A212959 -id:A212959 - cp's OEIS frontend

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413

Offset: 0

Clark Kimberling, Apr 27 2012

nonn

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or xA212750 ... w=R or x=R or yA212751 ... w=R or x=R or yA212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

		Examples
					a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
		
	
References
			A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

		
Links
			Bo Gyu Jeong, Table of n, a(n) for n = 0..200

		
CrossrefsCf. A000583 (n^4), A210000, A211809, A212959.
Programs
			
				Mathematica
				t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)
				

		
Formulaa(n) = n^4 - A211809(n).

A032528 Concentric hexagonal numbers: floor(3*n^2/2).

Original entry on oeis.org

0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601, 3750

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Aug 20 2011: (Start)
Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.
A033581 and A003154 interleaved.
Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example).  (End)
Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - Omar E. Pol, Sep 08 2011

			From _Omar E. Pol_, Aug 20 2011: (Start)
Using the numbers A008458 we can write:
  0, 1, 6, 12, 18, 24, 30, 36, 42,  48,  54, ...
  0, 0, 0,  1,  6, 12, 18, 24, 30,  36,  42, ...
  0, 0, 0,  0,  0,  1,  6, 12, 18,  24,  30, ...
  0, 0, 0,  0,  0,  0,  0,  1,  6,  12,  18, ...
  0, 0, 0,  0,  0,  0,  0,  0,  0,   1,   6, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ...
...
Illustration of initial terms as concentric hexagons:
.
.                                         o o o o o
.                         o o o o        o         o
.             o o o      o       o      o   o o o   o
.     o o    o     o    o   o o   o    o   o     o   o
. o  o   o  o   o   o  o   o   o   o  o   o   o   o   o
.     o o    o     o    o   o o   o    o   o     o   o
.             o o o      o       o      o   o o o   o
.                         o o o o        o         o
.                                         o o o o o
.
. 1    6        13           24               37
.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to cellular automata.

Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040.
Cf. A004524, A004525, A033436, A212959, A238410, A227017, A282513.
Cf. A033581, A003154, A011934, A184533.

a032528 n = a032528_list !! n
a032528_list = scanl (+) 0 a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[Floor(3*n^2/2): n in [0..50]]; // Vincenzo Librandi, Aug 21 2011

f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* Clark Kimberling, Apr 20 2012 *)

a(n)=3*n^2\2 \\ Charles R Greathouse IV, Sep 24 2015

From Joerg Arndt, Aug 22 2011: (Start)
G.f.: (x+4*x^2+x^3)/(1-2*x+2*x^3-x^4) = x*(1+4*x+x^2)/((1+x)*(1-x)^3).
a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End)
a(n) = (6*n^2+(-1)^n-1)/4. - Bruno Berselli, Aug 22 2011
a(n) = A184533(n), n >= 2. - Clark Kimberling, Apr 20 2012
First differences of A011934: a(n) = A011934(n) - A011934(n-1) for n>0. - Franz Vrabec, Feb 17 2013
From Paul Curtz, Mar 31 2019: (Start)
a(-n) = a(n).
a(n) = a(n-2) + 6*(n-1) for n > 1.
a(2*n) = A033581(n).
a(2*n+1) = A003154(n+1). (End)
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Aug 19 2022
Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Jan 16 2023

New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011

A047241 Numbers that are congruent to {1, 3} mod 6.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57, 61, 63, 67, 69, 73, 75, 79, 81, 85, 87, 91, 93, 97, 99, 103, 105, 109, 111, 115, 117, 121, 123, 127, 129, 133, 135, 139, 141, 145, 147, 151, 153, 157, 159, 163, 165, 169, 171, 175, 177, 181, 183

Offset: 1

N. J. A. Sloane

Also the numbers k such that 10^p+k could possibly be prime. - Roderick MacPhee, Nov 20 2011 This statement can be written as follows. If 10^m + k = prime, for any m >= 1, then k is in this sequence. See the pink box comments by Roderick MacPhee from Dec 09 2014. - Wolfdieter Lang, Dec 09 2014
The odd-indexed terms are one more than the arithmetic mean of their neighbors; the even-indexed terms are one less than the arithmetic mean of their neighbors. - Amarnath Murthy, Jul 29 2003
Partial sums are A212959. - Philippe Deléham, Mar 16 2014
12*a(n) is conjectured to be the length of the boundary after n iterations of the hexagon and square expansion shown in the link. The squares and hexagons have side length 1 in some units. The pattern is supposed to become the planar Archimedean net 4.6.12 when n -> infinity. - Kival Ngaokrajang, Nov 30 2014
Positive numbers k for which 1/2 + k/3 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

