A094013 -id:A094013 - cp's OEIS frontend

A221463 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, starting with 0.

0, 0, 1, 0, 2, 1, 0, 3, 4, 2, 0, 4, 9, 12, 3, 0, 5, 16, 36, 32, 5, 0, 6, 25, 80, 135, 88, 8, 0, 7, 36, 150, 384, 513, 240, 13, 0, 8, 49, 252, 875, 1856, 1944, 656, 21, 0, 9, 64, 392, 1728, 5125, 8960, 7371, 1792, 34, 0, 10, 81, 576, 3087, 11880, 30000, 43264, 27945, 4896, 55, 0, 11

Offset: 1

R. H. Hardin, general recursion proved by Robert Israel in the Sequence Fans Mailing List, Jan 17 2013

Table starts
..0.....0.......0........0.........0..........0..........0...........0
..1.....2.......3........4.........5..........6..........7...........8
..1.....4.......9.......16........25.........36.........49..........64
..2....12......36.......80.......150........252........392.........576
..3....32.....135......384.......875.......1728.......3087........5120
..5....88.....513.....1856......5125......11880......24353.......45568
..8...240....1944.....8960.....30000......81648.....192080......405504
.13...656....7371....43264....175625.....561168....1515031.....3608576
.21..1792...27945...208896...1028125....3856896...11949777....32112640
.34..4896..105948..1008640...6018750...26508384...94253656...285769728
.55.13376..401679..4870144..35234375..182191680..743424031..2543058944
.89.36544.1522881.23515136.206265625.1252200384.5863743809.22630629376

			Some solutions for n=6 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....2....1....3....3....2....2....1....4....2....3....1....3....2....3....3
..1....4....0....1....2....4....3....0....2....0....3....3....2....0....0....4
..0....0....3....4....2....3....2....1....0....3....2....4....3....3....4....1
..1....2....1....2....1....0....3....4....0....1....3....1....3....0....2....0
..0....3....2....3....3....1....4....0....4....2....0....0....0....4....0....3

R. H. Hardin, Table of n, a(n) for n = 1..10000

Column 1 is A000045(n-1)
Column 2 is A028860(n+1)
Column 3 is A106435(n-1)
Column 4 is A094013
Column 5 is A106565(n-1)
Row 2 is A000027
Row 3 is A000290
Row 4 is A011379

Recursion for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +2*a(n-2)
k=3: a(n) = 3*a(n-1) +3*a(n-2)
k=4: a(n) = 4*a(n-1) +4*a(n-2)
k=5: a(n) = 5*a(n-1) +5*a(n-2)
k=6: a(n) = 6*a(n-1) +6*a(n-2)
k=7: a(n) = 7*a(n-1) +7*a(n-2)
Empirical for row n:
n=2: a(k) = 1*k
n=3: a(k) = 1*k^2
n=4: a(k) = 1*k^3 + 1*k^2
n=5: a(k) = 1*k^4 + 2*k^3
n=6: a(k) = 1*k^5 + 3*k^4 + 1*k^3
n=7: a(k) = 1*k^6 + 4*k^5 + 3*k^4
n=8: a(k) = 1*k^7 + 5*k^6 + 6*k^5 + 1*k^4
n=9: a(k) = 1*k^8 + 6*k^7 + 10*k^6 + 4*k^5
n=10: a(k) = 1*k^9 + 7*k^8 + 15*k^7 + 10*k^6 + 1*k^5
n=11: a(k) = 1*k^10 + 8*k^9 + 21*k^8 + 20*k^7 + 5*k^6
n=12: a(k) = 1*k^11 + 9*k^10 + 28*k^9 + 35*k^8 + 15*k^7 + 1*k^6
n=13: a(k) = 1*k^12 + 10*k^11 + 36*k^10 + 56*k^9 + 35*k^8 + 6*k^7
n=14: a(k) = 1*k^13 + 11*k^12 + 45*k^11 + 84*k^10 + 70*k^9 + 21*k^8 + 1*k^7
n=15: a(k) = 1*k^14 + 12*k^13 + 55*k^12 + 120*k^11 + 126*k^10 + 56*k^9 + 7*k^8
Apparently then T(n,k) = sum { binomial(n-2-i,i)*k^(n-1-i) , 0<=2*i<=n-2 }.
The formula reduces to T(n,k) = [4*k^(n-1)*(1+G)^(2*n-2) +4^n] /[2^(n+1) *G *(1+G)^(n-1)] for even n and to T(n,k) = [4*k^(n-1) *(1+G)^(2*n-2) -4^n] /[2^(n+1) *G *(1+G)^(n-1)] for odd n, where G=sqrt(1+4/k). - R. J. Mathar, Jan 21 2013

A094014 Expansion of (1-2x)/(1-8x^2).

Original entry on oeis.org

1, -2, 8, -16, 64, -128, 512, -1024, 4096, -8192, 32768, -65536, 262144, -524288, 2097152, -4194304, 16777216, -33554432, 134217728, -268435456, 1073741824, -2147483648, 8589934592, -17179869184, 68719476736, -137438953472

Offset: 0

Paul Barry, Apr 21 2004

Second inverse binomial transform of A094013. Third inverse binomial transform of A000129(2n-1).

The unsigned sequence has g.f. (1+2*x)/(1-8*x^2) and abs(a(n)) = 2^(3*n/2)*(1/2 + sqrt(2)/4 + (1/2 - sqrt(2)/4)*(-1)^n).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A000129, A016116, A094013, A113836, A158020.

[n le 2 select (-2)^(n-1) else 8*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 04 2021

LinearRecurrence[{0,8}, {1,-2}, 40] (* G. C. Greubel, Dec 04 2021 *)

[(-2)^n*2^(n//2) for n in (0..40)] # G. C. Greubel, Dec 04 2021

a(n) = (2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (-2*sqrt(2))^n*(1/2 + sqrt(2)/4).
a(n) = (-2)^n * A016116(n). - R. J. Mathar, Apr 28 2008
Abs(a(n)) = A113836(n+1) - A113836(n) for n > 0. - Reinhard Zumkeller, Feb 22 2010
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = Sum_{k=0..n} A158020(n,k)*3^k. - Philippe Deléham, Dec 01 2011
E.g.f.: cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x). - G. C. Greubel, Dec 04 2021

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

Original entry on oeis.org

2, 4, 24, 112, 544, 2624, 12672, 61184, 295424, 1426432, 6887424, 33255424, 160571392, 775307264, 3743514624, 18075287552, 87275208704, 421401985024, 2034708774912, 9824443039744, 47436607258624, 229044201193472

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010

Sequence gives the rational part of the radii of the circles in nested circles and squares inspired by Vitruvian Man, starting with a square whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(2)), namely R(n) = A(n-1) + B(n)*sqrt(2) with A(-1)=1, for n >= 1, A(n-1) = A170931(n-1)*-1^(n-1); and B(n) = A094013(n)*-1^n. See illustrations in the links. - Kival Ngaokrajang, Feb 15 2015

			L(n,0)=2, L(n,1)=n.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man
Index entries for linear recurrences with constant coefficients, signature (4, 4).

Cf. similar sequences with d=sqrt(n*(n+k)): A000032 (k=1, classic Lucas), A080040 (k=2), A085480 (k=3).

Cf. A094013, A174968.

I:=[2,4]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[2 (1 - 2 x) / (1 - 4 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 16 2015 *)
LinearRecurrence[{4,4},{2,4},30] (* Harvey P. Dale, Sep 03 2016 *)

x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 2*A084128(n) = 4*a(n-1) + 4*a(n-2).
G.f.: 2*(1-2*x)/(1 - 4*x - 4*x^2). (End)

A255162 Rational part of circle radii in nested circles and hexagons (see comment).

Original entry on oeis.org

2, 0, 24, -288, 3744, -48384, 625536, -8087040, 104550912, -1351655424, 17474476032, -225913577472, 2920656642048, -37758842634240, 488153991315456, -6310954007396352, 81589295984541696, -1054802999903256576, 13636707550653579264

Offset: 0

Kival Ngaokrajang, Feb 15 2015

sign

Inspired by Vitruvian Man, but using hexagons instead of squares, starting with a hexagon whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(3)), namely R(n) = A(n) + B(n)*sqrt(3) with A(0)=2, A(n) = a(n), and B(0) = 1, B(n) = A255163(n). See illustrations in the links.

Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man

Cf. A174968, A170931, A094013, A255163.

{a=2;b=1;print1(a,", ");for(n=1,30,c=12*b-6*a;d=4*a-6*b;print1(c,", ");a=c;b=d)}

Conjectures from Colin Barker, Feb 15 2015: (Start)
a(n) = -12*a(n-1) + 12*a(n-2).
G.f.: -2*(12*x+1) / (12*x^2 - 12*x - 1).
(End)

A255163 Irrational parts of circle radii in nested circles and hexagons (see comment).

Original entry on oeis.org

1, 2, -12, 168, -2160, 27936, -361152, 4669056, -60362496, 780378624, -10088893440, 130431264768, -1686241898496, 21800077959168, -281835838291968, 3643630995013632, -47105601999667200, 608990795936169984, -7873156775230046208

Offset: 0

Kival Ngaokrajang, Feb 15 2015

sign

Inspired by Vitruvian Man, but using hexagons instead of squares, starting with a hexagon whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(3)), namely R(n) = A(n) + B(n)*sqrt(3) with A(0)=2, A(n) = A255162(n), and B(0) = 1, B(n) = a(n). See illustrations in the links.

Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man

Cf. A174968, A170931, A094013, A255162.

{a=2;b=1;print1(b,", ");for(n=1,30,c=12*b-6*a;d=4*a-6*b;print1(d,", ");a=c;b=d)}

Conjectures from Colin Barker, Feb 15 2015: (Start)
a(n) = -12*a(n-1) + 12*a(n-2).
G.f.: -(14*x+1) / (12*x^2-12*x-1).
(End)

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

cp's OEIS Frontend

A221463 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, starting with 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A094014 Expansion of (1-2*x)/(1-8*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A170931 Extended Lucas L(n,i) = n*(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A255162 Rational part of circle radii in nested circles and hexagons (see comment).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A255163 Irrational parts of circle radii in nested circles and hexagons (see comment).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A094014 Expansion of (1-2x)/(1-8x^2).

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

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