A166517 -id:A166517 - cp's OEIS frontend

A166577 Inverse binomial transform of A166517.

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

A166542 a(n) = 6*n - a(n-1), with n>1, a(1)=2.

Original entry on oeis.org

2, 10, 8, 16, 14, 22, 20, 28, 26, 34, 32, 40, 38, 46, 44, 52, 50, 58, 56, 64, 62, 70, 68, 76, 74, 82, 80, 88, 86, 94, 92, 100, 98, 106, 104, 112, 110, 118, 116, 124, 122, 130, 128, 136, 134, 142, 140, 148, 146, 154, 152, 160, 158, 166, 164, 172, 170, 178, 176, 184

Offset: 1

Vincenzo Librandi, Oct 16 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

[n eq 1 select 2 else 6*n-Self(n-1): n in [1..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[-(- 2 - 8 x + 4 x^2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 13 2013 *)
LinearRecurrence[{1,1,-1}, {2, 10, 8}, 50] (* G. C. Greubel, May 16 2016 *)

G.f.: -x*(-2-8*x+4*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 13 2013
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: (1/2)*(3*(1 + 2*x)*exp(x) + 5*exp(-x) - 8).
a(n) = a(n-1) + a(n-2) - a(n-3). (End)
a(n) = 2*A166517(n). - Philippe Deléham, Nov 29 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/4 + Pi/(6*sqrt(3)). - Amiram Eldar, Feb 23 2023

A226293 Class of sequences of (p-1)-tuples of reverse order of natural numbers for p = 7.

Original entry on oeis.org

6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 34, 33, 32, 31, 30, 29, 41, 40, 39, 38, 37, 36, 48, 47, 46, 45, 44, 43, 55, 54, 53, 52, 51, 50, 62, 61, 60, 59, 58, 57, 69, 68, 67, 66, 65, 64, 76, 75, 74, 73, 72, 71, 83

Offset: 1

Sam Vaseghi, Jun 02 2013

nonn

Given a prime p, the class of sequences a(n,p) can be constructed from linear combination of the two sequences b(n,p) (A010885) and c(n,p) (A226233), according to a(n,p) = c(n,p)*p - b(n,p) (see Formula below) that ensures uniqueness of the form q = a(n,p)*p^m according to the decomposition theorem Vaseghi 2013 (see link and reference below), for p prime, q a positive integer and m a positive integer or zero. The above example is for p=7. The class is crucial and will be applied to define other number theoretic sequences, that will be submitted to OEIS as well a posterior.

			for p=2: 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,...
for p=3: 2,1,5,4,8,7,11,10,14,13,17,16,20,19,23,22,26,25,29,28,...
for p=5: 4,3,2,1,9,8,7,6,14,13,12,11,19,18,17,16,24,23,22,21,...
for p=7: 6,5,4,3,2,1,13,12,11,10,9,8,20,19,18,17,16,15,27,26,...

S. Vaseghi (alias al-Hwarizmi), Combination of positive integers in terms of primes (sophisticated version 2)

Cf. A166517, A226233.

p = 7; k = p - 1; c = (k + n - 1 - Mod[n - 1, k])/k; b = 1 + Mod[n - 1, k]; Table[c*p - b, {n, 68}]

a(n,p) = c(n,p)*p - b(n,p), where b(n,p) = (1+[(n-1)mod(p-1)]) (see A010885) and c(n,p) = ((p-1)+n-(1+[(n-1)mod(p-1)]))/(p-1) (see A226233), with p = 7.

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A166577 Inverse binomial transform of A166517.

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A166542 a(n) = 6*n - a(n-1), with n>1, a(1)=2.

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A226293 Class of sequences of (p-1)-tuples of reverse order of natural numbers for p = 7.

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