L. Lovasz, J. Pelikan, K. Vesztergombi, Discrete Mathematics, Springer (2003); 14.4, p. 225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics, Elementary and Beyond, Springer (2003); 14.4, p. 225.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047233, A056970, A007310, A047228, A047261, A047273, A212959.
Subsequence of A186422.
Union of A016921 and A016945. - Wesley Ivan Hurt, Sep 28 2013

a047241 n = a047241_list !! (n-1)
a047241_list = 1 : 3 : map (+ 6) a047241_list
-- Reinhard Zumkeller, Feb 19 2013

seq(3*k-2-((k+1) mod 2), k=1..100); # Wesley Ivan Hurt, Sep 28 2013

Table[{2, 4}, {30}] // Flatten // Prepend[#, 1]& // Accumulate (* Jean-François Alcover, Jun 10 2013 *)
Select[Range[200], MemberQ[{1, 3}, Mod[#, 6]]&] (* or *) LinearRecurrence[{1, 1, -1}, {1, 3, 7}, 70] (* Harvey P. Dale, Oct 01 2013 *)

a(n)=bitor(3*n-3,1) \\ Charles R Greathouse IV, Sep 28 2013

for n in range(1,10**5):print(3*n-2-((n+1)%2)) # Soumil Mandal, Apr 14 2016

From Paul Barry, Sep 04 2003: (Start)
O.g.f.: (1 + 2*x + 3*x^2)/((1 + x)*(1 - x)^2) = (1 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)).
E.g.f.: (6*x + 1)*exp(x)/2 + exp(-x)/2;
a(n) = 3*n - 5/2 - (-1)^n/2. (End)
a(n) = 2*floor((n-1)/2) + 2*n - 1. - Gary Detlefs, Mar 18 2010
a(n) = 6*n - a(n-1) - 8 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 05 2010
a(n) = 3*n - 2 - ((n+1) mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(1)=1, a(2)=3, a(3)=7; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Oct 01 2013
From Benedict W. J. Irwin, Apr 13 2016: (Start)
A005408(a(n)+1) = A016813(A001651(n)),
A007310(a(n)) = A005408(A087444(n)-1),
A007310(A005408(a(n)+1)) = A017533(A001651(n)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Dec 11 2021

Formula corrected by Bruno Berselli, Jun 24 2010

A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 26, 44, 68, 100, 140, 190, 250, 322, 406, 504, 616, 744, 888, 1050, 1230, 1430, 1650, 1892, 2156, 2444, 2756, 3094, 3458, 3850, 4270, 4720, 5200, 5712, 6256, 6834, 7446, 8094, 8778, 9500, 10260, 11060, 11900, 12782, 13706

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.
Magic numbers of nucleons in a biaxially deformed nucleus at oscillator ratio 1:2 (oblate ellipsoid) under the simple harmonic oscillator model. - Jess Tauber, May 14 2013
a(n) is the number of Sidon subsets of {1,...,n+1} of size 3. - Carl Najafi, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
Ikuko Hamamoto, One-particle motion in nuclear many-body problem, Division of Mathematical Physics, LTH, University of Lund, Sweden.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A212959, A334187.

First differences: A007590, is first differences of 2*A001752(n-4) for n > 3; partial sums: 2*A001752(n-3) for n > 2, is partial sums of A007590(n-1) for n > 0. - Guenther Schrack, Mar 19 2018

[(2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Jul 25 2014

A212964:=n->add(floor(i^2/2) - 2*floor(i/2), i=1..n): seq(A212964(n), n=0..50); # Wesley Ivan Hurt, Jul 23 2014

