A168380 -id:A168380 - cp's OEIS frontend

A000982 a(n) = ceiling(n^2/2).

0, 1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405

Offset: 0

N. J. A. Sloane

a(n) = number of pairs (i,j) in [1..n] X [1..n] with integral arithmetic mean. Cf. A132188, A362931. - N. J. A. Sloane, Aug 28 2023
Also, floor( (n^2+1)/2 ). - N. J. A. Sloane, Feb 08 2019
Floor(arithmetic mean of next n numbers). - Amarnath Murthy, Mar 11 2003
Pairwise sums of repeated squares (A008794).
Also, number of topologies on n+1 unlabeled elements with exactly 4 elements in the topology. a(3) gives 4 elements a,b,c,d; the valid topologies are (0,a,ab,abcd), (0,a,abc,abcd), (0,ab,abc,abcd), (0,a,bcd,abcd) and (0,ab,cd,abcd), with a count of 5. - Jon Perry, Mar 05 2004
Partition n into two parts, say, r and s, so that r^2 + s^2 is minimal, then a(n) = r^2 + s^2. Geometrical significance: folding a rod with length n units at right angles in such a way that the end points are at the least distance, which is given by a(n)^(1/2) as the hypotenuse of a right triangle with the sum of the base and height = n units. - Amarnath Murthy, Apr 18 2004
Convolution of A002061(n)-0^n and (-1)^n. Convolution of n (A001477) with {1,0,2,0,2,0,2,...}. Partial sums of repeated odd numbers {0,1,1,3,3,5,5,...}. - Paul Barry, Jul 22 2004
The ratio of the sum of terms over the total number of terms in an n X n spiral. The sum of terms of an n X n spiral is A037270, or Sum_{k=0..n^2} k = (n^4 + n^2)/2 and the total number of terms is n^2. - William A. Tedeschi, Feb 27 2008
Starting with offset 1 = row sums of triangle A158946. - Gary W. Adamson, Mar 31 2009
Partial sums of A109613. - Reinhard Zumkeller, Dec 05 2009
Also the number of compositions of even natural numbers into 2 parts < n. For example a(3)=5 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2) of even natural numbers into 2 parts < 3. a(4)=8 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2), (1,3), (3,1), (3,3) of even natural numbers into 2 parts < 4. - Adi Dani, Jun 05 2011
A001105 and A001844 interleaved. - Omar E. Pol, Sep 18 2011
Number of (w,x,y) having all terms in {0,...,n} and w=average(x,y). - Clark Kimberling, May 15 2012
For n > 0, minimum number of lines necessary to get through all unit cubes of an n X n X n cube (see Kantor link). - Michel Marcus, Apr 13 2013
Sum_{n > 0} 1/a(n) = Sum_{n > 0} 1/(2*n^2) + Sum_{n >= 0} 1/(2*n + 2*n^2 + 1) = (zeta(2) + (Pi* tanh(Pi/2)))/2 = 2.26312655.... - Enrique Pérez Herrero, Jun 17 2013
For n > 1, a(n) is the edge cover number of the n X n king graph. - Eric W. Weisstein, Jun 20 2017
Also the number of vertices in the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
The same sequence arises in the triangular array of the integers >= 1, according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array, and the second row of that sub-array (with apex a(n-1)) contains just two numbers, one odd, one even. The one with opposite parity to a(n-1) is a(n). - David James Sycamore, Jul 29 2018
Size of minimal ternary 1-covering code with code length n, i.e., K_n(3,1). See Kalbfleisch and Stanton. - Patrick Wienhöft, Jan 29 2019
For n > 1, a(n-1) is the maximum number of inversions in a permutation consisting of a single n-cycle on n symbols. - M. Ryan Julian Jr., Sep 10 2019
Also the number of classes of convex inscribed polyominoes in a (2,n) rectangular grid; two polyominoes are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other. - Jean-Luc Manguin, Jan 29 2020
a(n) is the number of pairs (p,q) such that 1 <= p, p+1 < q <= n+2 and q <> 2*p. - César Eliud Lozada, Oct 25 2020
a(n) is the maximum number of copies of a 12 permutation pattern in an alternating (or zig-zag) permutation of length n+1. The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous. - Lara Pudwell, Dec 01 2020
It appears that a(n) is the largest number of nodes of an induced path in the n X n king graph. An induced path going in a simple spiraling pattern, starting in a corner, has a(n) nodes. For even n this is optimal, because an induced path can have at most two nodes in any 2 X 2 subsquare. For odd n, I cannot see how to prove that (n^2+1)/2 is best possible. See also A357501. - Pontus von Brömssen, Oct 02 2022 [Proved by Beluhov (2023). - Pontus von Brömssen, Jan 30 2023]
a(n) = n + 2*(n-2) + 2*(n-4) + 2*(n-6) + ... number of black squares on an n X n chessboard. - R. J. Mathar, Dec 03 2022

