A057087 -id:A057087 - cp's OEIS frontend

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A084169 A Pell Jacobsthal product.

Original entry on oeis.org

0, 1, 2, 15, 60, 319, 1470, 7267, 34680, 168435, 810898, 3921103, 18918900, 91381991, 441150502, 2130258075, 10285325040, 49663079099, 239791814010, 1157823924167, 5590452446700, 26993130847215, 130334271942158

Offset: 0

Paul Barry, May 18 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,13,4,-4).

Cf. A000129, A001045, A007985, A057087.

[0] cat [(2^n-(-1)^n)*Evaluate(DicksonSecond(n-1,-1), 2)/3: n in [1..40]]; // G. C. Greubel, Oct 11 2022

LinearRecurrence[{2,13,4,-4}, {0,1,2,15}, 41] (* G. C. Greubel, Oct 11 2022 *)

def A084169(n): return (2^n-(-1)^n)*lucas_number1(n,2,-1)/3
[A084169(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

a(n) = (2^n - (-1)^n)*( (1+sqrt(2))^n - (1-sqrt(2))^n )/(6*sqrt(2)).
a(n) = A001045(n)*A000129(n).
G.f.: x*(1-2*x^2)/((1+2*x-x^2)*(1-4*x-4*x^2)). - Colin Barker, May 01 2012
a(n) = (A007985 + 2*A057087(n))/3. - R. J. Mathar, Sep 29 2020

A106568 Expansion of 4x/(1 - 4x - 4*x^2).

Original entry on oeis.org

0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360, 782005818097664, 3775858103812096, 18231455687639040

Offset: 0

Roger L. Bagula, May 30 2005

This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - G. C. Greubel, Sep 06 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A000129, A057087, A009013.

Sequences of the form a(n) = m*(a(n-1) + a(n-2)): A000045 (m=1), A028860 (m=2), A106435 (m=3), A094013 (m=4), A106565 (m=5), A221461 (m=6), A221462 (m=7).

[n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021

A106568 := n -> ifelse(n=0, 0, 4^(n)*hypergeom([(1-n)/2, 1-n/2], [1-n], -1)):
seq(simplify(A106568(n)), n = 0..24);  # Peter Luschny, Mar 30 2025

LinearRecurrence[{4,4}, {0,4}, 40] (* G. C. Greubel, Sep 06 2021 *)

[2^(n+1)*lucas_number1(n,2,-1) for n in (0..40)] # G. C. Greubel, Sep 06 2021

a(n) = 4 * A057087(n).
a(n) = A094013(n+1). - R. J. Mathar, Aug 24 2008
From Philippe Deléham, Sep 19 2009: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4.
G.f.: 4*x/(1 - 4*x - 4*x^2). (End)
G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) = 2^(n+1)*A000129(n). - G. C. Greubel, Sep 06 2021
a(n) = 4^n*hypergeom([(1-n)/2, 1-n/2], [1-n], -1) for n > 0. - Peter Luschny, Mar 30 2025

Edited by N. J. A. Sloane, Apr 30 2006

Simpler name using o.g.f. by Joerg Arndt, Oct 05 2013

A182436 Triangle T(n,k), read by rows, given by (2, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 4, 8, 11, 4, 4, 20, 25, 24, 8, 8, 28, 70, 69, 52, 16, 8, 60, 126, 213, 178, 112, 32, 16, 80, 288, 460, 599, 440, 240, 64, 16, 160, 472, 1128, 1489, 1600, 1056, 512, 128, 32, 208, 976, 2152, 3914, 4457, 4120, 2480, 1088, 256

Offset: 0

Philippe Deléham, Apr 28 2012

Row sums are the powers of 3.

