A002001 -id:A002001 - cp's OEIS frontend

A196665 Expansion of g.f. (1-6x)/(1-19x).

1, 13, 247, 4693, 89167, 1694173, 32189287, 611596453, 11620332607, 220786319533, 4194940071127, 79703861351413, 1514373365676847, 28773093947860093, 546688785009341767, 10387086915177493573, 197354651388372377887, 3749738376379075179853

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (19).

Cf. A002001, A193722, A196661, A196662, A196663, A196664.

a(0) = 1, a(n) = 13*19^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*6^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (13*exp(19*x) + 6)/19.
a(n) = 19*a(n-1) for n > 1. (End)

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).

Original entry on oeis.org

1, 0, 4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 0

Paul Barry, Mar 12 2004

Partial sums are A092896.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A002001, A092896, A110594.

[1,0,4] cat [3*4^(n-2): n in [3..30]]; // G. C. Greubel, Feb 21 2021

a:= n-> 3*4^n/16+13*0^n/16+add(binomial(n,k)*(-1)^k*(3*k/4+k*(k-1)/2), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2018

Join[{1, 0, 4}, LinearRecurrence[{4}, {12}, 22]] (* Jean-François Alcover, Sep 16 2019 *)

Vec((1 -4*x +4*x^2 -4*x^3)/(1-4*x) + O(x^30)) \\ Andrew Howroyd, Nov 03 2018

[1,0,4]+[3*4^(n-2) for n in (3..30)] # G. C. Greubel, Feb 21 2021

a(n+2) = 4 * A002001(n).
a(n) = (3*4^n + 13*0^n)/16 + Sum_{k=0..n} binomial(n, k)*(-1)^k*(3*k/4 + k*(k-1)/2).
G.f.: 1 - x + 8*x^2 + 2*x/G(0), where G(k) = 1 + 1/(1 - x*(3*k+4)/(x*(3*k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
a(n) = A110594(n-1) for n >= 2. - Georg Fischer, Nov 03 2018
From G. C. Greubel, Feb 21 2021: (Start)
a(n) = (3*4^n +16*[n=2] -12*[n=1] +13*0^n)/16.
E.g.f.: (13 -12*x + 8*x^2 + 3*exp(4*x))/16. (End)

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

Original entry on oeis.org

1, 2, 3, 0, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 0, 17, 0, 19, 20, 21, 0, 23, 0, 25, 0, 27, 28, 29, 0, 31, 32, 33, 0, 35, 36, 37, 0, 39, 0, 41, 0, 43, 44, 45, 0, 47, 48, 49, 0, 51, 52, 53, 0, 55, 0, 57, 0, 59, 60, 61, 0, 63, 0, 65, 0, 67, 68, 69, 0, 71, 0, 73, 0, 75, 76, 77, 0, 79, 80

Offset: 1

Adriano Caroli, Dec 05 2010

If n > 0 and n is in the sequence, then a(2*n) = 0. Example: 5 is in the sequence, so a(2*5) = a(10) = 0.

Is this a(n) = n*A039982(n-1), n > 1? [R. J. Mathar, Dec 07 2010]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053661.

A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

import Data.List (delete)
a175880 n = a175880_list !! (n-1)
a175880_list = 1 : f [2..] [2..] where
   f (x:xs) (y:ys) | x == y    = x : (f xs $ delete (2*x) ys)
                   | otherwise = 0 : (f xs (y:ys))
for_bFile = take 10000 a175880_list
-- Reinhard Zumkeller, Feb 09 2011

A110654 := proc(n) 2*n+1-(-1)^n ; %/4 ;end proc:
A175880 := proc(n) if n <=2 then n; else if type(n,'even') then n-2*procname(A110654(n)) ; else n; end if; end if; end proc:
seq(A175880(n),n=1..40) ; # R. J. Mathar, Dec 07 2010

a(n) = n - (1 + (-1)^n) * a((2*n + 1 - (-1)^n)/4), n >= 3.

a(n) = n - A010673(n+1)*a(A110654(n)).

A196666 Expansion of g.f. (1 - 7x)/(1 - 22x).

Original entry on oeis.org

1, 15, 330, 7260, 159720, 3513840, 77304480, 1700698560, 37415368320, 823138103040, 18109038266880, 398398841871360, 8764774521169920, 192825039465738240, 4242150868246241280, 93327319101417308160, 2053201020231180779520, 45170422445085977149440

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (22).