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 13*x^5 + 18*x^6 + 25*x^7 + 32*x^8 + ...
Centrosymmetric 3 X 3 matrix: [[a,b,c],[d,e,d],[c,b,a]], a(3) = 3*(3-1)/2 + (3-1)/2 + 1 = (3^2+1)/2 = 5 from a,b,c,d,e. 4 X 4 case: [[a,b,c,d],[e,f,g,h],[h,g,f,e],[d,c,b,a]], a(4) = 4*4/2 = 8. - _Wolfdieter Lang_, Oct 12 2015
a(3) = 5. The alternating permutation of length 3 + 1 = 4 with the maximum number of copies of 123 is 1324. The five copies are 12, 13, 14, 23, and 24. - _Lara Pudwell_, Dec 01 2020

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,4) in theorem 5.
Andrea C. Burgess, Caleb W. Jones, and David A. Pike, Extending Graph Burning to Hypergraphs, arXiv:2403.01001 [math.CO], 2024. See p. 9.
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Ronald Cools, Ian H. Sloan, Minimial cubature formulae of trigonometric degree, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 2.
John Elias, Illustration of Initial Terms: Intersection of a double spaced square grid and centrally aligned triangle.
J. G. Kalbfleisch and R. G. Stanton, A combinatorial problem in matching, J. London Math. Soc. Vol. 1, No. 1 (1969), 60-64. [Corrected by _N. J. A. Sloane_, Feb 08 2019]
J. M. Kantor, Mathématiques venues d'ailleurs: divertissements mathématiques en U.R.S.S., Le cube transpercé, pp. 56-62, Belin, Paris, 1982.
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta, and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society, 67.7 (2020), 994-1001.
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Edge Cover Number
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Topology
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000096, A000217, A001105, A001477, A001844, A002061, A004526, A005843, A007590, A008794, A037270, A081352, A109613, A110654, A116940, A134444, A158946, A168380, A357501.
Column 2 of A195040.
Cf. also A132188, A362931.

a000982 = (`div` 2) . (+ 1) . (^ 2)  -- Reinhard Zumkeller, Jun 27 2013

[(2*n^2 + 1 - (-1)^n) / 4: n in [0..60]]; // Vincenzo Librandi, Jun 16 2011

seq( ceil(n^2/2),n=0..30) ; # R. J. Mathar, Jun 05 2011

Table[Ceiling[n^2/2], {n, 0, 120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
Accumulate[Join[{0}, (# - Boole[EvenQ[#]] &) /@ Range[80]]] (* Alonso del Arte, Sep 11 2019 *)

a(n)=(n^2+1)\2 \\ Charles R Greathouse IV, Sep 13 2013

x='x+O('x^100); concat([0], Vec(x*(1+x^2)/((1+x)*(1-x)^3))) \\ Altug Alkan, Oct 12 2015

