A163888 -id:A163888 - cp's OEIS frontend

A240039 T(n,k)=Number of nXk 0..2 arrays with no element equal to zero plus the sum of elements to its left or zero plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 3.

2, 2, 2, 4, 2, 4, 4, 6, 6, 4, 8, 4, 16, 4, 8, 8, 10, 16, 16, 10, 8, 16, 8, 42, 18, 42, 8, 16, 16, 20, 44, 52, 52, 44, 20, 16, 32, 16, 114, 62, 154, 62, 114, 16, 32, 32, 40, 122, 162, 178, 178, 162, 122, 40, 32, 64, 32, 314, 204, 494, 282, 494, 204, 314, 32, 64, 64, 80, 340, 530, 600

Offset: 1

R. H. Hardin, Mar 31 2014

Table starts
..2..2...4...4....8....8....16....16....32.....32.....64.....64.....128.....128
..2..2...6...4...10....8....20....16....40.....32.....80.....64.....160.....128
..4..6..16..16...42...44...114...122...314....340....872....950....2432....2658
..4..4..16..18...52...62...162...204...530....672...1736...2198....5706....7202
..8.10..42..52..154..178...494...600..1606...2014...5262...6690...17464...22360
..8..8..44..62..178..282...710..1074..2770...4162..10836..15764...41374...59680
.16.20.114.162..494..710..1976..2884..7958..12074..31824..49078..130758..197208
.16.16.122.204..600.1074..2884..5706.14686..25224..66774.113794..307698..508002
.32.40.314.530.1606.2770..7958.14686.42470..72446.200916.360716..984770.1730698
.32.32.340.672.2014.4162.12074.25224.72446.147092.407954.778294.2194512.4046228

			Some solutions for n=3 k=4
..2..1..1..2....2..1..2..1....2..1..2..1....1..2..1..2....1..2..1..2
..1..0..0..0....1..0..0..0....1..0..0..0....2..0..0..0....2..0..0..0
..1..0..0..0....1..0..0..0....2..0..0..0....2..0..0..0....2..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..511

Column 1 is A016116(n+1)

Column 2 is A163888(n-2) for n>3

Empirical for column k:
k=1: a(n) = 2*a(n-2)
k=2: a(n) = 2*a(n-2) for n>5
k=3: a(n) = 4*a(n-2) -3*a(n-4) -a(n-6) for n>7
k=4: [order 24] for n>27
k=5: [order 86] for n>89

A163609 a(n) = ((5 + 2sqrt(2))(3 + sqrt(2))^n + (5 - 2sqrt(2))(3 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 19, 79, 341, 1493, 6571, 28975, 127853, 564293, 2490787, 10994671, 48532517, 214232405, 945666811, 4174374031, 18426576509, 81338840837, 359047009459, 1584910170895, 6996131959157, 30882420558677, 136321599637963

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163608. Third binomial transform of A163888. Inverse binomial transform of A163610.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A163608, A163610, A163888.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(3+r)^n+(5-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{6, -7}, {5, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((5-11*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 19.
G.f.: (5-11*x)/(1-6*x+7*x^2).
a(n) = 5*A081179(n+1) - 11*A081179(n). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(3*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163610 a(n) = ((5 + 2sqrt(2))(4 + sqrt(2))^n + (5 - 2sqrt(2))(4 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 24, 122, 640, 3412, 18336, 98920, 534656, 2892368, 15653760, 84736928, 458742784, 2483625280, 13446603264, 72802072192, 394164131840, 2134084044032, 11554374506496, 62557819435520, 338701312393216, 1833801027048448

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

nonn

Binomial transform of A163609. Fourth binomial transform of A163888. Inverse binomial transform of A163611.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A163609, A163611, A163888.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(4+r)^n+(5-2*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{8,-14},{5,24},30] (* Harvey P. Dale, May 29 2016 *)

x='x+O('x^50); Vec((5-16*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 24.
G.f.: (5-16*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163607 a(n) = ((5 + 2sqrt(2))(1 + sqrt(2))^n + (5 - 2sqrt(2))(1 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 9, 23, 55, 133, 321, 775, 1871, 4517, 10905, 26327, 63559, 153445, 370449, 894343, 2159135, 5212613, 12584361, 30381335, 73347031, 177075397, 427497825, 1032071047, 2491639919, 6015350885, 14522341689, 35060034263, 84642410215

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163888. Inverse binomial transform of A163608.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A163608, A163888.

Cf. A000129, A001333.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(1+r)^n+(5-2*r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{2, 1}, {5, 9}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((5-x)/(1-2*x-x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 9.
G.f.: (5-x)/(1-2*x-x^2).
a(n) = 5*A000129(n+1) - A000129(n). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017
a(n) = 2*A001333(n) + A001333(n+2). - Philippe Deléham, Mar 06 2023

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163611 a(n) = ((5 + 2sqrt(2))(5 + sqrt(2))^n + (5 - 2sqrt(2))(5 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 29, 175, 1083, 6805, 43141, 274895, 1756707, 11244485, 72040589, 461782735, 2960893803, 18987935125, 121778793781, 781065429935, 5009742042387, 32132915535365, 206105088378749, 1321993826474095, 8479521232029723

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

nonn

Binomial transform of A163610. Fifth binomial transform of A163888.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A163610, A163888.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(5+r)^n+(5-2*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{10, -23}, {5, 29}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((5-21*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 29.
G.f.: (5-21*x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

cp's OEIS Frontend

A240039 T(n,k)=Number of nXk 0..2 arrays with no element equal to zero plus the sum of elements to its left or zero plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 3.

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A163609 a(n) = ((5 + 2*sqrt(2))*(3 + sqrt(2))^n + (5 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

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A163610 a(n) = ((5 + 2*sqrt(2))*(4 + sqrt(2))^n + (5 - 2*sqrt(2))*(4 - sqrt(2))^n)/2.

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A163607 a(n) = ((5 + 2*sqrt(2))*(1 + sqrt(2))^n + (5 - 2*sqrt(2))*(1 - sqrt(2))^n)/2.

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Programs

Magma

Mathematica

PARI

Formula

Extensions

A163611 a(n) = ((5 + 2*sqrt(2))*(5 + sqrt(2))^n + (5 - 2*sqrt(2))*(5 - sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163609 a(n) = ((5 + 2sqrt(2))(3 + sqrt(2))^n + (5 - 2sqrt(2))(3 - sqrt(2))^n)/2.

A163610 a(n) = ((5 + 2sqrt(2))(4 + sqrt(2))^n + (5 - 2sqrt(2))(4 - sqrt(2))^n)/2.

A163607 a(n) = ((5 + 2sqrt(2))(1 + sqrt(2))^n + (5 - 2sqrt(2))(1 - sqrt(2))^n)/2.

A163611 a(n) = ((5 + 2sqrt(2))(5 + sqrt(2))^n + (5 - 2sqrt(2))(5 - sqrt(2))^n)/2.