A010855 -id:A010855 - cp's OEIS frontend

A010709 Constant sequence: the all 4's sequence.

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 25 2010: (Start)
Continued fraction expansion of 2+sqrt(5).
Decimal expansion of 4/9.
Inverse binomial transform of A020707. (End)

Tom Edgar, Proof without words: sum of powers of 4/9, Math. Mag. 89 no. 3 (2016) 191.
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1012
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

From Klaus Brockhaus, May 25 2010: (Start)
Equals 4*A000012, 2*A007395, A010731/2, A010855/4, A010871/8.
Cf. A098317 (decimal expansion of 2+sqrt(5)), A020707 (2^(n+2)). (End)

makelist(4, n, 0, 30); /* Martin Ettl, Nov 09 2012 */

a(n) = 4 \\ Charles R Greathouse IV, Apr 07 2012

def A010709(n): return 4 # Chai Wah Wu, Mar 22 2023

From Klaus Brockhaus, May 25 2010: (Start)
a(n) = 4.
G.f.: 4/(1-x). (End)
E.g.f.: 4*e^x. - Vincenzo Librandi, Jan 29 2012

A040056 Continued fraction for sqrt(65).

Original entry on oeis.org

8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

N. J. A. Sloane

			8.06225774829854965236661... = 8 + 1/(16 + 1/(16 + 1/(16 + 1/(16 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010517 (decimal expansion), A041112/A041113 (convergents), A248289 (Egyptian fraction).

Cf. A040000, A040002, A040012.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[65],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16}] (* Harvey P. Dale, Nov 27 2020 *)

{ allocatemem(932245000); default(realprecision, 49000); x=contfrac(sqrt(65)); for (n=0, 20000, write("b040056.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

From Elmo R. Oliveira, Feb 10 2024: (Start)
a(n) = 16 = A010855(n) for n >= 1.
G.f.: 8*(1+x)/(1-x).
E.g.f.: 16*exp(x) - 8.
a(n) = 8*A040000(n) = 4*A040002(n) = 2*A040012(n). (End)

A177499 Period 4: repeat [1, 16, 4, 16].

Original entry on oeis.org

1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16

Offset: 0

Paul Curtz, May 10 2010

From Klaus Brockhaus, May 14 2010: (Start)
Interleaving of A000012, A010855, A010709 and A010855.
Continued fraction expansion of (44+sqrt(2442))/88. (End)

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

Cf. A010674, A177838. [Klaus Brockhaus, May 14 2010]

Cf. A000012, A010709, A010855, A176895.

&cat[[1, 16, 4, 16]^^26]; // Klaus Brockhaus, May 14 2010

seq(op([1, 16, 4, 16]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

Table[{1, 16, 4, 16}, {21}] // Flatten (* Jean-François Alcover, May 24 2013 *)

From Klaus Brockhaus, May 14 2010: (Start)
a(n+2) - a(n) = A010674(n).
a(n) = a(n-4) for n > 3.
G.f.: (1+16*x+4*x^2+16*x^3)/(1-x^4). (End)
a(n) = A176895(n)^2. - Paul Curtz, Mar 21 2011
a(n) = (37 - 6*cos(n*Pi/2) - 27*cos(n*Pi) - 27*I*sin(n*Pi))/4. - Wesley Ivan Hurt, Jul 09 2016

A023014 Number of partitions of n into parts of 16 kinds.

Original entry on oeis.org

1, 16, 152, 1088, 6460, 33440, 155584, 663936, 2636326, 9845040, 34861152, 117809728, 381946360, 1193074144, 3603543040, 10556065152, 30068145905, 83466484112, 226236086512, 599785472000, 1557643542308, 3967888347232, 9926348625408, 24413219138816

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010855. - Alois P. Heinz, Oct 17 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
N. J. A. Sloane, Transforms

Cf. 16th column of A144064. - Alois P. Heinz, Oct 17 2008

with (numtheory): a:= proc(n) option remember; `if`(n=0, 1, add (add (d*16, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

CoefficientList[Series[1/QPochhammer[x]^16, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

Vec(1/eta(x)^16 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

G.f.: Product_{m>=1} 1/(1-x^m)^16.
a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 16. - Vaclav Kotesovec, Jun 28 2025

A187541 a(4n+2) = 2n+1, otherwise a(n) = 4n.

Original entry on oeis.org

0, 4, 1, 12, 16, 20, 3, 28, 32, 36, 5, 44, 48, 52, 7, 60, 64, 68, 9, 76, 80, 84, 11, 92, 96, 100, 13, 108, 112, 116, 15, 124, 128, 132, 17, 140, 144, 148, 19, 156, 160, 164, 21, 172, 176, 180, 23, 188, 192, 196, 25, 204, 208, 212, 27, 220, 224, 228, 29, 236, 240, 244, 31, 252

Offset: 0

Paul Curtz, Mar 11 2011

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A008586, A010855, A060819, A061038.

A187541:=n->16*n/(11+7*(I^(2*n)-I^(-n)-I^n)): seq(A187541(n), n=0..100); # Wesley Ivan Hurt, Jul 05 2016

Table[16n/(11+7*(I^(2*n)-I^(-n)-I^n)), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 05 2016 *)

a(n) = 2*a(n-4) - a(n-8) for n>7.
G.f.: x*(4+x+12*x^2+16*x^3+12*x^4+x^5+4*x^6)/(1-x^4)^2; a(n) = (n/8)*(32 -7*(1+(-1)^n)*(1-i^n)) where i=sqrt(-1). - Bruno Berselli, Mar 15 2011
From Paul Curtz, Mar 22 2011: (Start)
A060819(n)*a(n) = 0,4,1,36,16,100, = 0,4, followed by A061038(n+2).
a(n) = a(n-4) + period 4: repeat [16, 16, 2, 16]. Note that a(n) = 4*n/(period 4: repeat [1, 1, 8, 1]), Hence 16's = A010855. (End)
a(n) = 16*n/(11+7*(I^(2*n)-I^(-n)-I^n)). - Wesley Ivan Hurt, Jul 05 2016

Edited by N. J. A. Sloane, Mar 15 2011

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A010709 Constant sequence: the all 4's sequence.

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A040056 Continued fraction for sqrt(65).

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A177499 Period 4: repeat [1, 16, 4, 16].

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A023014 Number of partitions of n into parts of 16 kinds.

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A187541 a(4n+2) = 2n+1, otherwise a(n) = 4n.

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