apply( A000982(n)=n^2\/2, [0..55]) \\ M. F. Hasler, Feb 29 2020

def A000982(n): return n**2+1>>1 # Chai Wah Wu, Aug 28 2023

(((1 to 49) by 2) flatMap { List.fill(2)() }).scanLeft(0)( + ) // _Alonso del Arte, Sep 11 2019

a(2*n) = 2*n^2, a(2*n+1) = 2*n^2 + 2*n + 1.
G.f.: -x*(1+x^2) / ( (1+x)*(x-1)^3 ). - Simon Plouffe in his 1992 dissertation
From Benoit Cloitre, Nov 06 2002: (Start)
a(n) = (2*n^2 + 1 - (-1)^n) / 4.
a(0)=0, a(1)=1; for n>1, a(n+1) = n + 1 + max(2*floor(a(n)/2), 3*floor(a(n)/3)). (End)
G.f.: (x + x^2 + x^3 + x^4)/((1 - x)*(1 - x^2)^2), not reduced. - Len Smiley
a(n) = a(n-2) + 2n - 2. - Paul Barry, Jul 17 2004
From Paul Barry, Jul 22 2004: (Start)
G.f.: x*(1+x^2)/((1-x^2)*(1-x)^2) = x*(1+x^2)/((1+x)*(1-x)^3);
a(n) = Sum_{k=0..n} (k^2 - k + 1 - 0^k)*(-1)^(n-k);
a(n) = Sum_{k=0..n} (1 + (-1)^(n-k) - 0^(n-k))*k. (End)
From Reinhard Zumkeller, Feb 27 2006: (Start)
a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1.
a(n) = A116940(n) - A005843(n). (End)
Starting with offset 1, = row sums of triangle A134444. Also, with offset 1, = binomial transform of [1, 1, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
a(n) = floor((n^2+1)/2). - William A. Tedeschi, Feb 27 2008
a(n) = A004526(n+1) + A000217(n-1). - Yosu Yurramendi, Sep 12 2008, corrected by Klaus Purath, Jun 15 2021
From Jaume Oliver Lafont, Dec 05 2008: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End)
a(n) = A004526(n)^2 + A110654(n)^2. - Philippe Deléham, Mar 12 2009
a(n) = n^2 - floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
Euler transform is length 4 sequence [2, 2, 0, -1].
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
a(n) is also the number of independent entries in a centrosymmetric n X n matrix: M(i, j) = M(n-i+1, n-j+1). - Wolfdieter Lang, Oct 12 2015
For n > 1, a(n+1)/a(n) = 3 - A081352(n-2)/a(n). - Miko Labalan, Mar 26 2016
E.g.f.: (1/2)*(x*(1 + x)*cosh(x) + (1 + x + x^2)*sinh(x)). - Stefano Spezia, Feb 03 2020
a(n) = binomial(n+1,2) - floor(n/2). - César Eliud Lozada, Oct 25 2020
From Klaus Purath, Jun 15 2021: (Start)
a(n-1) + a(n) = A002061(n).
a(n) = (a(n-1)^2 + 1) / a(n-2), n >= 3 odd.
a(n) = (a(n-1)^2 - (n-1)^2) / a(n-2), n >= 4 even. (End)

A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

Original entry on oeis.org

0, 4, 20, 56, 120, 220, 364, 560, 816, 1140, 1540, 2024, 2600, 3276, 4060, 4960, 5984, 7140, 8436, 9880, 11480, 13244, 15180, 17296, 19600, 22100, 24804, 27720, 30856, 34220, 37820, 41664, 45760, 50116, 54740, 59640, 64824, 70300, 76076, 82160

