A010697 -id:A010697 - cp's OEIS frontend

A074600 a(n) = 2^n + 5^n.

2, 7, 29, 133, 641, 3157, 15689, 78253, 390881, 1953637, 9766649, 48830173, 244144721, 1220711317, 6103532009, 30517610893, 152587956161, 762939584197, 3814697527769, 19073486852413, 95367432689201, 476837160300277

Offset: 0

Robert G. Wilson v, Aug 25 2002

Digital root of a(n) is A010697(n). - Peter M. Chema, Oct 24 2016

Miller, Steven J., ed. Benford's Law: Theory and Applications. Princeton University Press, 2015. See page 14.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, pp. 2211-2217.
Index entries for linear recurrences with constant coefficients, signature (7,-10).

Cf. A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074601-A074624, A010697, A094475, A337429.

[2^n + 5^n: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Table[2^n + 5^n, {n, 0, 25}]
LinearRecurrence[{7,-10},{2,7},30] (* Harvey P. Dale, May 09 2019 *)

a(n)=2^n+5^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*a(n-1)-3*2^(n-1) = 7*a(n-1)- 10*a(n-2). [Corrected by Zak Seidov, Oct 24 2009]

G.f.: 1/(1-2*x)+1/(1-5*x). E.g.f.: e^(2*x)+e^(5*x). - Mohammad K. Azarian, Jan 02 2009

A056020 Numbers that are congruent to +-1 mod 9.

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A141430 a(n) = A000111(n) mod 9.

Original entry on oeis.org

1, 1, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2, 2, 7

Offset: 0

Paul Curtz, Aug 06 2008

After the initial 1,1, the sequence is periodic with period 12.

This sequence's periodic part is a shuffled version of the two period-6 sequences A070366 and A010697. The sequence contains only the digits 1, 2, 4, 5, 7 and 8 (those of A141425).

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,-1,1).

Cf. A000111, A004099, A010686, A014021.

def A141430(n): return (2, 7, 1, 2, 5, 7, 7, 2, 8, 7, 4, 2)[n%12] if n>1 else 1 # Chai Wah Wu, Apr 17 2023

a(n) = A000111(n) mod 9 = A004099(n) mod 9.
a(n+12) = a(n), n > 1.
a(n) + a(n+6) = 9, n > 1.
a(n+11-p) - a(n+p) = 6 (p=0 or 5), 0 (p=1 or 4), -3 (p=2 or 3), any n > 1.
G.f.: (6x^8-5x^7+x^6+2x^5+3x^4+x^3+1) / ((1-x)(x^2+1)(x^4-x^2+1)). - R. J. Mathar, Dec 05 2008
a(n) = 9/2 +(-1)^floor(n/2)*A010686(n)/2 - 3*A014021(n), n > 1. - R. J. Mathar, Dec 05 2008
a(n) = 9/2 - (3/2)*cos(Pi*n/6) + (1/2)*3^(1/2)*sin(Pi*n/6) - (1/2)*cos(Pi*n/2) - (5/2)*sin(Pi*n/2) - (3/2)*cos(5*Pi*n/6) - (1/2)*3^(1/2)*sin(5*Pi*n/6). - Richard Choulet, Dec 12 2008

Edited by R. J. Mathar, Dec 05 2008

A021444 Decimal expansion of 1/440.

Original entry on oeis.org

0, 0, 2, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7

Offset: 0

N. J. A. Sloane

			0.002272727272727272727272727272727272727272727272727...

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010697.

First[RealDigits[1/440, 10, 100, -1]] (* or *)
PadRight[{0, 0, 2}, 100, {7, 2}] (* Paolo Xausa, Aug 14 2025 *)

1/440. \\ Charles R Greathouse IV, Jul 13 2016

(define (A021444 n) (cond ((<= n 1) 0) ((= 2 n) n) ((even? n) 7) (else 2))) ;; Antti Karttunen, Sep 14 2017

A176054 Decimal expansion of (7+3*sqrt(7))/7.

Original entry on oeis.org

2, 1, 3, 3, 8, 9, 3, 4, 1, 9, 0, 2, 7, 6, 8, 1, 6, 8, 1, 6, 4, 3, 5, 4, 9, 6, 0, 8, 7, 0, 2, 5, 4, 0, 1, 8, 2, 4, 4, 7, 2, 5, 3, 9, 3, 5, 6, 0, 6, 7, 6, 4, 3, 6, 3, 0, 1, 5, 0, 0, 0, 4, 8, 2, 5, 1, 4, 7, 4, 3, 7, 8, 1, 3, 8, 4, 4, 0, 7, 2, 6, 9, 0, 4, 0, 1, 6, 8, 3, 7, 9, 9, 1, 7, 6, 6, 1, 5, 4, 7, 4, 0, 6, 4, 5

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (7+3*sqrt(7))/7 is A010697.

			(7+3*sqrt(7))/7 = 2.13389341902768168164...

Cf. A010465 (decimal expansion of sqrt(7)), A010697 (repeat 2, 7).

A176434 Decimal expansion of (7+3*sqrt(7))/2.

Original entry on oeis.org

7, 4, 6, 8, 6, 2, 6, 9, 6, 6, 5, 9, 6, 8, 8, 5, 8, 8, 5, 7, 5, 2, 4, 2, 3, 6, 3, 0, 4, 5, 8, 8, 9, 0, 6, 3, 8, 5, 6, 5, 3, 8, 8, 7, 7, 4, 6, 2, 3, 6, 7, 5, 2, 7, 0, 5, 5, 2, 5, 0, 1, 6, 8, 8, 8, 0, 1, 6, 0, 3, 2, 3, 4, 8, 4, 5, 4, 2, 5, 4, 4, 1, 6, 4, 0, 5, 8, 9, 3, 2, 9, 7, 1, 1, 8, 1, 5, 4, 1, 5, 9, 2, 2, 5, 9

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (7+3*sqrt(7))/2 is A010697 preceded by 7.

			(7+3*sqrt(7))/2 = 7.46862696659688588575...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010465 (decimal expansion of sqrt(7)), A010697 (repeat 2, 7).

cp's OEIS Frontend

A074600 a(n) = 2^n + 5^n.

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A056020 Numbers that are congruent to +-1 mod 9.

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A141430 a(n) = A000111(n) mod 9.

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A021444 Decimal expansion of 1/440.

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A176054 Decimal expansion of (7+3*sqrt(7))/7.

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A176434 Decimal expansion of (7+3*sqrt(7))/2.

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