A328791 Triangular numbers of the form k^2 + 3.

3, 28, 903, 30628, 1040403, 35343028, 1200622503, 40785822028, 1385517326403, 47066803275628, 1598885794044903, 54315050194251028, 1845112820810490003, 62679520857362409028, 2129258596329511416903, 72332112754346025765628, 2457162575051435364614403

Offset

1,1

Comments

There exist triangular numbers of the form k^2 + j for j=0 (A001110), j=1 (A164055), j=2 (A214838), and j=3 (this sequence), but not for j=4,7,8,13,16,18,... (A328792).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

a(1) = 3, a(2) = 28; for n > 2, a(n) = 34*a(n-1) - a(n-2) - 46.

G.f.: x*(3 - 77*x + 28*x^2)/((1 - x)*(1 - 34*x + x^2)). - Andrew Howroyd, Nov 14 2025

Prog

(Python)
limit = 10**7 # rough limit for k
A000217 = set(k*(k+1)//2 for k in range(14*limit//10))
A117950 = set(k**2 + 3 for k in range(limit))
print(sorted(A000217 & A117950)) # Michael S. Branicky, Mar 28 2021

Crossrefs

Cf. A001110, A164055, A214838, A328792.

Intersection of A000217 and A117950.

Cf. A276598 (the k's).

Sequence in context: A015474 A324462 A053601 * A140990 A196735 A395573

Adjacent sequences: A328788 A328789 A328790 * A328792 A328793 A328794

Keyword

nonn,easy

Author

Jon E. Schoenfield, Oct 27 2019

Status

approved