59, 74, 89, 104, 119, 134, 149, 164, 179, 194, 209, 224, 239, 254, 269, 284, 299, 314, 329, 344, 359, 374, 389, 404, 419, 434, 449, 464, 479, 494, 509, 524, 539, 554, 569, 584, 599, 614, 629, 644, 659, 674, 689, 704, 719, 734, 749, 764, 779

Offset: 0

Arkadiusz Wesolowski, Nov 05 2015

A123159(a(n)) <= 4.
This is not a subsequence of A047725 (for example, 239 is missing in A047725). - Bruno Berselli, Nov 06 2015
Equivalently, intersection of A016897 and A016789. - Bruno Berselli, Jan 24 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A016897, A175887, A123159, A204542.

Magma
```
[15*n+14: n in [0..51]];
```
Maple
```
seq(15*n+14, n=0..51);
```
Mathematica
```
15 Range[50] - 1
```
PARI
```
for(n=0, 51, print1(15*n+14, ", "));
```

G.f.: (14 + x)/(1 - x)^2.
a(n) = A008597(n+1) - 1. - Omar E. Pol, Nov 05 2015
a(n) = A016897(3n+2) = A175887(2n+2) = A204542(4n+4). - Bruno Berselli, Nov 06 2015
E.g.f.: (15*x + 14)*exp(x). - G. C. Greubel, Apr 23 2018
a(n) = 2*a(n-1)-a(n-2). - Wesley Ivan Hurt, Dec 27 2023

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A204221 Integers of the form (N^2 - 1) / 15.

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A379210 List of integers of the form (N^2 - 4)/15.

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A379211 List of positive integers that are congruent to {2, 7, 8, 13} mod 15.

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A257645 a(n) = 15*n + 14.

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