A195040 -id:A195040 - 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)

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

A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

Original entry on oeis.org

0, 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, 108, 127, 147, 169, 192, 217, 243, 271, 300, 331, 363, 397, 432, 469, 507, 547, 588, 631, 675, 721, 768, 817, 867, 919, 972, 1027, 1083, 1141, 1200, 1261, 1323, 1387, 1452, 1519, 1587, 1657, 1728, 1801, 1875, 1951

Offset: 0

Henry Bottomley, Oct 22 2002

Triangular numbers plus quarter squares: (n+1)*(n+2)/2 + floor(n^2/4) (i.e., A000217(n+1) + A002620(n)).
Largest coefficient in the expansion of (1+x+x^2+...+x^(n-1))^3=((1-x^n)/(1-x))^3, i.e., the coefficient of x^floor[3(n-1)/2] and of x^ceiling[3(n-1)/2]; also number of compositions of [3(n+1)/2] into exactly 3 positive integers each no more than n.
A set of n independent statements a,b,c,d..., produces n^2 conditional statements of the form "If a, then b" (including self-implications such as "If a, then a"). If such statements are taken as equivalent to "It is not the case that the first statement is true and the second is false" (material implication), A077043(n) is the minimum number of the conditional statements that can be true. (The maximum number of false conditional statements is A002620(n), the maximum product of two integers whose sum is n.) - Matthew Vandermast, Mar 04 2003
This is also the maximum number of triple intersections between three sets of n lines, where the lines in each set are parallel to each other. E.g., for n=3:
\.\.\.../././
.\.\.\./././.
..\.\.x././..
---+-*-*-+---
----*-*-*----
---+-*-*-+---
.././.x.\.\..
./././.\.\.\.
/././...\.\.\
where '*' = triple intersection, '+' and 'x' = double intersection.
I am pretty sure that the hexagonal configuration of intersections shown above is the optimum and I get the formulas a(n) = (3n^2)/4 for n even and (3n^2+1)/4 for n odd. - Gabriel Nivasch (gnivasch(AT)yahoo.com), Jan 13 2004
For n > 1 the sequence represents the maximum number of points that can be placed in a plane such that the largest distances between any two points does not exceed the shortest of the distances between any two points by more than a factor n-1. - Johannes Koelman (Joc_kay(AT)hotmail.com), Apr 27 2006
This is also the number of distinct noncongruent isosceles triangles with side length up to n. - Patrick Hurst (patrick(AT)imsa.edu), May 14 2008
Also concentric triangular numbers. A033428 and A003215 interleaved. - Omar E. Pol, Sep 28 2011
Number of (w,x,y) with all terms in {0,...,n} and w=x>range{w,x,y}. - Clark Kimberling, Jun 02 2012
Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x<=y. - Clark Kimberling, Jul 02 2012
From Bob Selcoe, Aug 05 2013: (Start)
a(n) is the number of 3-member sets with non-repeating positive integer values (x,y,z) whose sums equal 3(n+1). Example: a(4)=12; thus there are 12 sets where x+y+z = 15: (1,2,12), (1,3,11), (1,4,10), (1,5,9), (1,6,8), (2,3,10), (2,4,9), (2,5,8), (2,6,7), (3,4,8), (3,5,7) and (4,5,6).
From above, the number of sets sharing minimum values (minvals) equals a(1)-a(0), a(2)-a(1), a(3)-a(2),... a(n)-a(n-1) which are the numbers not divisible by 3, in sequence (A001651), range n to 1. So in the above example, there is one set with minval 4, two sets with minval 3, four sets with minval 2 and five sets with minval 1. (End)
Number of partitions of 3n into exactly 3 parts. - Wesley Ivan Hurt, Jan 21 2014
Number of partitions of 3(n-1) into at most 3 parts. - Colin Barker, Mar 31 2015
Number of possible positions after n-1 steps on the lines of a hexagonal grid. - Reg Robson, Mar 08 2014
12*a(n) is a perfect square when n is even and 12*a(n) - 3 is a perfect square when n is odd. - Miquel Cerda, Jun 30 2016
Square of largest Euclidean distance from start point reachable by an n-step walk on a honeycomb lattice. - Hugo Pfoertner, Jun 21 2018

			G.f. = x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...
a(4)=12 since the compositions of floor(3*(4+1)/2) = 7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+3, 2+4+1, 3+1+3, 3+2+2, 3+3+1, 4+1+2, 4+2+1.
From _Philippe Deléham_, Dec 17 2011: (Start)
a(1) = 1 = 1^3;
a(1) + a(3) = 1 + 7 = 2^3;
a(1) + a(3) + a(5) = 1 + 7 + 19 = 3^3;
a(1) + a(3) + a(5) + a(7) = 1 + 7 + 19 + 37 = 4^3;
a(1) + a(3) + a(5) + a(7) + a(9) = 1 + 7 + 19 + 37 + 61 = 5^3; ... (End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 4.4.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7 (2004), Article 04.1.6.
Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See page 402.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to compositions.

