A008598 -id:A008598 - cp's OEIS frontend

A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.

1, 5, 11, 16, 21, 27, 32, 37, 43, 48, 53, 59, 64, 69, 75, 80, 85, 91, 96, 101, 107, 112, 117, 123, 128, 133, 139, 144, 149, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 213, 219, 224, 229, 235, 240, 245, 251, 256, 261, 267, 272, 277, 283, 288, 293, 299

Offset: 0

Allan C. Wechsler, Nov 21 2012

a(n) is the number of vertices of the 3.3.4.3.4 tiling (which has three triangles and two squares, in the given cyclic order, meeting at each vertex) whose shortest path connecting them to a given origin vertex contains n edges.
This is the dual tiling to the Cairo tiling (cf. A296368). - N. J. A. Sloane, Nov 02 2018
First few terms provided by Allan C. Wechsler; Fred Lunnon and Fred Helenius gave the next few; Fred Lunnon suggested that the recurrence was a(n+3) = a(n) + 16 for n > 1. [This conjecture is true - see the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 31 2017]
Appears also to be coordination sequence for node of type V2 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 25 2018
Appears also to be coordination sequence for node of type V1 in "krj" 2-D tiling (or net). This also should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 26 2018
First differences of A301696. - Klaus Purath, May 23 2020

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 2nd tiling, also 2nd row, third tiling.

Joseph Myers, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Colouring tiles in an isohedral tiling: automaton, defects and grain boundaries, arXiv:2301.10975 [math.CO], 2023.
Brian Galebach, Collection of n-Uniform Tilings. See Numbers 14 and 17 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Chaim Goodman-Strauss and N. J. A. Sloane, Trunks and branches coloring (taken from preceding reference)
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Kival Ngaokrajang, Illustration of initial terms
Reticular Chemistry Structure Resource, tts
Reticular Chemistry Structure Resource (RCSR), The krd tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The krj tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Cf. A296368, A301694, A301697.

-- Very slow, could certainly be accelerated.  SST stands for Snub Square Tiling.
setUnion [] l2 = l2
setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setUnion rst l2
setDifference [] l2 = []
setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setDifference rst l2
adjust k = (if (even k) then 1 else -1)
weirdAdjacent (x,y) = (x+(adjust y),y+(adjust x))
sstAdjacents (x,y) = [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(weirdAdjacent (x,y))]
sstNeighbors core = foldl setUnion core (map sstAdjacents core)
sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
sstHalo core = setDifference (sstNeighbors core) core
origin = [(0,0)]
a219529 n = length (sstHalo (sstGlob (n-1) origin))
-- Allan C. Wechsler, Nov 30 2012

A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1,3))/3);
seq(A219529(n), n = 0..60); # G. C. Greubel, May 27 2020

Join[{1}, LinearRecurrence[{1,0,1,-1}, {5,11,16,21}, 60]] (* Jean-François Alcover, Dec 13 2018 *)
Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* G. C. Greubel, May 27 2020 *)
CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2),{x,0,70}],x] (* Harvey P. Dale, Jul 03 2021 *)

[1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # G. C. Greubel, May 27 2020

Conjectured to be a(n) = floor((16n+1)/3) for n>0; a(0) = 1; this is a consequence of the suggested recurrence due to Lunnon (see comments). [This conjecture is true - see the CGS-NJAS link in A296368 for a proof. - N. J. A. Sloane, Dec 31 2017]
G.f.: (x+1)^4/((x^2+x+1)*(x-1)^2). - N. J. A. Sloane, Feb 07 2018
From G. C. Greubel, May 27 2020: (Start)
a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.
a(n) = (A008598(n) - A049347(n-1))/3 for n >0 with a(0)=1. (End)

Corrected attributions and epistemological status in Comments; provided slow Haskell code - Allan C. Wechsler, Nov 30 2012

Extended by Joseph Myers, Dec 04 2014

A121036 Multiples of 16 containing a 16 in their decimal representation.

Original entry on oeis.org

16, 160, 416, 816, 1168, 1216, 1600, 1616, 1632, 1648, 1664, 1680, 1696, 2016, 2160, 2416, 2816, 3168, 3216, 3616, 4016, 4160, 4416, 4816, 5168, 5216, 5616, 6016, 6160, 6416, 6816, 7168, 7216, 7616, 8016, 8160, 8416, 8816, 9168, 9216, 9616, 10016

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A121041, A008598, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121037, A121038, A121039, A121040.

seqMemberQ[lst_,seq_]:=MemberQ[Partition[lst,Length[seq],1],seq]
m16Q[n_]:=seqMemberQ[IntegerDigits[n],{1,6}]
Select[16Range[1000],m16Q]  (* Harvey P. Dale, Feb 26 2011 *)
Select[16Range[1000],SequenceCount[IntegerDigits[#],{1,6}]>0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Apr 12 2016 *)

is(n)=if(n%16,return(0));n=eval(Vec(Str(n)));for(i=2,#n,if(n[i]==6&&n[i-1]==1,return(1)));0

a(n) ~ 16n. - Charles R Greathouse IV, Jul 16 2011

Corrected by Harvey P. Dale, Feb 26 2011

A008600 Multiples of 18.

Original entry on oeis.org

0, 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360, 378, 396, 414, 432, 450, 468, 486, 504, 522, 540, 558, 576, 594, 612, 630, 648, 666, 684, 702, 720, 738, 756, 774, 792, 810, 828, 846, 864, 882, 900, 918, 936

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 330.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A008598, A044102, A154612.

Range[0, 1000, 18] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[18 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)
18*Range[0,60] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=18*n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 18*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 18*x*exp(x).
a(n) = 18*n = 2*A008591(n) = A044102(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A195040 Square array read by antidiagonals with T(n,k) = kn^2/4+(k-4)((-1)^n-1)/8, n>=0, k>=0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 3, 2, 1, 0, 1, 4, 5, 3, 1, 0, 0, 7, 8, 7, 4, 1, 0, 1, 9, 13, 12, 9, 5, 1, 0, 0, 13, 18, 19, 16, 11, 6, 1, 0, 1, 16, 25, 27, 25, 20, 13, 7, 1, 0, 0, 21, 32, 37, 36, 31, 24, 15, 8, 1, 0, 1, 25, 41, 48, 49, 45, 37, 28, 17, 9, 1, 0

Offset: 0

Omar E. Pol, Sep 27 2011

Also, if k >= 2 and m = 2*k, then column k lists the numbers of the form k*n^2 and the centered m-gonal numbers interleaved.
For k >= 3, this is also a table of concentric polygonal numbers. Column k lists the concentric k-gonal numbers.
It appears that the first differences of column k are the numbers that are congruent to {1, k-1} mod k, if k >= 3.

			Array begins:
  0,   0,   0,   0,   0,   0,   0,   0,   0,   0, ...
  1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5,   6,   7,   8,   9, ...
  1,   3,   5,   7,   9,  11,  13,  15,  17,  19, ...
  0,   4,   8,  12,  16,  20,  24,  28,  32,  36, ...
  1,   7,  13,  19,  25,  31,  37,  43,  49,  55, ...
  0,   9,  18,  27,  36,  45,  54,  63,  72,  81, ...
  1,  13,  25,  37,  49,  61,  73,  85,  97, 109, ...
  0,  16,  32,  48,  64,  80,  96, 112, 128, 144, ...
  1,  21,  41,  61,  81, 101, 121, 141, 161, 181, ...
  0,  25,  50,  75, 100, 125, 150, 175, 200, 225, ...
  ...

Muniru A Asiru, Rows n=0..100, flattened

Rows n: A000004 (n=0), A000012 (n=1), A001477 (n=2), A005408 (n=3), A008586 (n=4), A016921 (n=5), A008591 (n=6), A017533 (n=7), A008598 (n=8), A215145 (n=9), A008607 (n=10).

Columns k: A000035 (k=0), A004652 (k=1), A000982 (k=2), A077043 (k=3), A000290 (k=4), A032527 (k=5), A032528 (k=6), A195041 (k=7), A077221 (k=8), A195042 (k=9), A195142 (k=10), A195043 (k=11), A195143 (k=12), A195045 (k=13), A195145 (k=14), A195046 (k=15), A195146 (k=16), A195047 (k=17), A195147 (k=18), A195048 (k=19), A195148 (k=20), A195049 (k=21), A195149 (k=22), A195058 (k=23), A195158 (k=24).

nmax:=13;; T:=List([0..nmax],n->List([0..nmax],k->k*n^2/4+(k-4)*((-1)^n-1)/8));; b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 19 2018

A195040 := proc(n,k)
        k*n^2/4+((-1)^n-1)*(k-4)/8 ;
end proc:
for d from 0 to 12 do
        for k from 0 to d do
                printf("%d,",A195040(d-k,k)) ;
        end do:
end do; # R. J. Mathar, Sep 28 2011

t[n_, k_] := k*n^2/4+(k-4)*((-1)^n-1)/8; Flatten[ Table[ t[n-k, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011 *)

A016802 a(n) = (4*n)^2.

Original entry on oeis.org

0, 16, 64, 144, 256, 400, 576, 784, 1024, 1296, 1600, 1936, 2304, 2704, 3136, 3600, 4096, 4624, 5184, 5776, 6400, 7056, 7744, 8464, 9216, 10000, 10816, 11664, 12544, 13456, 14400, 15376, 16384, 17424, 18496, 19600, 20736, 21904, 23104, 24336, 25600, 26896, 28224

Offset: 0

N. J. A. Sloane

A bisection of A016742. Sequence arises from reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Also, sequence found by reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000 (first 200 terms from Ivan Panchenko).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001105, A001539, A016742, A016754, A016814, A016826, A016838, A112628.

Cf. A008598, A074377, A139098, A195146.

a(n) = (4*n)^2; \\ Michel Marcus, Mar 04 2014

def A016802(n): return (4*n)**2 # Karl-Heinz Hofmann, Sep 11 2024

a(n) = 16*n^2 = 16*A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 8*A001105(n) = 4*A016742(n) = 2*A139098(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 16*(2*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/192.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/4)/(Pi/4).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/4)/(Pi/4) = 2*sqrt(2)/Pi (A112628). (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 16*x*(1 + x)/(1-x)^3.
E.g.f.: 16*x*(1 + x)*exp(x).
a(n) = n*A008598(n) = A195146(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A008599 Multiples of 17.

Original entry on oeis.org

0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, 578, 595, 612, 629, 646, 663, 680, 697, 714, 731, 748, 765, 782, 799, 816, 833, 850, 867, 884

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Michael Penn, Divisibility by 17: The Lesser Known Rule, YouTube video, 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 329.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A008598, A008600, A244630.

Range[0, 1003, 17] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=17*n \\ Charles R Greathouse IV, Oct 07 2015

for (n <- 0 to 1003 by 17) print(n + ", ") // Alonso del Arte, Jun 17 2018

(floor(a(n)/10) - 5*(a(n) mod 10)) == 0 (mod 17), see A076311. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 17*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 17*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 17*x*exp(x).
a(n) = (A008598(n) + A008600(n))/2. (End)

A051062 a(n) = 16*n + 8.

Original entry on oeis.org

8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, 248, 264, 280, 296, 312, 328, 344, 360, 376, 392, 408, 424, 440, 456, 472, 488, 504, 520, 536, 552, 568, 584, 600, 616, 632, 648, 664, 680, 696, 712, 728, 744, 760, 776, 792, 808, 824, 840

Offset: 0

N. J. A. Sloane, Gary W. Adamson, Dec 11 1999

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).
n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/8). - Benoit Cloitre, Dec 17 2002
If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009
a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - Carmine Suriano, Jun 01 2014
For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - Bernard Schott, Sep 24 2017
Numbers k such that 3^k + 1 is divisible by 17*193. - Bruno Berselli, Aug 22 2018

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999.

Mia Boudreau, Table of n, a(n) for n = 0..10000
Mihaly Bencze, Problem 11508, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.
Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A008598, A017113, A106839, A139098, A244978.
Cf. A003484, A007814, A037227, A118413, A209675.
Cf. A003629, A016825, A017197, A017233, A098502, A119413.