Offset: 0

N. J. A. Sloane

Total number of possible bishop moves on an n+1 X n+1 chessboard, if the bishop is placed anywhere. E.g., on a 3 X 3-Board: bishop has 8 X 2 moves and 1 X 4 moves, so a(2)=20. - Ulrich Schimke (ulrschimke(AT)aol.com)
Let M_n denote the n X n matrix M_n(i,j)=(i+j)^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Partial sums of A016742. - Lekraj Beedassy, Jun 19 2004
0,4,20,56,120 gives the number of electrons in closed shells in the double shell periodic system of elements. This is a new interpretation of the periodic system of the elements. The factor 4 in the formula 4*n(n+1)(2n+1)/6 plays a significant role, since it designates the degeneracy of electronic states in this system. Closed shells with more than 120 electrons are not expected to exist. - Karl-Dietrich Neubert (kdn(AT)neubert.net)
Inverse binomial transform of A240434. - Wesley Ivan Hurt, Apr 13 2014
Atomic number of alkaline-earth metals of period 2n. - Natan Arie Consigli, Jul 03 2016
a(n) are the negative cubic coefficients in the expansion of sin(kx) into powers of sin(x) for the odd k: sin(kx) = k sin(x) - c(k) sin^3(x) + O(sin^5(x)); a(n) = c(2n+1) = A000292(2n). - Mathias Zechmeister, Jul 24 2022
Also the number of distinct series-parallel networks under series-parallel reduction on three unlabeled edges of n element kinds. - Michael R. Hayashi, Aug 02 2023

A. O. Barut, Group Structure of the Periodic System, in Wybourne, Ed., The Structure of Matter, University of Canterbury Press, Christchurch, 1972, p. 126.
Edward G. Mazur, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press, Alabama, 1974.
W. Permans and J. Kemperman, "Nummeringspribleem van S. Dockx, Mathematisch Centrum. Amsterdam," Rapport ZW; 1949-005, 4 leaves, 19.8 X 34 cm.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 32.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article #14.3.5.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
D. Neubert, Double Shell Structure of the Periodic System of the Elements, Z. Naturforschung, 25A (1970), p. 210.
Karl-Dietrich Neubert, Double-Shell PSE: Metals - Nonmetals.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2219-2226.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000330, A005408, A006331, A053120, A081277.
Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A049450 (Pawn).
Cf. A002412, A016061.

[2*n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

A002492:=n->2*n*(n+1)*(2*n+1)/3; seq(A002492(n), n=0..50); # Wesley Ivan Hurt, Apr 04 2014

Table[2n(n+1)(2n+1)/3, {n,0,40}] (* or *) Binomial[2*Range[0,40]+2,3] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,4,20,56},40] (* Harvey P. Dale, Aug 15 2012 *)
Accumulate[(2*Range[0,40])^2] (* Harvey P. Dale, Jun 04 2019 *)

PARI
```
a(n)=2*n*(n+1)*(2*n+1)/3
```

G.f.: 4*x*(1+x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(-1-n) = -a(n).
a(n) = 4*A000330(n) = 2*A006331(n) = A000292(2*n).
a(n) = (-1)^(n+1)*A053120(2*n+1,3) (fourth unsigned column of Chebyshev T-triangle, zeros omitted).
a(n) = binomial(2*n+2, 3). - Lekraj Beedassy, Jun 19 2004
A035005(n+1) = a(n) + A035006(n+1) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010
a(n) - a(n-1) = 4*n^2. - Joerg Arndt, Jun 16 2011
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Aug 15 2012
a(n) = Sum_{k=0..3} C(n-2+k,n-2)*C(n+3-k,n), for n>2. - J. M. Bergot, Jun 14 2014
a(n) = 2*A006331(n). - R. J. Mathar, May 28 2016
From Natan Arie Consigli Jul 03 2016: (Start)
a(n) = A166464(n) - 1.
a(n) = A168380(2*n). (End)
a(n) = Sum_{i=0..n} A005408(i)*A005408(i-1)+1 with A005408(-1):=-1. - Bruno Berselli, Jan 09 2017
a(n) = A002412(n) + A016061(n). - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 9/2 - 6*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/2 - 9/2. (End)
a(n) = A081277(3, n-1) = (1+2*n)*binomial(n+1, n-2)*2^2/(n-1) for n > 0. - Mathias Zechmeister, Jul 26 2022
E.g.f.: 2*exp(x)*x*(6 + 9*x + 2*x^2)/3. - Stefano Spezia, Jul 31 2022

Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010

Title modified by Charles R Greathouse IV at the suggestion of J. M. Bergot, Apr 05 2014

A172002 A permutation of the natural numbers in groups of 2*k^2, k=1,2,....