Column 3 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A019298 (partial sums).
Cf. A002620, A033428, A003215.
Equals one more than A331952 and one less than A084684. - Greg Dresden, Feb 22 2020

a077043 n = a077043_list !! n
a077043_list = scanl (+) 0 a001651_list
-- Reinhard Zumkeller, Jan 06 2014

[Ceiling(n^2*3/4): n in [0..60]]; // Vincenzo Librandi, Jun 29 2011

A077043:=n->ceil(3*n^2/4); seq(A077043(n), n=0..60); # Wesley Ivan Hurt, Jan 21 2014

Table[Ceiling[(3n^2)/4], {n,0,60}] (* or *) LinearRecurrence[{2,0,-2,1}, {0,1,3,7}, 60] (* Harvey P. Dale, Dec 16 2012 *)

{a(n) = n^2 - (n^2 \ 4)}; /* Michael Somos, Jun 29 2011 */

a(n) = ceiling(n^2*3/4) = A077042(n, 3); a(-n) = a(n).
Also can be computed from 1 * C(n,0) + 2 * C(n,1) + 2 * C(n,2) - Sum((-2)^(k-3) C(n, k)). - Joshua Zucker, Nov 10 2002
a(n) = A002620(n-1) + A002620(n) + A002620(n+1). - Jon Perry, May 29 2003
From Jon Perry, May 29 2003: (Start)
a(2k)   = a(2k-2) + 6k - 3,
a(2k+1) = a(2k-1) + 6k,
a(4n)   = 12n^2,
a(4n+1) = a(4n)   + 6n + 1,
a(4n+2) = a(4n+1) + 6n + 2,
a(4n+3) = a(4n+2) + 6n + 4,
a(4n+4) = a(4n+3) + 6n + 5.
Differences between alternate terms give 3, 6, 9, 12, ... (End)
a(n+1) - a(n) = A001651(n), partial sums of A001651. - Reinhard Zumkeller, Dec 28 2007
From R. J. Mathar, Nov 10 2008: (Start)
G.f.: x*(1+x+x^2)/((1+x)*(1-x)^3).
a(n) + a(n+1) = A005448(n+1).
The inverse binomial transform yields 0 followed by A141531. (End)
Euler transform of length 3 sequence [3, 1, -1]. - Michael Somos, Jun 29 2011
a(n) = 3*n^2/4 - ((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
Sum_{k=0..n} a(2k+1) = partial sums of A003215 = (n+1)^3 (see example). - Philippe Deléham, Dec 17 2011
a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Dec 16 2012
a(0)=0, a(1)=1, a(n) = 3*(n-1) + a(n-2). - Reg Robson, Mar 08 2014
a(2k) = 3k^2 = A033428(k), a(2k+1) = 3k^2 + 3k + 1 = A003215(k). - Jon Perry, Oct 25 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/2). - Wesley Ivan Hurt, Mar 12 2015
a(n) = (3*n)^2/12 for n even and a(n) = ((3*n)^2 + 3)/12 for n odd. - Miquel Cerda, Jun 30 2016
a(n) = Sum_{k=1..n} floor((n+k)/2). - Wesley Ivan Hurt, Mar 31 2017
0 = 1 +a(n)*(+a(n+1) -a(n+2)) +a(n+1)*(-3 -a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Apr 02 2017
E.g.f.: (1/8)*exp(-x)*(-1 + exp(2*x)*(1 + 6*x + 6*x^2)). - Stefano Spezia, Nov 29 2019
Sum_{n>=1} 1/a(n) = Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Jan 16 2023

A077221 a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.

