A202391 Indices of the smallest of four consecutive triangular numbers summing up to a square.

5, 39, 237, 1391, 8117, 47319, 275805, 1607519, 9369317, 54608391, 318281037, 1855077839, 10812186005, 63018038199, 367296043197, 2140758220991, 12477253282757, 72722761475559, 423859315570605, 2470433131948079, 14398739476117877, 83922003724759191, 489133282872437277

Offset

1,1

Comments

Positive integers n such that A000217(n) + A000217(n + 1) + A000217(n + 2) + A000217(n + 3) is a square (=A075870(n+1)^2).

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Formula

a(n) = A002315(n) - 2.

G.f.: x*(1+x)*(x-5) / ( (x-1)*(1-6*x+x^2) ). - R. J. Mathar, Dec 19 2011

a(n+2) = 6*a(n+1) - a(n) + 8; a(n+3) = 7*a(n+2) - 7*a(n+1) + a(n); a(n+1) = (-4 + (7 + 5*r)*(3 + 2*r)^n + (7 - 5*r)*(3 - 2*r)^n)/2 where r = sqrt(2). - Paul Weisenhorn, Jan 13 2013

Mathematica

LinearRecurrence[{7, -7, 1}, {5, 39, 237}, 25] (* Paolo Xausa, May 19 2026 *)

Prog

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -7, 7]^(n-1)*[5; 39; 237])[1, 1] \\ Charles R Greathouse IV, May 16 2026

Crossrefs

Cf. A176541, A176542, A165517, A116476, A075870.

Sequence in context: A123614 A218918 A075135 * A053573 A003482 A221357

Adjacent sequences: A202388 A202389 A202390 * A202392 A202393 A202394

Keyword

nonn,easy

Author

Max Alekseyev, Dec 18 2011

Extensions

More terms from Paolo Xausa, May 19 2026

Status

approved