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 7, 10, 6, 11, 5, 12, 16, 17, 15, 18, 14, 19, 13, 20, 29, 30, 28, 31, 27, 32, 26, 33, 25, 34, 24, 35, 23, 36, 22, 37, 21, 38, 47, 48, 46, 49, 45, 50, 44, 51, 43, 52, 42, 53, 41, 54, 40, 55, 39, 56, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67, 78, 66, 79, 65, 80

Offset: 1

Paul Curtz, Jan 22 2010

nonn

The idea is based on the Janet table of the elements (see A138509 and A171710). Arrange the atomic numbers as if the rows of the table were centered. There are two rows with 2 =2*1^2 elements, 2 rows with 8=2*2^2 elements, 2 rows with 18=2*3^2 elements, and this is extended infinitely by adding 2 rows with 2*k^2 elements (see A137583), incrementing k:
...........................1...2.........................
...........................3...4.........................
..................5..6..7..8...9.10.11.12................
.................13.14.15.16..17.18.19.20................
..21.22.23.24.25.26.27.28.29..30.31.32.33.34.35.36.37.38.
..39.40.41.42.43.44.45.46.47..48.49.50.51.52.53.54.55.56.
The sequence is obtained by reading the numbers in each of the rows (top-down), starting with the center left column, then the center right column, and then alternating from the left to the right, increasing the distance to the center until all 2*k^2 numbers of the block are exhausted.

Michael De Vlieger, Table of n, a(n) for n = 1..10680 (First 76 rows)
Anonymous, Janet periodic table, Web Elements Chemistry
Anonymous, Periodic Table: Formulations, Chemogenesis web book
Albert Tarantola, PSE of Elements (Janet form).

Table[(Riffle[Reverse@ #, Length@ # + #] &@ Range[Ceiling[n/2]^2]) + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 7}] // Flatten (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

Edited by R. J. Mathar, Mar 02 2010

A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

Original entry on oeis.org

2, 1, 4, 3, 12, 11, 10, 9, 8, 7, 6, 5, 20, 19, 18, 17, 16, 15, 14, 13, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89

Offset: 1

Paul Curtz, May 10 2008

Cf. A137583, A168380 (first number in each row), A171710, A172002.

Table[Reverse@ Range[2 Ceiling[n/2]^2] + (# + 1) (3 + 2 #^2 + 4 # - 3 (-1)^#)/12 &[n - 1], {n, 8}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

There are respectively A137583 = 2, 2, 8, 8, 18, 18, 32, 32 terms from right to left.

A160914 Extended s-block elements for Janet table.

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 19, 20, 37, 38, 55, 56, 87, 88, 119, 120, 169, 170, 219, 220, 291, 292, 363, 364, 461, 462, 559, 560, 687, 688, 815, 816, 977, 978, 1139, 1140, 1339, 1340, 1539, 1540, 1781, 1782, 2023, 2024, 2311, 2312, 2599, 2600, 2937, 2938, 3275, 3276

Offset: 1

Paul Curtz, Oct 15 2011

See A168342. Must be included in A167268. From right to left, first vertical is A168380 from 1 to 8. Second vertical is A168380-1. In (1) page 12, introducing elements 93 to 120, Janet says that there is a probable 8th row. For row 8, he proposes, like for row 7, 32 elements (89 to 120). Page 16 he presents 4 blocks: first has 2*8 elements, second: 6*6, third: 10*4, fourth: 14*2. Today, blocks are s,p,d,f for Mendeleyev-Moseley-Seaborg 118 elements periodic table. See (2), (3), A173592 and A138509. In 1927, only 88 on the first 92 elements were known; 41 (1937 discovered), 61 (1947), 85 (1940) and 87 (1939) were missing. Since 2010 (117 discovered) the first 118 elements are known. Janet predicted only 120 elements.