Original entry on oeis.org

0, 1, 8, 17, 32, 49, 72, 97, 128, 161, 200, 241, 288, 337, 392, 449, 512, 577, 648, 721, 800, 881, 968, 1057, 1152, 1249, 1352, 1457, 1568, 1681, 1800, 1921, 2048, 2177, 2312, 2449, 2592, 2737, 2888, 3041, 3200, 3361, 3528, 3697, 3872, 4049, 4232

Offset: 0

Amarnath Murthy, Nov 03 2002

This sequence arises from reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the triangular numbers A000217. Cf. A139591, etc. - Omar E. Pol, May 03 2008
The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=2, thus a(k) = |(P(2,0)-(-1)^k*P(2,2*k))/2|. - Peter Luschny, Jul 12 2009
Axis perpendicular to A046092 in the square spiral whose vertices are the triangular numbers A000217. See the comment above. - Omar E. Pol, Sep 14 2011
Column 8 of A195040. - Omar E. Pol, Sep 28 2011
Concentric octagonal numbers. A139098 and A069129 interleaved. - Omar E. Pol, Sep 17 2011
Subsequence of A194274. - Bruno Berselli, Sep 22 2011
Partial sums of A047522. - Reinhard Zumkeller, Jan 07 2012
Alternating sum of the first n odd squares in decreasing order, n >= 1. Also number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton. The rules are: on the infinite square grid, start with all cells OFF, so a(0) = 0. Turn a single cell to the ON state, so a(1) = 1. At each subsequent step, the neighbor cells of each cell of the old generation are turned ON, and the cells of the old generation are turned OFF. Here "neighbor" refers to the eight adjacent cells of each ON cell. See example. - Omar E. Pol, Feb 16 2014

			From _Omar E. Pol_, Feb 16 2014: (Start)
Illustration of initial terms as a cellular automaton:
.
.                                   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       8           17              32
.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, An origin of A077221 (illustration) (see Pol's comment).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to cellular automata.

Cf. A032528, A069129, A077222, A131925, A139098, A194274, A195040, A195041, A195042, A195142.

a077221 n = a077221_list !! n
a077221_list = scanl (+) 0 a047522_list
-- Reinhard Zumkeller, Jan 07 2012

[2*n^2 - (n mod 2): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011

a := n -> 2*n^2 - (n mod 2); # Peter Luschny, Jul 12 2009

a=1;lst={a};Do[b=n^2-a;AppendTo[lst,b];a=b,{n,3,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, May 18 2009 *)

a(2n) = 8*n^2, a(2n+1) = 8*n(n+1) + 1.
From Ralf Stephan, Mar 31 2003: (Start)
a(n) = 2*n^2 + 4*n + 1 [+1 if n is odd] with a(0)=1.
G.f.: x*(x^2+6*x+1)/(1-x)^3/(1+x). (End)
Row sums of triangle A131925; binomial transform of (1, 7, 2, 4, -8, 16, -32, ...). - Gary W. Adamson, Jul 29 2007
a(n) = a(-n); a(n+1) = A195605(n) - (-1)^n. - Bruno Berselli, Sep 22 2011
a(n) = 2*n^2 + ((-1)^n-1)/2. - Omar E. Pol, Sep 28 2011
Sum_{n>=1} 1/a(n) = Pi^2/48 + tan(Pi/(2*sqrt(2)))*Pi /(4*sqrt(2)). - Amiram Eldar, Jan 16 2023

Extended by Ralf Stephan, Mar 31 2003

A032527 Concentric pentagonal numbers: floor( 5*n^2 / 4 ).

Original entry on oeis.org

0, 1, 5, 11, 20, 31, 45, 61, 80, 101, 125, 151, 180, 211, 245, 281, 320, 361, 405, 451, 500, 551, 605, 661, 720, 781, 845, 911, 980, 1051, 1125, 1201, 1280, 1361, 1445, 1531, 1620, 1711, 1805, 1901, 2000, 2101, 2205, 2311, 2420, 2531, 2645, 2761, 2880, 3001

Offset: 0

N. J. A. Sloane

Also A033429 and A062786 interleaved. - Omar E. Pol, Sep 28 2011
Partial sums of A047209. - Reinhard Zumkeller, Jan 07 2012
From Wolfdieter Lang, Aug 06 2013: (Start)
a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*5)/4, the norm for integers a + b*omega(5), a, b rational integers, in the quadratic number field Q(sqrt(5)), where omega(5) = (1 + sqrt(5))/2 (golden section).
a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = n^2 + n*floor(n/2) - floor(n/2)^2 = floor(5*n^2/4) (the last eq. checks for even and odd n). (End)

			From _Omar E. Pol_, Sep 28 2011 (Start):
Illustration of initial terms (In a precise representation the pentagons should appear strictly concentric):
.
.                                             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    5        11          20                31
.
(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).

