A120741 -id:A120741 - cp's OEIS frontend

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

Klaus Brockhaus, Apr 04 2010

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

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).

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).

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

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

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

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

A350992 Triangular numbers that are palindromes in base 7.

Original entry on oeis.org

0, 1, 3, 6, 78, 171, 300, 2850, 8256, 9453, 14706, 120786, 208335, 399171, 405450, 416328, 448878, 720600, 5877306, 6046503, 6835753, 9350650, 10122750, 18431556, 19130205, 22596003, 35309406, 499169406, 934394835, 969430528, 999335571, 1059265378, 1730160900

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((7^k-1)/2) is a term for all k >= 0 (Trigg, 1974).

			78 is a term since 78 = A000217(12) is a triangular number and also a palindromic number in base 7: 78 = 141_7.
171 is a term since 171 = A000217(18) is a triangular number and also a palindromic number in base 7: 171 = 333_7.

Amiram Eldar, Table of n, a(n) for n = 1..103
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.

Intersection of A000217 and A029954.
The septenary version of A003098.
Cf. A120741, A350987, A350990, A350991, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 7]] &]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula