A170931 -id:A170931 - cp's OEIS frontend

A094013 Expansion of (1-4x)/(1-4x-4*x^2).

1, 0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360

Offset: 0

Paul Barry, Apr 21 2004

Inverse binomial transform of A000129(2n-1). a(n+2)/4 = A057087(n).

a(n) is the irrational part of circle radii 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(m)*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

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
Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A000129, A001653, A170931, A174968.

[n le 2 select 2-n else 4*(Self(n-1) + Self(n-2)): n in [1..41]]; // G. C. Greubel, Dec 04 2021

CoefficientList[Series[(1-4x)/(1-4x-4x^2),{x,0,40}],x] (* or *) LinearRecurrence[{4,4},{1,0},40] (* Harvey P. Dale, May 21 2012 *)
Table[2^n*Fibonacci[n-1, 2], {n, 0, 40}] (* G. C. Greubel, Dec 04 2021 *)

Vec((1-4*x)/(1-4*x-4*x^2) + O(x^30)) \\ Michel Marcus, Feb 15 2015

[2^n*lucas_number1(n-1, 2, -1) for n in (0..40)] # G. C. Greubel, Dec 04 2021

a(n) = (2 + 2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (2 - 2*sqrt(2))^n*(1/2 + sqrt(2)/4).
a(n) = 4*a(n-1) + 4*a(n-2); a(0)=1, a(1)=0. - Philippe Deléham, Nov 03 2008
a(n) = A057087(n) - 4*A057087(n-1). - R. J. Mathar, Jan 15 2013
From G. C. Greubel, Dec 04 2021: (Start)
a(n) = 2^n * A000129(n-1).
E.g.f.: exp(2*x)*( cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x) ). (End)

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

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)

cp's OEIS Frontend

A094013 Expansion of (1-4*x)/(1-4*x-4*x^2).

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A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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

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

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A094013 Expansion of (1-4x)/(1-4x-4*x^2).