Cf. A000290, A032528, A077043, A195041. Column 5 of A195040. [Omar E. Pol, Sep 28 2011]

a032527 n = a032527_list !! n
a032527_list = scanl (+) 0 a047209_list
-- Reinhard Zumkeller, Jan 07 2012

[5*n^2/4+((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

A032527:=n->5*n^2/4+((-1)^n-1)/8: seq(A032527(n), n=0..100); # Wesley Ivan Hurt, Mar 12 2015

Table[Round[5n^2/4], {n, 0, 39}] (* Alonso del Arte, Sep 28 2011 *)

a(n)=5*n^2>>2 \\ Charles R Greathouse IV, Sep 28 2011

def A032527(n): return 5*n**2>>2  # Chai Wah Wu, Jul 30 2022

a(n) = 5*n^2/4+((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
G.f.: x*(1+3*x+x^2)/(1-2*x+2*x^3-x^4). - Colin Barker, Jan 06 2012
a(n) = a(-n); a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>0, a(-1) = 1, a(0) = 0, a(1) = 1, a(2) = 5, n >= 3. (See the Bruno Berselli recurrence and a general comment for primes 1 (mod 4) under A227541). - Wolfdieter Lang, Aug 08 2013
a(n) = Sum_{j=1..n} Sum{i=1..n} ceiling((i+j-n+1)/2). - Wesley Ivan Hurt, Mar 12 2015
Sum_{n>=1} 1/a(n) = Pi^2/30 + tan(Pi/(2*sqrt(5)))*Pi/sqrt(5). - Amiram Eldar, Jan 16 2023

New name from Omar E. Pol, Sep 28 2011

A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 9, 13, 16, 21, 25, 31, 36, 43, 49, 57, 64, 73, 81, 91, 100, 111, 121, 133, 144, 157, 169, 183, 196, 211, 225, 241, 256, 273, 289, 307, 324, 343, 361, 381, 400, 421, 441, 463, 484, 507, 529, 553, 576, 601, 625, 651, 676, 703, 729, 757, 784, 813

Offset: 0

N. J. A. Sloane

As a Molien series this arises as (1+x^12)/((1-x^4)*(1-x^8)^2).
Starting (1, 3, 4, ...) = row sums of an infinite triangle with alternate columns of (1, 2, 3, ...) and (1, 0, 0, 0, ...). - Gary W. Adamson, May 14 2010
a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and one square has one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016
Also the genus of the complete bipartite graph K_{n+2,n+2}. - Eric W. Weisstein, Jan 19 2018

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the generating triangle =
1;
2, 1;
3, 0, 1;
4, 0, 2, 1;
5, 0, 3, 0, 1;
6, 0, 4, 0, 2, 1;
7, 0, 5, 0, 3, 0, 1;
8, 0, 6, 0, 4, 0, 2, 1;
...
Example: a(7) = 13 = (6 + 0 + 4 + 0 + 2 + 1). (End)
x + x^2 + 3*x^3 + 4*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 16*x^8 + 21*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
J. E. Strapasson, S. I. R. Costa, and M. M. S. Alves, On Genus of Circulant Graphs, arXiv:1004.0244 [math.GN], 2010-2016. - _Jonathan Vos Post_, Apr 05 2010
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Graph Genus
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

First differences give A028242. Cf. A035104, A035106.
A002061(n)=a(2*n-1). A035104(n)=a(n+7)-12. A035106(n)=a(n+3)-1.
Cf. A135840, A000290.
Column 1 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A054772, column 2.

a004652 = ceiling . (/ 4) . fromIntegral . (^ 2)
a004652_list = 0 : 1 : zipWith (+) a004652_list [1..]
-- Reinhard Zumkeller, Dec 18 2013

[Ceiling(n^2/4): n in [0..60] ]; // Vincenzo Librandi, Aug 19 2011

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=2)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m+3),m=0..57) ; # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[x (1 - x + x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 57}], x] (* Michael De Vlieger, Oct 03 2016 *)
Table[Ceiling[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)
Ceiling[Range[0, 20]^2/4] (* Eric W. Weisstein, Jan 19 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 3, 4}, {0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)