Cf. A002001, A193722, A196661, A196662, A196663, A196664, A196665.

a(n)=if(n,15*22^n--,1) \\ Charles R Greathouse IV, Oct 06 2011

a(0) = 1, a(n) = 15*22^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*7^k.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (15*exp(22*x) + 7)/22.
a(n) = 22*a(n-1) for n > 1. (End)

A196731 Expansion of g.f. (1-x)/(1-12*x).

Original entry on oeis.org

1, 11, 132, 1584, 19008, 228096, 2737152, 32845824, 394149888, 4729798656, 56757583872, 681091006464, 8173092077568, 98077104930816, 1176925259169792, 14123103110037504, 169477237320450048, 2033726847845400576, 24404722174144806912, 292856666089737682944, 3514279993076852195328

Offset: 0

Philippe Deléham, Oct 05 2011

Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A002001, A005054, A025192, A052934, A055272, A055274, A055276, A193722.

a:= n-> ceil(11*12^(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 25 2025

Join[{1},NestList[12#&,11,20]] (* Harvey P. Dale, Sep 19 2018 *)

a(n)=if(n,11*12^n--,1) \\ Charles R Greathouse IV, Oct 05 2011

a(n)=(11+!n)*12^(n-1)  \\ M. F. Hasler, Oct 05 2011

a(n) = Sum_{k=0..n} A193722(n,k)*9^(n-k).
a(n+1) = 12*a(n) for n > 0. - M. F. Hasler, Oct 05 2011
From Elmo R. Oliveira, Mar 18 2025: (Start)
a(n) = 11*12^(n-1) with a(0)=1.
E.g.f.: (11*exp(12*x) + 1)/12. (End)

More terms from Elmo R. Oliveira, Mar 25 2025

A067327 Triangle related to generalized Catalan numbers A064340.

Original entry on oeis.org

1, 1, 3, 4, 12, 12, 28, 84, 96, 48, 256, 768, 912, 576, 192, 2704, 8112, 9792, 6720, 3072, 768, 31168, 93504, 113856, 81408, 42240, 15360, 3072, 380608, 1141824, 1397760, 1023744, 568320, 242688, 73728

Offset: 0

Wolfdieter Lang, Feb 05 2002

The row polynomials Z(2,2; n,y)= sum(a(n,m)*y^m,m=0..n) appear in c(2,2; x) (the g.f. of C(2,2; n) := A064340(n)) with the first (n+1) expansion terms subtracted, as follows: c(2,2; x)-sum(C(2,2; k)*x^k,k=0..n) = x^(n+1)*G(2,2; x)*Z(2,2; n,y), n>=0, where y=c(4*x) and c(x) is the g.f. of A000108 (Catalan) and G(2,2; x) is the g.f. of C(2,2; n+1), that is G(2,2; x)= (c(2,2; x)-1)/x. Hence G(2,2; x)*Z(2,2; k,c(4*x)) is, for k=0,1,..., the g.f. for C(2,2; n+k), n>=0.

Column sequences are: A064340(n), 3*A064340(n+1), Main diagonal gives A002001(n). Row sums give C(2,2; n+1)= A064340(n+1).

Cf. A067328 (scaled triangle with 1's in main diagonal).

a(n, 0)= C(2, 2; n) := A064340(n), n>=0; a(n, 1)= 3*C(2, 2; n), n>=1; a(n, m)=4*sum(a(n-1, k), k=(m-1)..(n-1)) if n>=m>=2, else 0.

A067328 Scaled triangle A067327.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 28, 28, 8, 1, 256, 256, 76, 12, 1, 2704, 2704, 816, 140, 16, 1, 31168, 31168, 9488, 1696, 220, 20, 1, 380608, 380608, 116480, 21328, 2960, 316, 24, 1, 4840960, 4840960, 1486784, 276544, 40016

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(n, m)=A067327(n, m)/A067327(m, m), with A067327(m, m)=A002001(n), n>=m>=0, else 0.

A069946 Numbers k such that phi(k) mod core(k) = 1 where core(k) is the squarefree part of k.