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] < Abs[x - y] < Abs[y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A212964 *)
m/2 (* essentially A002623 *)
CoefficientList[Series[2 x^3/((1 + x) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8;
vector (100, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=2*x^3 and g(x)=(1+x)(1-x)^4.
a(n+3) = 2*A002623(n).
a(n) = Sum_{k=0..n} floor((k-1)^2/2). - Enrique Pérez Herrero, Dec 28 2013
a(n) = Sum_{i=1..n} floor(i^2/2) - 2*floor(i/2). - Wesley Ivan Hurt, Jul 23 2014
a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8. - Robert Israel, Jul 23 2014
E.g.f.: (x*(2*x^2 + 3*x - 3)*cosh(x) + (2*x^3 + 3*x^2 - 3*x + 3)*sinh(x))/12. - Stefano Spezia, Jul 06 2021

A250229 T(n,k)=Number of length n+1 0..k arrays with the sum of the cubes of adjacent differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

2, 3, 4, 4, 11, 8, 5, 20, 27, 16, 6, 33, 52, 79, 32, 7, 48, 89, 208, 223, 64, 8, 67, 132, 473, 704, 651, 128, 9, 88, 187, 872, 1785, 2720, 1907, 256, 10, 113, 248, 1519, 3496, 9437, 10952, 5639, 512, 11, 140, 321, 2392, 6367, 24888, 47953, 45888, 16967, 1024, 12, 171

Offset: 1

R. H. Hardin, Nov 14 2014

			Table starts
....2.....3......4.......5........6.........7.........8..........9.........10
....4....11.....20......33.......48........67........88........113........140
....8....27.....52......89......132.......187.......248........321........400
...16....79....208.....473......872......1519......2392.......3617.......5184
...32...223....704....1785.....3496......6367.....10640......16909......25152
...64...651...2720....9437....24888.....59415....120412.....222037.....374712
..128..1907..10952...47953...144624....371227....838604....1732385....3243544
..256..5639..45888..264473..1019568...3347259...8983896...21295973...45095084
..512.16967.195516.1440243..6717892..25280899..78435176..215244983..519836920
.1024.52131.852260.8079297.47046932.217539879.789142896.2486304965.6802360404
...
Some solutions for n=6 k=4
..4....2....2....2....3....2....0....2....2....2....1....3....1....2....2....4
..3....1....1....3....0....3....2....1....0....0....4....0....0....4....1....0
..1....0....3....2....2....3....3....4....0....4....1....1....0....3....1....4
..3....3....1....1....0....1....3....1....2....0....3....0....3....3....0....2
..0....1....2....0....0....0....3....3....4....2....2....3....4....2....2....0
..3....4....1....3....3....2....1....4....2....4....0....0....4....2....2....3
..2....2....0....0....3....2....2....2....2....2....1....3....1....0....4....0

R. H. Hardin, Table of n, a(n) for n = 1..181

Column 1 is A000079.

Row 2 is A212959.

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: [linear recurrence of order 9] for n>12
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
n=3: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2

A250277 T(n,k)=Number of length n+1 0..k arrays with the sum of the squares of adjacent differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

2, 3, 4, 4, 11, 8, 5, 20, 27, 16, 6, 33, 52, 79, 32, 7, 48, 89, 240, 255, 64, 8, 67, 140, 581, 984, 843, 128, 9, 88, 207, 1132, 2909, 4412, 2763, 256, 10, 113, 288, 1991, 6732, 17885, 20252, 8903, 512, 11, 140, 389, 3156, 14003, 51884, 107387, 91808, 28215, 1024, 12

Offset: 1

R. H. Hardin, Nov 16 2014

Table starts
....2.....3.......4........5.........6.........7..........8...........9
....4....11......20.......33........48........67.........88.........113
....8....27......52.......89.......140.......207........288.........389
...16....79.....240......581......1132......1991.......3156........4841
...32...255.....984.....2909......6732.....14003......25964.......45303
...64...843....4412....17885.....51884....130335.....281552......564985
..128..2763...20252...107387....381812...1154141....2908232.....6704631
..256..8903...91808...636197...2783500..10172515...30143732....80256473
..512.28215..406748..3664311..19762916..86975297..301550620...925066871
.1024.88195.1759740.20397261.135821156.715749943.2901853512.10244309701

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

R. H. Hardin, Table of n, a(n) for n = 1..181

Column 1 is A000079

Row 2 is A212959

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 9*a(n-1) -31*a(n-2) +51*a(n-3) -40*a(n-4) +12*a(n-5) for n>6
k=3: [order 15] for n>18
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2