[16*n+8: n in [0..50]]; // Wesley Ivan Hurt, Jun 01 2014

A051062:=n->16*n+8; seq(A051062(n), n=0..50); # Wesley Ivan Hurt, Jun 01 2014

Range[8, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
Table[16n+8, {n,0,50}] (* Wesley Ivan Hurt, Jun 01 2014 *)
LinearRecurrence[{2,-1},{8,24},60] (* or *) NestList[#+16&,8,60] (* Harvey P. Dale, Aug 18 2019 *)

a(n)=16*n+8 \\ Charles R Greathouse IV, May 09 2016

a(n) = A118413(n+1,4) for n>3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 32*n - a(n-1) for n>0, a(0)=8. - Vincenzo Librandi, Aug 06 2010
A003484(a(n)) = 8; A209675(a(n)) = 9. - Reinhard Zumkeller, Mar 11 2012
A007814(a(n)) = 3; A037227(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012
a(-1 - n) = - a(n). - Michael Somos, Jun 02 2014
Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 8*(1+x)/(1-x)^2.
E.g.f.: 8*exp(x)*(1 + 2*x).
a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).
a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(7*Pi/32).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(7*Pi/32). (End)

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

Original entry on oeis.org

0, 1, 1, 4, 3, 2, 12, 8, 5, 3, 32, 20, 12, 7, 4, 80, 48, 28, 16, 9, 5, 192, 112, 64, 36, 20, 11, 6, 448, 256, 144, 80, 44, 24, 13, 7, 1024, 576, 320, 176, 96, 52, 28, 15, 8, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 9, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 10

Offset: 0

Henry Bottomley, May 30 2001

From Philippe Deléham, Apr 15 2007: (Start)
This triangle can be found in the Laisant reference in the following form:
  .......................5...11..
  ...................4...9...20..
  ...............3...7..16...36..
  ...........2...5..12..28.......
  .......1...3...8..20..48.......
  ...0...1...4..12..32..80....... (End)
Triangle A152920 reversed. - Philippe Deléham, Apr 21 2009

			As a lower triangle (T(n, k)):
    0;
    1,   1;
    4,   3,   2;
   12,   8,   5,  3;
   32,  20,  12,  7,  4;
   80,  48,  28, 16,  9,  5;
  192, 112,  64, 36, 20, 11,  6;
  448, 256, 144, 80, 44, 24, 13, 7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
F. Ellermann, Illustration of binomial transforms
C.-A. Laisant, Sur les tableaux de sommes - Nouvelles applications, Compt. Rendus de l'Association Francaise pour l'Avancement des Sciences, Aout 04 1893, pp. 206-216 (table given on p. 212).

Rows include (essentially) A001787, A001792, A034007, A045623, A045891.
Diagonals include (essentially) A001477, A005408, A008586, A008598, A017113.
Column sums are A058877.
Cf. A111297, A159694, A159695, A159696, A159697. - Philippe Deléham, Apr 21 2009
Cf. A058877, A073371, A130129, A152920.

[2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 28 2022

Table[2^(n-k-1)*(n+k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 28 2022 *)

def A062111(n,k): return 2^(n-k-1)*(n+k)
flatten([[A062111(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 28 2022

A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.
A(n, k) = (k+n)*2^(k-n-1) if k >= n.
T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 21 2009
From G. C. Greubel, Sep 28 2022: (Start)
T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.
T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.
T(2*n-1, n-1) = A130129(n-1).
T(2*n+1, n-1) = 12*A001787(n).
Sum_{k=0..n} T(n, k) = A058877(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.
T(n, k) = A152920(n, n-k). (End)

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

cp's OEIS Frontend

A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.

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A121036 Multiples of 16 containing a 16 in their decimal representation.

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A008600 Multiples of 18.

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A195040 Square array read by antidiagonals with T(n,k) = k*n^2/4+(k-4)*((-1)^n-1)/8, n>=0, k>=0.

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A016802 a(n) = (4*n)^2.

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A008599 Multiples of 17.

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A051062 a(n) = 16*n + 8.

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A195040 Square array read by antidiagonals with T(n,k) = kn^2/4+(k-4)((-1)^n-1)/8, n>=0, k>=0.