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

1,2

Comments

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

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).

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.

Example

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

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 *)

Prog

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

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.

Sequence in context: A326161 A354838 A038253 * A287350 A294091 A032379

Adjacent sequences: A274678 A274679 A274680 * A274682 A274683 A274684

Keyword

nonn,easy

Author

Colin Barker, Jul 02 2016

Status

approved