Original entry on oeis.org

2, 12, 48, 60, 63, 75, 175, 192, 363, 405, 468, 704, 768, 816, 867, 891, 960, 980, 1008, 1020, 1587, 1875, 2023, 2107, 2331, 2475, 2523, 2527, 2800, 2835, 3072, 3075, 3185, 3332, 3757, 4100, 4335, 4477, 4851, 5043, 5780, 6171, 6292, 6627, 6727, 6877, 7220

Offset: 1

Benoit Cloitre, Apr 27 2002

This sequence is infinite. For example, 3*4^k is a term for all k > 0, since core(3*4^k) = 3, phi(3*4^k) = 4^k and 4^k == 1 (mod 3). - Amiram Eldar, Sep 03 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002001, A007913.

core[n_] := Times @@ (First[#]^Mod[Last[#], 2] & /@ FactorInteger[n]); Select[Range[10^4], Mod[EulerPhi[#], core[#]] == 1 &] (* Amiram Eldar, Sep 03 2020 *)

for(n=1,15000,if(eulerphi(n)%core(n)==1,print1(n,",")))

Name corrected by Amiram Eldar, Sep 05 2020

A096646 Triangle (read by rows) where the number of row entries increases by steps of 2 and the entries are stacked in a rectangular fashion. The end entries = 1. Rest of entries in the n-th row are the sum of the entries directly above and to the left and right in all previous rows (total of 3*(n-1) entries).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 11, 14, 11, 5, 1, 1, 7, 22, 41, 50, 41, 22, 7, 1, 1, 9, 37, 92, 154, 182, 154, 92, 37, 9, 1, 1, 11, 56, 175, 375, 582, 672, 582, 375, 175, 56, 11, 1

Offset: 1

Gerald McGarvey, Aug 14 2004

The row sums are 1,3, then 2^(2*(n-2)) * 3. (I.e., A002001 a(n) = 3*4^(n-1), n>0; a(0)=1.) The n-th row is the (2n-1)st row of A072405 (Triangle of C(n,k)-C(n-2,k-1)).

			......................1....................
..................1...1...1................
..............1...3...4...3...1............
..........1...5..11..14..11...5...1........
......1...7..22..41..50..41..22...7..1.....
...1..9..37..92.154.182.154..92..37..9..1..
1.11.56.175.375.582.672.582.375.175.56.11.1

Cf. A002001, A072405.

G.f.: 1/[(1-z(1+w+w^2))(1-wz)]. Partial sums of trinomial array A027907. - Ralf Stephan, Jan 09 2005

A099489 Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 3, 11, 42, 157, 585, 2183, 8148, 30409, 113487, 423539, 1580670, 5899141, 22015893, 82164431, 306641832, 1144402897, 4270969755, 15939476123, 59486934738, 222008262829, 828546116577, 3092176203479, 11540158697340

Offset: 0

Paul Barry, Oct 18 2004

A Chebyshev transform of the sequence A002001 which has with g.f. (1-x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Matthew House, Table of n, a(n) for n = 0..1739
Index entries for linear recurrences with constant coefficients, signature (4,-2,4,-1).

Cf. A099486, A099487, A099488.

CoefficientList[Series[(1-x+x^2)/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *_)
LinearRecurrence[{4,-2,4,-1},{1,3,11,42},30] (* Harvey P. Dale, Dec 28 2019 *)

a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-4). - corrected by Matthew House, Oct 22 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(3*4^(n-2*k)+0^(n-2*k))/4.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3)).
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*A001353(n-k+1).
a(n) = 3*A001353(n+1)/4 +A056594(n)/4. - R. J. Mathar, Sep 21 2012

cp's OEIS Frontend

A196665 Expansion of g.f. (1-6*x)/(1-19*x).

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A092898 Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).

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A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

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A196666 Expansion of g.f. (1 - 7*x)/(1 - 22*x).

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A196731 Expansion of g.f. (1-x)/(1-12*x).

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A067327 Triangle related to generalized Catalan numbers A064340.

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A067328 Scaled triangle A067327.

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A069946 Numbers k such that phi(k) mod core(k) = 1 where core(k) is the squarefree part of k.

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A196665 Expansion of g.f. (1-6x)/(1-19x).

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).

A196666 Expansion of g.f. (1 - 7x)/(1 - 22x).