			The following is an s-block, 2*8=16 elements, i.e., a(n) written vertically, after p-block, 6*6, (A138469).
                          1   2
                          3   4
5   6   7   8   9   10   11  12
13  14  15  16  17  18   19  20
31  32  33  34  35  36   37  38
49  50  51  52  53  54   55  56
81  82  83  84  85  86   87  88
113 114 115 116 117 118  119 120

Charles JANET, La structure du Noyau de l'atome,considérée dans la Classification périodique, des éléments chimiques, 1927 (Novembre) N. 2 Beauvais, 67 pages, 3 leaflets.

Cf. A099955.

A168234 A138100(n) + A168142(n).

Original entry on oeis.org

3, 3, 13, 13, 13, 13, 13, 13, 13, 13, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 171, 171, 171, 171, 171, 171, 171

Offset: 1

Paul Curtz, Nov 21 2009

A138100 counts upwards in blocks of 2*k^2 numbers, restarting from 1,5,21,57,.. = A166464(k-1) = (2*k+1)*(2*k^2-4*k+3)/3, k>=1.
A168142 counts downwards in blocks of 2*k^2 numbers, restarting from 2*k^2, k>=1.
In consequence, the sequence here contains 2*k^2 copies of the number 1+2*k*(1+2*k^2)/3 = 1+A035597(k), k>=1,
where the sequence A035597 is a bisection of A168380.

Cf. A134984, A138725, A168208.

Edited by R. J. Mathar, Feb 15 2010

A168388 First number in the n-th row of A172002.

Original entry on oeis.org

1, 3, 5, 13, 21, 39, 57, 89, 121, 171, 221, 293, 365, 463, 561, 689, 817, 979, 1141, 1341, 1541, 1783, 2025, 2313, 2601, 2939, 3277, 3669, 4061, 4511, 4961, 5473, 5985, 6563, 7141, 7789, 8437, 9159, 9881, 10681, 11481, 12363, 13245, 14213, 15181, 16239, 17297

Offset: 1

Paul Curtz, Nov 24 2009

Harvey P. Dale, 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. A168234.

[(12+n+3*(-1)^n*n+2*n^3)/12: n in [1..60]]; // Vincenzo Librandi, Jul 20 2016

LinearRecurrence[{2,1,-4,1,2,-1},{1,3,5,13,21,39},50] (* Harvey P. Dale, Nov 29 2014 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 0, 46}] (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

a(2*n+1) = A166464(n)        a(2*n) = A166911(n).
a(n+1) - a(n) = A093907(n-1), n>1.
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
G.f.: x*(1 - x^2 + 2*x)*(1 - x + x^2 + x^3)/( (1+x)^2 * (x-1)^4).
a(n+1) = A168380(n)+1.
From G. C. Greubel, Jul 19 2016: (Start)
a(n) = (12 + n + 3*(-1)^n*n + 2*n^3)/12.
E.g.f.: (1/12)*( -3*x - 12*exp(x) + (12 + 3*x + 6*x^2 + 2*x^3)*exp(2*x) )*exp(-x). (End)

Edited and extended by R. J. Mathar, Mar 25 2010

A168582 a(n) = (4n^3 - 6n^2 + 8n + 9 + 3(-1)^n)/12.

Original entry on oeis.org

1, 1, 3, 7, 17, 33, 59, 95, 145, 209, 291, 391, 513, 657, 827, 1023, 1249, 1505, 1795, 2119, 2481, 2881, 3323, 3807, 4337, 4913, 5539, 6215, 6945, 7729, 8571, 9471, 10433, 11457, 12547, 13703, 14929, 16225, 17595, 19039, 20561

Offset: 0

Paul Curtz, Nov 30 2009

Starting with a(2), the sum of the first and last term in row n-1 of the Janet table A172002.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A137928 (first differences).

