A169634 a(n) = 3*7^n.

3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021

Offset: 0

Author

Klaus Brockhaus, Apr 04 2010

Keywords

• nonn
• easy

Comments

Essentially first differences of A120741.

Binomial transform of A169604.

Second binomial transform of A005053 without initial term 1.

Inverse binomial transform of A103333 without initial term 1.

Second inverse binomial transform of A013708.

Except for first term 3, these are the integers that satisfy phi(n) = 4*n/7. - Michel Marcus, Jul 14 2015

Number of distinct quadratic residues (QR) over Z_7^n such that gcd(QR, 7^n) = 1 where n >= 1. - Param Mayurkumar Parekh, Feb 11 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Shalosh B. Ekhad and Doron Zeilberger, A Bijective Proof of Richard Stanley's Observation that the sum of the cubes of the n-th row of Stern's Diatomic array equals 3 times 7 to the power n-1, arXiv:2103.12852 [math.CO], 2021.
• Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
• Index entries for linear recurrences with constant coefficients, signature (7).

Crossrefs

Cf. A120741, A169604 (3*6^n), A005053 (expand (1-2x)/(1-5x)), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2*n+1)), A007283 (3*2^n), A164346 (3*4^n).

Programs

Magma

[ 3*7^n: n in [0..19] ];

Mathematica

3*7^Range[0, 25] (* Paolo Xausa, Jan 17 2025 *)

Formula

a(n) = 7*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-7*x).