			Triangle begins :
1
2, 1
2, 5, 2
4, 8, 11, 4
4, 20, 25, 24, 8
8, 28, 70, 69, 52, 16
8, 60, 126, 213, 178, 112, 32
16, 80, 288, 460, 599, 440, 240, 64
16, 160, 472, 1128, 1489, 1600, 1056, 512, 128
32, 208, 976, 2152, 3914, 4457, 4120, 2480, 1088, 256

Cf. A016116, A011782, A111297, A000244

G.f.: (1+2*x-y*x)/(1-2*y*x-(2+y)*x^2).
T(n,k) = 2*T(n-1,k-1) + 2*T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = T(2,0) = T(2,2) = 2, T(2,1) = 5 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A123335(n-1), A016116(n+1), A000244(n), A057087(n), A091928(n) for x = -2, -1, 0, 1, 2, 3 respectively.

A228603 a(1) = 9, a(2) = 44, a(n) = 4*(a(n-1) + a(n-2)) (n >=3).

Original entry on oeis.org

9, 44, 212, 1024, 4944, 23872, 115264, 556544, 2687232, 12975104, 62649344, 302497792, 1460588544, 7052345344, 34051735552, 164416323584, 793872236544, 3833154240512, 18508105908224, 89365040594944, 431492586012672, 2083430506430464, 10059692369772544

Offset: 1

Emeric Deutsch, Nov 02 2013

a(n) = number of independent vertex subsets (i.e. the Merrifield-Simmons index) of the normal alkyl radical of n carbons (i.e. CH_3(CH_2)_{n-1}).

R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.

H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20,1982, 16-21.
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A228602.

a := proc (n) if n = 1 then 9 elif n = 2 then 44 else 4*a(n-1)+4*a(n-2) end if end proc: seq(a(n), n = 1 .. 25);

LinearRecurrence[{4,4},{9,44},30] (* Harvey P. Dale, Oct 30 2016 *)

a(n) = (8 - 5*sqrt(2))*(2 - 2*sqrt(2))^(n)/8 + (8 + 5*sqrt(2))*(2 + 2*sqrt(2))^(n)/8.
G.f.: x*(9+8*x)/(1-4*x-4*x^2).
a(n) = 9*A057087(n-1)+8*A057087(n-2). - R. J. Mathar, Nov 24 2013

A267652 a(n) = 4a(n - 1) + 4a(n - 2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 20, 92, 448, 2160, 10432, 50368, 243200, 1174272, 5669888, 27376640, 132186112, 638251008, 3081748480, 14879997952, 71846985728, 346907934720, 1675019681792, 8087710466048, 39050920591360, 188554524229632, 910421779283968, 4395905214054400, 21225307973353472, 102484852749631488

Offset: 0

Ilya Gutkovskiy, Jan 19 2016

Generalized Fibonacci sequence.

Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A057087, A084128.

Table[2^(n - 5/2) ((1 + 4 Sqrt[2]) (1 - Sqrt[2])^n - (1 - 4 Sqrt[2]) (1 + Sqrt[2])^n), {n, 0, 25}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 4 a[n - 1] + 4 a[n - 2]}, a, {n, 25}] (* Bruno Berselli, Jan 19 2016 *)
LinearRecurrence[{4, 4}, {2, 3}, 20] (* Vincenzo Librandi, Jan 19 2016 *)

Vec((2-5*x)/(1-4*x-4*x^2) + O(x^100)) \\ Altug Alkan, Jan 19 2016

G.f.: (2 - 5*x)/(1 - 4*x - 4*x^2).
a(n) = 2^(n-5/2)*((1+4*sqrt(2))*(1-sqrt(2))^n - (1-4*sqrt(2))*(1+sqrt(2))^n).
Lim_{n -> infinity} a(n)/a(n - 1) = 2 + 2*sqrt(2) = 2*A014176 = 4.82842712...
a(n) = 2*A057087(n)-5*A057087(n-1). - R. J. Mathar, Jun 07 2016

A320660 Number of business cards required to build an origami level n Jerusalem cube.