A250167 T(n,k)=Number of length n+1 0..k arrays with the sum of adjacent differences multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

2, 3, 4, 4, 11, 8, 5, 20, 37, 16, 6, 33, 96, 119, 32, 7, 48, 211, 436, 373, 64, 8, 67, 380, 1269, 1880, 1151, 128, 9, 88, 639, 2860, 7109, 7836, 3517, 256, 10, 113, 976, 5831, 19896, 37881, 32032, 10679, 512, 11, 140, 1437, 10460, 49037, 129648, 195927

Offset: 1

R. H. Hardin, Nov 13 2014

Table starts
....2.....3.......4........5.........6..........7..........8...........9
....4....11......20.......33........48.........67.........88.........113
....8....37......96......211.......380........639........976........1437
...16...119.....436.....1269......2860.......5831......10460.......17765
...32...373....1880.....7109.....19896......49037.....103556......203615
...64..1151....7836....37881....129648.....380939.....938128.....2121089
..128..3517...32032...195927....810964....2810751....7989940....20567199
..256.10679..129572...996933...4962056...20169871...65768448...191480917
..512.32293..521256..5029417..30034672..142786013..532548628..1748028901
.1024.97391.2091052.25262121.180893724.1004527983.4281269376.15822382297

			Some solutions for n=5 k=4
..3....0....3....4....0....3....4....4....2....4....4....2....0....4....3....1
..2....0....4....2....0....4....1....4....4....1....3....1....1....2....1....1
..4....4....0....4....4....2....2....2....4....3....4....3....1....2....3....3
..0....2....0....1....2....1....3....2....1....0....3....2....3....1....0....4
..1....2....4....1....3....1....3....3....3....0....0....2....0....4....3....3
..1....0....3....2....0....1....2....4....0....4....0....2....0....2....3....1

R. H. Hardin, Table of n, a(n) for n = 1..263

Column 1 is A000079
Column 2 is A084171
Row 2 is A212959

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 5*a(n-1) -6*a(n-2)
k=3: a(n) = 8*a(n-1) -21*a(n-2) +22*a(n-3) -8*a(n-4)
k=4: [order 8]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4); also a quadratic polynomial plus a constant quasipolynomial with period 2
n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6); also a cubic polynomial plus a linear quasipolynomial with period 2
n=4: [order 12; also a quartic polynomial plus a quadratic quasipolynomial with period 12]
n=5: [order 24; also a polynomial of degree 5 plus a cubic quasipolynomialwith period 60]

A055232 Expansion of (1+2x+3x^2)/((1-x)^3*(1-x^2)).

Original entry on oeis.org

1, 5, 16, 36, 69, 117, 184, 272, 385, 525, 696, 900, 1141, 1421, 1744, 2112, 2529, 2997, 3520, 4100, 4741, 5445, 6216, 7056, 7969, 8957, 10024, 11172, 12405, 13725, 15136, 16640, 18241, 19941, 21744, 23652, 25669, 27797, 30040, 32400, 34881, 37485, 40216, 43076

Offset: 0

N. J. A. Sloane, Jul 05 2000

a(n) is the number of (w,x,y) having all terms in {0..n} and w <= floor((x+y)/2). - Clark Kimberling, Jun 02 2012

First differences are in A212959. - Wesley Ivan Hurt, Apr 16 2016

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.28(c), y_3.

Michel Marcus, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A212959.

[(7+(-1)^n+16*n+14*n^2+4*n^3)/8 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016

A055232:=n->(7+(-1)^n+16*n+14*n^2+4*n^3)/8: seq(A055232(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016

Table[(7 + (-1)^n + 16*n + 14*n^2 + 4*n^3)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Apr 15 2016 *)
LinearRecurrence[{3,-2,-2,3,-1},{1,5,16,36,69},40] (* Harvey P. Dale, Oct 25 2020 *)

lista(nn) = for(n=0, nn, print1((7+(-1)^n+16*n+14*n^2+4*n^3)/8, ", ")); \\ Altug Alkan, Apr 16 2016

G.f.: (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Clark Kimberling, Jun 02 2012
a(n) = (7+(-1)^n+16*n+14*n^2+4*n^3)/8. - Eric W. Weisstein, Apr 04 2013
a(n) = ((n+1)^3 + ceiling((n+1)/2)^2 + floor((n+1)/2)^2)/2. - Wesley Ivan Hurt, Apr 15 2016
E.g.f.: ((7 + 34*x + 26*x^2 + 4*x^3)*exp(x) + exp(-x))/8. - Ilya Gutkovskiy, Apr 16 2016

A128624 Row sums of A128623.