PARI
```
{a(n) = ceil(n^2 / 4)}
```

a(n) = ceiling(n^2/4).
a(-n) = a(n).
G.f.: x * (1 - x + x^2) / ((1 - x)^2 * (1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 1. a(2*n) = n^2, a(2*n-1) = n^2 - n + 1. - Michael Somos, Apr 21 2000
Interleaves square numbers with centered polygonal numbers: a(2*n)=A000290(n), a(2*n+1)=A002061(n+1). - Paul Barry, Mar 13 2003
For n > 1: a(n) is the digit reversal of n in base A008619(n), where a(n) is written in base 10. - Naohiro Nomoto, Mar 15 2004
a(n) = a(n-2) + n - 1. - Paul Barry, Jul 14 2004
Euler transform of length 6 sequence [ 1, 2, 1, 0, 0, -1]. - Michael Somos, Apr 03 2007
Starting (1, 3, 4, 7, 9, 13, ...), row sums of triangle A135840. - Gary W. Adamson, Dec 01 2007
a(n) = (3/8)*(-1)^(n+1) + 5/8 - (3/4)*(n+1) + (1/4)*(n+2)*(n+1). - Richard Choulet, Nov 27 2008
a(n) = n^2/4 - 3*((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
a(n) = -n + floor( (n+1)(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = A054772(n, 1) = A054772(n, n^2-1), n >= 1. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(x + 1)*exp(x) + 3*sinh(x))/4. - Ilya Gutkovskiy, Oct 03 2016
a(n) = binomial(floor((n+3)/2),2) + binomial(floor((n+(-1)^n)/2),2). - Yuchun Ji, Feb 03 2021

A195143 a(n) = n-th concentric 12-gonal number.

Original entry on oeis.org

0, 1, 12, 25, 48, 73, 108, 145, 192, 241, 300, 361, 432, 505, 588, 673, 768, 865, 972, 1081, 1200, 1321, 1452, 1585, 1728, 1873, 2028, 2185, 2352, 2521, 2700, 2881, 3072, 3265, 3468, 3673, 3888, 4105, 4332, 4561, 4800, 5041, 5292, 5545, 5808, 6073, 6348

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric dodecagonal numbers. [corrected by Ivan Panchenko, Nov 09 2013]
Sequence found by reading the line from 0, in the direction 0, 12,..., and the same line from 1, in the direction 1, 25,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Main axis, perpendicular to A028896 in the same spiral.
Partial sums of A091998. - Reinhard Zumkeller, Jan 07 2012
Column 12 of A195040. - Omar E. Pol, Sep 28 2011

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

A135453 and A069190 interleaved.
Cf. A001082, A032527, A032528, A077221, A195043, A195045, A195040, A195142, A195145, A195146, A195147, A195148, A195149.
Cf. A016921 (6n+1), A016969 (6n+5), A091998 (positive integers of the form 12*k +- 1), A092242 (positive integers of the form 12*k +- 5).

a195143 n = a195143_list !! n
a195143_list = scanl (+) 0 a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[(3*n^2+(-1)^n-1): n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Table[Sum[2*(-1)^(n - k + 1) + 6*k - 3, {k, n}], {n, 0, 47}] (* L. Edson Jeffery, Sep 14 2014 *)

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 3*n^2+(-1)^n-1.
a(n) = -a(n-1) + 6*n^2 - 6*n + 1. (End)
G.f.: -x*(1+10*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = Sum_{k=1..n} (2*(-1)^(n-k+1) + 3*(2*k-1)), n>0, a(0) = 0. - L. Edson Jeffery, Sep 14 2014
Sum_{n>=1} 1/a(n) = Pi^2/72 + tan(Pi/sqrt(6))*Pi/(4*sqrt(6)). - Amiram Eldar, Jan 16 2023

A195145 Concentric 14-gonal numbers.

Original entry on oeis.org

0, 1, 14, 29, 56, 85, 126, 169, 224, 281, 350, 421, 504, 589, 686, 785, 896, 1009, 1134, 1261, 1400, 1541, 1694, 1849, 2016, 2185, 2366, 2549, 2744, 2941, 3150, 3361, 3584, 3809, 4046, 4285, 4536, 4789, 5054, 5321, 5600, 5881, 6174, 6469, 6776, 7085, 7406

Offset: 0

Omar E. Pol, Sep 17 2011

Also concentric tetradecagonal numbers or concentric tetrakaidecagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 14, ..., and the same line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Main axis, perpendicular to A024966 in the same spiral.

