A047399 -id:A047399 - cp's OEIS frontend

A143976 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having x + y == 1 (mod 3); then R(m,n) is the number of UNmarked squares in the rectangle [0,m] X [0,n].

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 4, 6, 6, 6, 4, 4, 7, 8, 8, 7, 4, 5, 8, 10, 11, 10, 8, 5, 6, 10, 12, 14, 14, 12, 10, 6, 6, 11, 14, 16, 17, 16, 14, 11, 6, 7, 12, 16, 19, 20, 20, 19, 16, 12, 7, 8, 14, 18, 22, 24, 24, 24, 22, 18, 14, 8, 8, 15, 20, 24, 27, 28, 28, 27, 24, 20, 15, 8

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...) = A000027.

			Northwest corner:
  1  2  2  3  4  4  5
  2  3  4  6  7  8 10
  2  4  6  8 10 12 14
  3  6  8 11 14 16 18
  4  7 10 14 17 20 24
See A143974.

Stefano Spezia, First 140 antidiagonals of the array, flattened

Rows: A004523, A004772, A005843, A047399, et al.
Main diagonal: A071619.
Cf. A143974, A143977, A143979.

T[m_,n_]:=m*n-Floor[m*n/3]; Flatten[Table[T[n-k+1,k],{n,12},{k,n}]] (* Stefano Spezia, Oct 25 2022 *)

R(m,n) = m*n - floor(m*n/3).

A232692 E.g.f. satisfies: A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ).

Original entry on oeis.org

1, 1, 3, 24, 213, 3096, 46071, 967608, 20251809, 555747048, 15004870731, 508165972056, 16810393586733, 677183788645704, 26523956467895103, 1238567261126084856, 56056407696184372281, 2976966230117448265128, 152872356339113679491859, 9098430770913969095416728

Offset: 0

Paul D. Hanna, Dec 06 2013

nonn

Compare e.g.f. to: B(x) = exp( 1/B(x)^3 * Integral B(x)^3 dx ) where B(y) = Bessel polynomial y_n(-3) (cf. A065923).
Note that G(x) = exp(1/G(x)^3 * Integral G(x)^7 dx) has negative coefficients.
CONJECTURE:
Given G(x,n,k) = G such that G = exp( 1/G^n * Integral G^k dx ) then G(x,n,k) consists solely of positive coefficients when k >= A047399(n) where A047399 lists numbers that are congruent to {0,3,6} mod 8.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 213*x^4/4! + 3096*x^5/5! +...
Related expansions:
log(A(x)) = x + 2*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1905*x^5/5! + 23640*x^6/6! +...
Integral A(x)^8 dx = x + 8*x^2/2! + 80*x^3/3! + 1032*x^4/4! + 16320*x^5/5! +...
1/A(x)^3 = 1 - 3*x + 3*x^2/2! - 24*x^3/3! + 117*x^4/4! - 2088*x^5/5! +...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A232690, A232691.

seq(n! * coeff(series((3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5), x, n+1), x, n), n=0..20) # Vaclav Kotesovec, Jan 05 2014

m = 20; A[] = 1; Do[A[x] = Exp[1/A[x]^3 Integrate[A[x]^8 + O[x]^m, x]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Nov 03 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A^3*intformal(A^8+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5). - Vaclav Kotesovec, Jan 05 2014

A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 141, 144, 146, 149, 152, 154, 157, 160, 162, 165, 168, 170, 173, 176, 178, 181, 184, 186, 189

Offset: 1

Kival Ngaokrajang, Jan 28 2014

nonn

Let points 2, 3, & 1 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the left hand side of point 1 (pattern 1); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 1, 2, 3, ... and so on.
Conjecture: All forms of 3 center points are non-expanded loops.
There are other sets of center points that give the same sequence, e.g.: [2,3,1,4]; [3,2,4,1]; [3,2,4,1,5]; [2,3,1,4,5,7,6]; [2,3,1,7,4,6,5]; [3,4,2,5,1,6,7]; [4,3,5,6,2,7,1]; [4,5,3,2,1,6,7]; [5,4,6,3,2,7,1].
Also, there are some similar patterns that give difference sequences, e.g.:
   A047622: [1,2,7,3,4,6,5]; [1,2,7,6,3,5,4]...
   A047399: [1,2,7,3,6,4,5]; [1,2,7,6,5,3,4]...
   A047395: [2,3,1,4 7,5,6]; [2,3,1,7,6,4,5]...
   A047464: [4,5,3,6,2,7,1]; [1,8,2,7,3,6,4,5];
            [9,1,8,2,7,3,6,4,5].
See illustration in links.
Appears to be basically a duplicate of A047618. - R. J. Mathar, Feb 03 2014

Kival Ngaokrajang, Illustration of irregular spirals (Center points 3..9)

Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points); A236326, A236327 (5 center points).

Conjecture from Colin Barker, Jul 12 2014: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4).
G.f.: x*(3*x^2+3*x+2) / ((x-1)^2*(x^2+x+1)). (End)

A047497 Numbers that are congruent to {1, 2, 4, 5, 7} mod 8.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 12, 13, 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 85, 87, 89, 90, 92, 93, 95, 97, 98, 100, 101, 103

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A047399 (complement).

I:=[1,2,4,5,7]; [n le 5 select I[n] else Self(n-5) + 8 : n in [1..70]]; // Vincenzo Librandi, Jun 06 2017

seq(floor((8*n-3)/5),n=1..56); # Gary Detlefs, Mar 07 2010

Select[Range[120],MemberQ[{1,2,4,5,7},Mod[#,8]]&] (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{1,2,4,5,7,9},100] (* Harvey P. Dale, Jun 05 2017 *)
Table[8 n + {1, 2, 4, 5, 7}, {n, 0, 20}]//Flatten (* Vincenzo Librandi, Jun 06 2017 *)

for (n=1, 80, print1((8*n-3)\5, ", ")) \\ Michel Marcus, Sep 10 2014

a(n) = floor((8n-3)/5). [Gary Detlefs, Mar 07 2010]
From R. J. Mathar, Mar 23 2010: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6).
G.f.: x*(1 + x + 2*x^2 + x^3 + 2*x^4 + x^5)/ ((x^4 + x^3 + x^2 + x + 1) * (x-1)^2). (End)

A198084 Ceiling(n*sqrt(7)).

Original entry on oeis.org

0, 3, 6, 8, 11, 14, 16, 19, 22, 24, 27, 30, 32, 35, 38, 40, 43, 45, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 75, 77, 80, 83, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 127, 130, 133, 135, 138, 141, 143, 146, 149

Offset: 0

Vincenzo Librandi, Oct 24 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A022841, A022850, A047399.

Magma
```
[Ceiling(n*Sqrt(7)): n in [0..60]];
```

Ceiling[Sqrt[7]Range[0,60]] (* Harvey P. Dale, May 12 2016 *)

cp's OEIS Frontend

A143976 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having x + y == 1 (mod 3); then R(m,n) is the number of UNmarked squares in the rectangle [0,m] X [0,n].

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A232692 E.g.f. satisfies: A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ).

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A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

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Formula

A047497 Numbers that are congruent to {1, 2, 4, 5, 7} mod 8.

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Magma

Maple

Mathematica

PARI

Formula

A198084 Ceiling(n*sqrt(7)).

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Keywords

Links

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Programs

Magma

Mathematica