Original entry on oeis.org

1, 4, 12, 24, 45, 72, 112, 160, 225, 300, 396, 504, 637, 784, 960, 1152, 1377, 1620, 1900, 2200, 2541, 2904, 3312, 3744, 4225, 4732, 5292, 5880, 6525, 7200, 7936, 8704, 9537, 10404, 11340, 12312, 13357, 14440, 15600, 16800, 18081, 19404, 20812, 22264, 23805

Offset: 1

Gary W. Adamson, Mar 14 2007

Also the number of (w,x,y) with all terms in {0,...,n-1} and w <= R <= x, where R = max(w,x,y)-min(w,x,y), see A212959. - Clark Kimberling, Jun 10 2012

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A128621, A128623, A212959.

Cf. A094728 (diagonal row sums).

[n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // G. C. Greubel, Mar 12 2024

Table[n*(n^2 +2*n +Mod[n,2])/4, {n,50}] (* G. C. Greubel, Mar 12 2024 *)

Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

[n*((n+1)^2-1+(n%2))//4 for n in range(1,51)] # G. C. Greubel, Mar 12 2024

G.f.: x*(1+2*x+3*x^2) / ((1+x)^2*(1-x)^4). - R. J. Mathar, Jun 27 2012
From Colin Barker, Jan 31 2016: (Start)
a(n) = n*(2*n^2 + 4*n + 1 - (-1)^n)/8.
a(n) = n^2*(n + 2)/4 for n even.
a(n) = n*(n^2 + 2*n + 1)/4 for n odd. (End)
From G. C. Greubel, Mar 12 2024: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} A094728(n, k).
E.g.f.: (1/8)*x*(exp(-x) + (7 + 10*x + 2*x^2)*exp(x)). (End)

Incorrect formula removed by R. J. Mathar, Jun 27 2012

A213399 Number of (w,x,y) with all terms in {0,...,n} and max(|w-x|,|x-y|) = x.

Original entry on oeis.org

1, 4, 14, 23, 43, 58, 88, 109, 149, 176, 226, 259, 319, 358, 428, 473, 553, 604, 694, 751, 851, 914, 1024, 1093, 1213, 1288, 1418, 1499, 1639, 1726, 1876, 1969, 2129, 2228, 2398, 2503, 2683, 2794, 2984, 3101, 3301, 3424, 3634, 3763, 3983, 4118

Offset: 0

Clark Kimberling, Jun 13 2012

For a guide to related sequences, see A212959.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A212959.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[x == Max[Abs[w - x], Abs[x - y]], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213399 *)

Vec((1+3*x+8*x^2+3*x^3+x^4) / ((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
G.f.: (1 + 3*x + 8*x^2 + 3*x^3 + x^4)/((1 - x)^3 * (1 + x)^2).
From Colin Barker, Jan 26 2016: (Start)
a(n) = (8*n^2+2*(-1)^n*n+8*n+(-1)^n+3)/4.
a(n) = (4*n^2+5*n+2)/2 for n even.
a(n) = (4*n^2+3*n+1)/2 for n odd.
(End)

Showing 1-10 of 76 results. Next

cp's OEIS Frontend

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

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A032528 Concentric hexagonal numbers: floor(3*n^2/2).

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A047241 Numbers that are congruent to {1, 3} mod 6.

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A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

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A250229 T(n,k)=Number of length n+1 0..k arrays with the sum of the cubes of adjacent differences multiplied by some arrangement of +-1 equal to zero.

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A250277 T(n,k)=Number of length n+1 0..k arrays with the sum of the squares of adjacent differences multiplied by some arrangement of +-1 equal to zero.

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A250167 T(n,k)=Number of length n+1 0..k arrays with the sum of adjacent differences multiplied by some arrangement of +-1 equal to zero.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

A055232 Expansion of (1+2x+3x^2)/((1-x)^3*(1-x^2)).