Partial sums of A113801. - Reinhard Zumkeller, Jan 07 2012

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

A144555 and A195314 interleaved.
Cf. A024966, A032527, A032528, A077221, A113801, A195045, A195046, A195142, A195143, A195146, A195147, A195148, A195149.
Column 14 of A195040. - Omar E. Pol, Sep 28 2011

a195145 n = a195145_list !! n
a195145_list = scanl (+) 0 a113801_list
-- Reinhard Zumkeller, Jan 07 2012

[(14*n^2+5*(-1)^n-5)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 14, 29}, 50] (* Amiram Eldar, Jan 16 2023 *)

G.f.: -x*(1+12*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = (14*n^2 + 5*(-1)^n - 5)/4;
a(n) = a(-n) = -a(n-1) + 7*n^2 - 7*n + 1. (End)
Sum_{n>=1} 1/a(n) = Pi^2/84 + tan(sqrt(5/7)*Pi/2)*Pi/(2*sqrt(35)). - Amiram Eldar, Jan 16 2023
E.g.f.: (7*x*(x + 1)*cosh(x) + (7*x^2 + 7*x - 5)*sinh(x))/2. - Stefano Spezia, Nov 30 2024

A195149 Concentric 22-gonal numbers.

Original entry on oeis.org

0, 1, 22, 45, 88, 133, 198, 265, 352, 441, 550, 661, 792, 925, 1078, 1233, 1408, 1585, 1782, 1981, 2200, 2421, 2662, 2905, 3168, 3433, 3718, 4005, 4312, 4621, 4950, 5281, 5632, 5985, 6358, 6733, 7128, 7525, 7942, 8361, 8800, 9241, 9702, 10165, 10648, 11133

Offset: 0

Omar E. Pol, Sep 17 2011

Sequence found by reading the line from 0, in the direction 0, 22,..., and the same line from 1, in the direction 1, 45,..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Main axis, perpendicular to A152740 in the same spiral.

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

A195323 and A195318 interleaved.
Cf. A032528, A077221, A195142, A195143, A195145, A195146, A195147, A195148.
Cf. A032527, A195049, A195058. Column 22 of A195040. - Omar E. Pol, Sep 29 2011

[(22*n^2+9*(-1)^n-9)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

A195149:=n->(22*n^2+9*(-1)^n-9)/4: seq(A195149(n), n=0..50); # Wesley Ivan Hurt, Jul 07 2014

Table[(22*n^2 + 9*(-1)^n - 9)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 07 2014 *)

a(n)=(22*n^2+9*(-1)^n-9)/4 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: -x*(1+20*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = (22*n^2+9*(-1)^n-9)/4; a(n) = -a(n-1)+11*n^2-11*n+1. - Vincenzo Librandi, Sep 27 2011
Sum_{n>=1} 1/a(n) = Pi^2/132 + tan(3*Pi/(2*sqrt(11)))*Pi/(6*sqrt(11)). - Amiram Eldar, Jan 17 2023
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Wesley Ivan Hurt, Jun 19 2025

A195142 Concentric 10-gonal numbers.

Original entry on oeis.org

0, 1, 10, 21, 40, 61, 90, 121, 160, 201, 250, 301, 360, 421, 490, 561, 640, 721, 810, 901, 1000, 1101, 1210, 1321, 1440, 1561, 1690, 1821, 1960, 2101, 2250, 2401, 2560, 2721, 2890, 3061, 3240, 3421, 3610, 3801, 4000, 4201, 4410, 4621, 4840, 5061, 5290

Offset: 0

Omar E. Pol, Sep 17 2011

Also concentric decagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 10, ..., and the same line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Main axis, perpendicular to A028895 in the same spiral.

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

A033583 and A069133 interleaved.
Cf. A028895, A032527, A032528, A077221, A085787, A195042, A195143, A195145, A195146, A195147, A195148, A195149.
Cf. A090771 (first differences).
Column 10 of A195040. - Omar E. Pol, Sep 28 2011

a195142 n = a195142_list !! n
a195142_list = scanl (+) 0 a090771_list
-- Reinhard Zumkeller, Jan 07 2012

[(10*n^2+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+10(n-1)},a[n],{n,50}] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,10,21},50] (* Harvey P. Dale, Sep 29 2011 *)

G.f.: -x*(1+8*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = -a(n-1) + 5*n^2 - 5*n + 1, a(0)=0. - Vincenzo Librandi, Sep 27 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = a(-n) = (10*n^2 + 3*(-1)^n - 3)/4.
a(n) = a(n-2) + 10*(n-1). (End)
a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=10, a(3)=21. - Harvey P. Dale, Sep 29 2011
Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(sqrt(3/5)*Pi/2)*Pi/(2*sqrt(15)). - Amiram Eldar, Jan 16 2023

cp's OEIS Frontend

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

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

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A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

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A077221 a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.

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A032527 Concentric pentagonal numbers: floor( 5*n^2 / 4 ).

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A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).