A378569 a(n) = 3*n*(n+1) + 7.

7, 13, 25, 43, 67, 97, 133, 175, 223, 277, 337, 403, 475, 553, 637, 727, 823, 925, 1033, 1147, 1267, 1393, 1525, 1663, 1807, 1957, 2113, 2275, 2443, 2617, 2797, 2983, 3175, 3373, 3577, 3787, 4003, 4225, 4453, 4687, 4927, 5173, 5425, 5683, 5947, 6217, 6493, 6775, 7063, 7357, 7657, 7963, 8275, 8593, 8917, 9247, 9583, 9925

Offset: 0

Author

M. F. Hasler, Feb 04 2025

Keywords

• nonn
• easy

Comments

The terms a(1) = 13 through a(7) = 175, coincide with A102724(2..8), cumulative sums of pairs of primes. From a(8) = 223, it differs from A102724(9) = 227.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Paul Bourgarel, Journal de mathématiques élémentaires, (1884), p. 111.
• Antreas Hatzipolakis, A sequence, mail to the SeqFan list / google group, Jan. 31, 2025.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001043, A102724.

Programs

Mathematica

LinearRecurrence[{3,-3,1},{7, 13, 25 }, 58] (* James C. McMahon, Feb 08 2025 *)

PARI

A378569(n)=3*n*(n+1)+7;
apply(A378569, [0..55])

Formula

a(n) = A102724(n+1) for 1 <= n <= 7, where A102724 = partial sums of A001043(n) = prime(n)+prime(n+1).

From Vincenzo Librandi, Feb 06 2025: (Start)

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

G.f.: (7-8x+7x^2)/ (1-3x+3x^2-x^3). (End)

E.g.f.: exp(x)*(7 + 6*x + 3*x^2). - Stefano Spezia, Feb 06 2025