A274681 -id:A274681 - cp's OEIS frontend

A299645 Numbers of the form m(8m + 5), where m is an integer.

0, 3, 13, 22, 42, 57, 87, 108, 148, 175, 225, 258, 318, 357, 427, 472, 552, 603, 693, 750, 850, 913, 1023, 1092, 1212, 1287, 1417, 1498, 1638, 1725, 1875, 1968, 2128, 2227, 2397, 2502, 2682, 2793, 2983, 3100, 3300, 3423, 3633, 3762, 3982, 4117, 4347, 4488, 4728, 4875

Offset: 1

Bruno Berselli, Feb 26 2018

Equivalently, numbers k such that 32*k + 25 is a square. This means that 4*a(n) + 3 is a triangular number.

Interleaving of A139277 and A139272 (without 0).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A139272, A139277.
Subsequence of A011861, A047222.
Cf. numbers of the form m*(8*m + h): A154260 (h=1), A014494 (h=2), A274681 (h=3), A046092 (h=4), this sequence (h=5), 2*A074377 (h=6), A274979 (h=7).

List([1..50], n -> (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4);

[div((8n*(n-1)-(2n-1)*(-1)^n-1), 4) for n in 1:50] # Peter Luschny, Feb 27 2018

[(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4: n in [1..50]];

seq((exp(I*Pi*x)*(1-2*x)+8*(x-1)*x-1)/4, x=1..50); # Peter Luschny, Feb 27 2018

Table[(8 n (n - 1) - (2 n - 1) (-1)^n - 1)/4, {n, 1, 50}]

makelist((8*n*(n-1)-(2*n-1)*(-1)^n-1)/4, n, 1, 50);

vector(50, n, nn; (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4)

concat(0, Vec(x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Feb 27 2018

[(8*n*(n-1)-(2*n-1)*(-1)**n-1)/4 for n in range(1, 60)]

def A299645(n): return (n>>1)*((n<<2)+(1 if n&1 else -5)) # Chai Wah Wu, Mar 11 2025

[(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4 for n in (1..50)]

O.g.f.: x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2).
E.g.f.: (1 + 2*x - (1 - 8*x^2)*exp(2*x))*exp(-x)/4.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (8*n*(n - 1) - (2*n - 1)*(-1)^n - 1)/4 = (2*n + (-1)^n - 1)*(4*n - 3*(-1)^n - 2)/4. Therefore, 3 and 13 are the only prime numbers in this sequence.
a(n) + a(n+1) = 4*n^2 for even n, otherwise a(n) + a(n+1) = 4*n^2 - 1.
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=2} 1/a(n) = 8/25 + (sqrt(2)-1)*Pi/5.
Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/5 - sqrt(2)*log(2*sqrt(2)+3)/5 - 8/25. (End)
a(n) = (n-1)*(4*n+1)/2 if n is odd and a(n) = n*(4*n-5)/2 if n is even. - Chai Wah Wu, Mar 11 2025

A274680 Values of n such that 2n+1 and 4n+1 are both triangular numbers.

Original entry on oeis.org

0, 16065, 545751, 21394547226, 726784809030, 28491418065071115, 967869505172593485, 37942420317086720855700, 1288925370210688376036076, 50528452330120333959563160501, 1716479960463788790499334882595, 67289447366315927998308608003134830

Offset: 1

Colin Barker, Jul 02 2016

			16065 is in the sequence because 2*16065+1 = 32131, 4*16065+1 = 64261, and 32131 and 64261 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..325
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

Cf. A124174 (2*n+1 and 9*n+1), A274579 (2*n+1 and 5*n+1), A274603 (2*n+1 and 3*n+1).

Rest@ CoefficientList[Series[459 x^2 (35 + 1154 x + 35 x^2)/((1 - x) (1 - 1154 x + x^2) (1 + 1154 x + x^2)), {x, 0, 12}], x] (* Michael De Vlieger, Jul 02 2016 *)

isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(4*n+1, 3)

concat(0, Vec(459*x^2*(35+1154*x+35*x^2)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)) + O(x^20)))

Intersection of A074377 and A274681.

G.f.: 459*x^2*(35+1154*x+35*x^2) / ((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)).