Original entry on oeis.org

12, 72, 672, 6048, 55488, 511872, 4738560, 43943424, 407890944, 3787941888, 35186122752, 326885842944, 3037038034944, 28217571901440, 262178452930560, 2436006721486848, 22634041833160704, 210303674768424960, 1954034324430913536, 18155901427591938048

Offset: 0

Franck Maminirina Ramaharo, Oct 18 2018

The actual Jerusalem cube fractal cannot be built using a simple integer grid. However, one can create an approximate one by choosing the cube side length to be a Pell number (see link).
In practice, the first two terms represent the level 0 because they both consist of cubes (1 X 1 X 1 and 2 X 2 X 2, respectively). The "cross" shape appears at index 2, which is usually considered as the first iteration (for example, the "hole" shape in the Menger Sponge is visible at level 1).
The limit of a(n+1)/a(n) is equal to 2*(2+sqrt(7)) as n approaches infinity.

			a(2) = 672 because 456 business cards are needed for the squeleton and 216 more for the panels.

Eric Baird, L'art fractal, Tangente 150 (2013), 45.
Thomas Hull, Project Origami: Activities for Exploring Mathematics, A K Peters/CRC Press, 2006.

Eric Baird, The Jerusalem Cube
Malachi B-J. Brown, Business Card Origami
Robert Dickau, Cross Menger (Jerusalem) Cube Fractal
Origami Resource Center, Jerusalem Cube Fractal (Level 1)
Franck Ramaharo, An approximate Jerusalem square whose side equals a Pell number, arXiv:1801.00466 [math.CO], 2018.
Wikipedia, Cube de Jérusalem [In French]
Index entries for linear recurrences with constant coefficients, signature (12, -16, -80, -48)

At the n-th level, the cube side length is A000129(n+1), the squeleton requires 6*A239549(n+1) business cards, and each face requires A057087(n) units for the panels.

Cf. A212596 (Origami Menger sponge), A304960 (Origami Mosely snowflake sponge).

LinearRecurrence[{12, -16, -80, -48}, {12, 72, 672, 6048}, 20]

makelist((3/14)*(7*(2 - 2*sqrt(2))^n + 7*(2 + 2*sqrt(2))^n + (21 - 5*sqrt(7))*(4 - 2*sqrt(7))^n + (21 + 5*sqrt(7))*(4 + 2*sqrt(7))^n), n, 0, 20), ratsimp;

a(n) = (3/14)*(7*(2 - 2*sqrt(2))^n + 7*(2 + 2*sqrt(2))^n + (21 - 5*sqrt(7))*(4 - 2*sqrt(7))^n + (21 + 5*sqrt(7))*(4 + 2*sqrt(7))^n).
a(n) = 12*a(n-1) - 16*a(n-2) - 80*a(n-3) - 48*a(n-4), n > 4.
G.f.: 12*(1 - 6*x + 8*x^3)/((1-4*x-4*x^2)*(1-8*x-12*x^2)) .
E.g.f.: (3/14)*(7*exp((2 - 2*sqrt(2))*x) + 7*exp((2 + 2*sqrt(2))*x) + (21 - 5*sqrt(7))*exp((4 - 2*sqrt(7))*x) + (21 + 5*sqrt(7))*exp((4 + 2*sqrt(7))*x)).
a(n) = 3*( A084128(n) -2*A239549(n) +3*A239549(n+1) ). - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A084169 A Pell Jacobsthal product.

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A106568 Expansion of 4*x/(1 - 4*x - 4*x^2).

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A182436 Triangle T(n,k), read by rows, given by (2, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A228603 a(1) = 9, a(2) = 44, a(n) = 4*(a(n-1) + a(n-2)) (n >=3).

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A267652 a(n) = 4*a(n - 1) + 4*a(n - 2) for n>1, a(0)=2, a(1)=3.

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A320660 Number of business cards required to build an origami level n Jerusalem cube.

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A106568 Expansion of 4x/(1 - 4x - 4*x^2).

A267652 a(n) = 4a(n - 1) + 4a(n - 2) for n>1, a(0)=2, a(1)=3.