[2*n/3 +3/4 -n^2/2 +n^3/3 +(-1)^n/4: n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[(4*n^3 - 6*n^2 + 8*n + 9 + 3*(-1)^n)/12, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)

a(n)=(4*n^3-6*n^2+8*n+9+3*(-1)^n)/12 \\ Charles R Greathouse IV, Jul 26 2016

a(n+2) = A168388(n) + A168380(n), n >= 0.
a(2n) = A168547(n);
a(2n+1) = A168574(n).
G.f.: (1 - 2*x + x^4 + 2*x^2 + 2*x^3)/((1+x)*(x-1)^4). - R. J. Mathar, Jun 27 2011
E.g.f.: (1/12)*((4*x^3 + 6*x^2 + 6*x + 9)*exp(x) + 3*exp(-x)). - G. C. Greubel, Jul 26 2016

A171710 Union of A168234 and A171219, sorted.

Original entry on oeis.org

3, 3, 5, 5, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89

Offset: 1

Paul Curtz, Dec 16 2009

Consider a table T(n,k) similar to A168142 = {2,1}, {8,7,6,...,2,1}, {18,17,...,2,1},... that repeats each row. Thus T(n,k) = {2,1}, {2,1}, {8,7,6,...,2,1}, {8,7,6,...,2,1}, {18,17,...,2,1}, etc. The rows of T(n,k) decrement from 2*ceiling(n/2)^2 to 1. Then we can construct the table of atomic numbers in the Janet periodic table A138509(n) = T(n,k) + a(n), with k=2*ceiling(n/2)^2 - 1 down to 1 by step -1.

Michael De Vlieger, Table of n, a(n) for n = 1..10680 (first 39 rows)

Cf. A138509 (Janet periodic table, rows n > 1 end in the repeated numbers in this sequence), A168234 (odd rows), A168380 (repeated numbers k - 1), A171219 (even rows), A172002 (smallest values of rows n > 1 are the repeated numbers in this sequence).

Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 7}, {k, 2 Ceiling[n/2]^2}] // Flatten (* Michael De Vlieger, Jul 20 2016 *)

May be regarded as an irregular triangle read by rows, defined by T(n,k) = A168380(n) + 1, with 1 <= k <= ceiling(n/2)^2. - Michael De Vlieger, Jul 20 2016

A217923 F-block elements for Janet periodic table.

Original entry on oeis.org

57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Jean-François Alcover, Oct 15 2012

From Paul Curtz, Oct 26 2012: (Start)
Tarantola's formulas for the Janet table:
Let D(n) = n*(n+1)*(n+2)/6 + (1-(-1)^n)*(n+1)/4 = A168380(n).
The row R at which the element with atomic number Z is to be placed is the unique value of R satisfying D(R-1) < Z <= D(R).
Once the row number R is determined, the column number is, from right to left, C = D(R) - Z + 1.
Example: Z=109. D(7) < Z <= D(8) = 120. C = 120 - 109 + 1 = 12. (End)

Wikipedia, The Janet periodic table

Cf. A160914 (s-block), A138469 (p-block), A199934 (d-block).

Mathematica
```
Join[Range[57,70], Range[89,102]]
```

cp's OEIS Frontend

A000982 a(n) = ceiling(n^2/2).

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A002492 Sum of the first n even squares: 2*n*(n+1)*(2*n+1)/3.

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A172002 A permutation of the natural numbers in groups of 2*k^2, k=1,2,....

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A138509 Janet (or left-step) periodic table from right to left. 120 terms. 8 rows, 32 columns without spaces.

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A160914 Extended s-block elements for Janet table.

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A168234 A138100(n) + A168142(n).

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A168388 First number in the n-th row of A172002.

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A168582 a(n) = (4*n^3 - 6*n^2 + 8*n + 9 + 3*(-1)^n)/12.

A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

A168582 a(n) = (4n^3 - 6n^2 + 8n + 9 + 3(-1)^n)/12.