A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

Original entry on oeis.org

0, 9, 15, 42, 54, 99, 117, 180, 204, 285, 315, 414, 450, 567, 609, 744, 792, 945, 999, 1170, 1230, 1419, 1485, 1692, 1764, 1989, 2067, 2310, 2394, 2655, 2745, 3024, 3120, 3417, 3519, 3834, 3942, 4275, 4389, 4740, 4860, 5229, 5355, 5742, 5874, 6279, 6417

Offset: 1

Colin Barker, Jul 04 2016

Numbers of the type floor(3*m*(m+1)/4) for which floor(3*m*(m+1)/4) = 3*floor(m*(m+1)/4). A014601 lists the values of m. - Bruno Berselli, Jan 13 2017

Numbers of the form 3*k*(4*k + 1) for k in Z. - Peter Bala, Nov 21 2024

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A014601.
Cf. A000096 (k+1), A074377 (2*k+1), A045943 (3*k+1), A274681 (4*k+1), A085787 (5*k+1).
Cf. similar sequences listed in A274830.

Table[3 (2 n - 1) (2 n + (-1)^n - 1)/4, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,9,15,42,54},50] (* Harvey P. Dale, Apr 13 2025 *)

PARI
```
isok(n) = ispolygonal(6*n+1, 3)
```

select(n->ispolygonal(6*n+1, 3), vector(7000, n, n-1))

concat(0, Vec(3*x^2*(3+2*x+3*x^2)/((1-x)^3*(1+x)^2) + O(x^60)))

G.f.: 3*x^2*(3 + 2*x + 3*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = 3*(2*n - 1)*(2*n + (-1)^n - 1)/4. Therefore:
a(n) = 3*n*(2*n - 1)/2 for n even,
a(n) = 3*(n-1)*(2*n - 1)/2 for n odd.

A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

Original entry on oeis.org

0, 2, 5, 11, 17, 27, 36, 50, 62, 80, 95, 117, 135, 161, 182, 212, 236, 270, 297, 335, 365, 407, 440, 486, 522, 572, 611, 665, 707, 765, 810, 872, 920, 986, 1037, 1107, 1161, 1235, 1292, 1370, 1430, 1512, 1575, 1661, 1727, 1817, 1886, 1980, 2052, 2150, 2225

Offset: 1

Colin Barker, Jul 08 2016

From Peter Bala, Nov 21 2024: (Start)
Numbers of the form n*(7*n + 3)/2 for n in Z. Cf. A057570.
The sequence terms occur as the exponents in the expansion of Product_{n >= 1} (1 - x^(7*n)) * (1 + x^(7*n-2)) * (1 + x^(7*n-5)) = 1 + x^2 + x^5 + x^11 + x^17 + x^27 + x^36 + .... Cf. A363800. (End)

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A057570, A363800.

Cf. similar sequences where k*n+1 is a triangular number: A000096 (k=1), A074377 (k=2), A045943 (k=3), A274681 (k=4), A085787 (k=5), A274757 (k=6).

Table[(14 (n - 1) n + (2 n - 1) (-1)^n + 1)/16, {n, 1, 60}] (* Bruno Berselli, Jul 08 2016 *)

select(n->ispolygonal(7*n+1, 3), vector(3000, n, n-1))

concat(0, Vec(x^2*(2+3*x+2*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

G.f.: x^2*(2 + 3*x + 2*x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = (14*(n - 1)*n + (2*n - 1)*(-1)^n + 1)/16. Therefore:
a(n) = n*(7*n - 6)/8 for n even,
a(n) = (n - 1)*(7*n - 1)/8 for n odd.
E.g.f.: (x*(7*x -1)*cosh(x) + (7*x^2 + x + 1)*sinh(x))/8. - Stefano Spezia, Nov 26 2024

Edited by Bruno Berselli, Jul 08 2016

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A299645 Numbers of the form m*(8*m + 5), where m is an integer.

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A274680 Values of n such that 2*n+1 and 4*n+1 are both triangular numbers.

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A274757 Numbers k such that 6*k+1 is a triangular number (A000217).

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A274830 Numbers k such that 7*k+1 is a triangular number (A000217).

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A299645 Numbers of the form m(8m + 5), where m is an integer.

A274680 Values of n such that 2n+1 and 4n+1 are both triangular numbers.