A195042 -id:A195042 - cp's OEIS frontend

A056020 Numbers that are congruent to +-1 mod 9.

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

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

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

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

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 *)

A016766 a(n) = (3*n)^2.

Original entry on oeis.org

0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876, 16641, 17424

Offset: 0

N. J. A. Sloane

Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002
Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014
Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022

Ivan Panchenko, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Numbers of the form 9*n^2 + k*n, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012

Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.

[(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014

A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014

(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)

A016766(n):=(3*n)^2$
makelist(A016766(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 9*n^2 = 9*A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 3*A033428(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
From Wesley Ivan Hurt, Sep 24 2014: (Start)
G.f.: 9*x*(1 + x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = A000290(A008585(n)). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
a(n) = A051624(n) + 8*A000217(n).  In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 9*x*(1 + x)*exp(x).
a(n) = n*A008591(n) = A195042(2*n). (End)

More terms from Zerinvary Lajos, May 30 2006

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

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A195041 Concentric heptagonal numbers.

Original entry on oeis.org

0, 1, 7, 15, 28, 43, 63, 85, 112, 141, 175, 211, 252, 295, 343, 393, 448, 505, 567, 631, 700, 771, 847, 925, 1008, 1093, 1183, 1275, 1372, 1471, 1575, 1681, 1792, 1905, 2023, 2143, 2268, 2395, 2527, 2661, 2800, 2941, 3087, 3235, 3388, 3543

Offset: 0

Omar E. Pol, Sep 27 2011

A033582 and A069127 interleaved.

Partial sums of A047336. - 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).

Column 7 of A195040.

Cf. A032527, A032528, A033582, A047336, A069099, A069127, A077221, A195042.

a195041 n = a195041_list !! n
a195041_list = scanl (+) 0 a047336_list
-- Reinhard Zumkeller, Jan 07 2012

[7*n^2/4+3*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

CoefficientList[Series[-((x (1+5 x+x^2))/((-1+x)^3 (1+x))),{x,0,80}],x] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,7,15},80] (* Harvey P. Dale, Jan 18 2021 *)

a(n)=7*n^2\4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*n^2/4 + 3*((-1)^n - 1)/8.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+5*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A069099(n+1). (End)
a(n) = n^2 + floor(3*n^2/4). - