A226308 -id:A226308 - cp's OEIS frontend

A047853 a(n) = A047848(5, n).

1, 2, 10, 74, 586, 4682, 37450, 299594, 2396746, 19173962, 153391690, 1227133514, 9817068106, 78536544842, 628292358730, 5026338869834, 40210710958666, 321685687669322, 2573485501354570, 20587884010836554, 164703072086692426, 1317624576693539402, 10540996613548315210

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000420(n-1) for n >= 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A000420, A047848, A226308.

[(8^n +6)/7: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=8*a[n-1]+1 od: seq(a[n]+1, n=0..18); # Zerinvary Lajos, Mar 20 2008

LinearRecurrence[{9, -8}, {1, 2}, 30] (* Harvey P. Dale, Dec 11 2016 *)
(8^Range[0,40] +6)/7 (* G. C. Greubel, Jan 12 2025 *)

def A047853(n): return (pow(8,n) +6)//7
print([A047853(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (8^n + 6)/7. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 05 2009: (Start)
a(0)=1, a(1)=2; a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
G.f.: (1 - 7*x)/(1 - 9*x + 8*x^2). (End)
a(n) = 8*a(n-1) - 6 for n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n+1) = A226308(3*n). - Philippe Deléham, Feb 24 2014
E.g.f.: exp(x)*(6 + exp(7*x))/7. - Stefano Spezia, Oct 16 2023

More terms from Vladimir Joseph Stephan Orlovsky, Nov 07 2008

A233328 a(n) = (2*8^(n+1) - 9) / 7.

Original entry on oeis.org

1, 17, 145, 1169, 9361, 74897, 599185, 4793489, 38347921, 306783377, 2454267025, 19634136209, 157073089681, 1256584717457, 10052677739665, 80421421917329, 643371375338641, 5146971002709137, 41175768021673105, 329406144173384849, 2635249153387078801

Offset: 0

Philippe Deléham, Feb 23 2014

Sum of n-th row of triangle of powers of 8: 1; 8 1 8; 64 8 1 8 64 ; 512 64 8 1 8 64 512; ...

			a(0) = 1;
a(1) = 8 + 1 + 8 = 17;
a(2) = 64 + 8 + 1 + 8 + 64 = 145;
a(3) = 512 + 64 + 8 + 1 + 8 + 64 + 512 = 1169; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A226308

[(2*8^(n+1)-9)/7: n in [0..30]]; // Vincenzo Librandi, Feb 25 2014

Table[(2 8^(n + 1) - 9)/7, {n, 0, 30}] (* Vincenzo Librandi, Feb 25 2014 *)
LinearRecurrence[{9,-8},{1,17},30] (* Harvey P. Dale, Apr 29 2019 *)

G.f.: (1+8*x)/((1-x)*(1-8*x)).
a(n) = 9*a(n-1) - 8*a(n-2) for n>1, a(0)=1, a(1)=17.
a(n) = 8*a(n-1) + 9 for n>0, a(0)=1.
a(n) = A226308(3n+1).

A046636 Number of cubic residues mod 8^n.

Original entry on oeis.org

1, 5, 37, 293, 2341, 18725, 149797, 1198373, 9586981, 76695845, 613566757, 4908534053, 39268272421, 314146179365, 2513169434917, 20105355479333, 160842843834661, 1286742750677285, 10293942005418277, 82351536043346213, 658812288346769701, 5270498306774157605, 42163986454193260837

Offset: 0

David W. Wilson

Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [broken link]
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [cached copy]
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A007583, A046530, A046630, A047853, A226308.

LinearRecurrence[{9, -8}, {1, 5}, 20] (* Jean-François Alcover, Jan 19 2019 *)

a(n) = (4*8^n + 3)/7.
a(n) = 8*a(n-1) - 3 (with a(0)=1). - Vincenzo Librandi, Nov 18 2010
From R. J. Mathar, Feb 28 2011: (Start)
a(n) = A046530(8^n) = A046630(3*n).
G.f.: (1-4*x)/((1-8*x)*(1-x)). (End)
a(n+1) = A226308(3*n+2). - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(4*exp(7*x) + 3)/7.
a(n) = 9*a(n-1) - 8*a(n-2).
a(n) = A047853(n+1)/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A341905 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.

Original entry on oeis.org

3, 0, 2, 8, 10, 22, 48, 90, 182, 368, 730, 1462, 2928, 5850, 11702, 23408, 46810, 93622, 187248, 374490, 748982, 1497968, 2995930, 5991862, 11983728, 23967450, 47934902, 95869808, 191739610, 383479222, 766958448, 1533916890, 3067833782, 6135667568, 12271335130

Offset: 0

Michael De Vlieger, Jun 04 2021

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A001045, A077947, A186575, A226308.

a:= n-> (<<0|1|0>, <0|0|1>, <2|1|1>>^n. <<3, 0, 2>>)[1,1]:
seq(a(n), n=0..34);  # Alois P. Heinz, Jun 04 2021

LinearRecurrence[{1, 1, 2}, {3, 0, 2}, 35] (* or *)
CoefficientList[Series[(-3 + 3 x + x^2)/(-1 + x + x^2 + 2 x^3), {x, 0, 34}], x]

G.f.: (-3 + 3*x + x^2)/(-1 + x + x^2 + 2*x^3).

a(n) = (10*2^(n-1) + 13*A049347(n) - 9*A079978(n+1) + 3)/7. - Greg Dresden, Jun 20 2021

cp's OEIS Frontend

A047853 a(n) = A047848(5, n).

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A233328 a(n) = (2*8^(n+1) - 9) / 7.

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A046636 Number of cubic residues mod 8^n.

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A341905 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.

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