A274681 Numbers k such that 4*k + 1 is a triangular number.

0, 5, 11, 26, 38, 63, 81, 116, 140, 185, 215, 270, 306, 371, 413, 488, 536, 621, 675, 770, 830, 935, 1001, 1116, 1188, 1313, 1391, 1526, 1610, 1755, 1845, 2000, 2096, 2261, 2363, 2538, 2646, 2831, 2945, 3140, 3260, 3465, 3591, 3806, 3938, 4163, 4301, 4536

Offset: 1

Author

Colin Barker, Jul 02 2016

Keywords

• nonn
• easy

Comments

Also, numbers of the form m*(8*m + 3) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

Examples

			5 is in the sequence since 4*5 + 1 = 21 is a triangular number (21 = 1 + 2 + 3 + 4 + 5 + 6). - _Michael B. Porter_, Jul 03 2016
		

Links

• 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).

Crossrefs

Cf. A000217, A000096 (n+1), A074377 (2*n+1), A045943 (3*n+1), A085787 (5*n+1).

Cf. A057029.

Cf. similar sequences listed in A299645.

Programs

Magma

[(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: n in [1..80]]; // Wesley Ivan Hurt, Jul 02 2016

Maple

A274681:=n->(1-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A274681(n), n=1..100); # Wesley Ivan Hurt, Jul 02 2016

Mathematica

Rest@ CoefficientList[Series[x^2 (5 + 6 x + 5 x^2)/((1 - x)^3 (1 + x)^2), {x, 0, 48}], x] (* Michael De Vlieger, Jul 02 2016 *)
Select[Range[0,5000],OddQ[Sqrt[8(4#+1)+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,11,26,38},50] (* Harvey P. Dale, Apr 21 2018 *)

PARI

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

PARI

select(n->ispolygonal(4*n+1, 3), vector(10000, n, n-1))

PARI

concat(0, Vec(x^2*(5+6*x+5*x^2)/((1-x)^3*(1+x)^2) + O(x^100)))

Python

def A274681(n): return (n>>1)*((n<<2)+(-1 if n&1 else -3)) # Chai Wah Wu, Mar 11 2025

Formula

G.f.: x^2*(5 + 6*x + 5*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) = A057029(n) - 1.

a(n) = (1 - (-1)^n + 2*(-4 + (-1)^n)*n + 8*n^2)/4.

a(n) = (4*n^2 - 3*n)/2 for n even, a(n) = (4*n^2 - 5*n + 1)/2 for n odd.