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A154153 Indices n such that 28 plus the n-th triangular number is a perfect square. 2
6, 8, 47, 57, 278, 336, 1623, 1961, 9462, 11432, 55151, 66633, 321446, 388368, 1873527, 2263577, 10919718, 13193096, 63644783, 76895001, 370948982, 448176912, 2162049111, 2612166473, 12601345686, 15224821928, 73446025007, 88736765097, 428074804358, 517195768656 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..30.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

David A. Corneth, Conjectured formula for a(n)

FORMULA

{k: 28+k*(k+1)/2 in A000290}.

Conjectures: (Start)

a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).

G.f.: x*(-6-2*x-3*x^2+2*x^3+7*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).

G.f.: ( 14 + 1/(x-1) + (14+29*x)/(x^2-2*x-1) + (-1-12*x)/(x^2+2*x-1) )/2. (End)

See also the Corneth link - David A. Corneth, Mar 18 2019

EXAMPLE

6*(6+1)/2+28 = 7^2. 8*(8+1)/2+28 = 8^2. 47*(47+1)/2+28 = 34^2. 57*(57+1)/2+28 = 41^2.

MATHEMATICA

Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* G. C. Greubel, Sep 03 2016 *)

PROG

(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) ); }

(PARI) a(n)=(15*g((n+1)\2)-1)/2+(-1)^n*f((n+1)\2)

g(n) = round((-1/2+sqrt(2)/2)*(3+sqrt(8))^n+(-1/2-sqrt(2)/2)*(3-sqrt(8))^n)

f(n) = round(((2+sqrt(2))/4)*(3-sqrt(8))^n+((2-sqrt(2))/4)*(3+sqrt(8))^n) \\ based on conjectured formula \\ David A. Corneth, Mar 18 2019

CROSSREFS

Cf. A000217, A000290.

Cf. A001108 (0), A006451 (1), A154138 (3), A154139 (4), A154140 (6), A154141 (8), A154142 (9), A154143 (10), A154144 (13), A154145 (15), A154146 (16), A154147 (19), A154148 (21), A154149 (22), A154150(24), A154151 (25), A154151 (26), this sequence (28), A154154 (30).

Sequence in context: A054102 A261063 A000380 * A164640 A223852 A227167

Adjacent sequences:  A154150 A154151 A154152 * A154154 A154155 A154156

KEYWORD

nonn

AUTHOR

R. J. Mathar, Oct 18 2009

EXTENSIONS

a(21)-a(30) from Amiram Eldar, Mar 18 2019

STATUS

approved

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Last modified September 21 03:13 EDT 2020. Contains 337266 sequences. (Running on oeis4.)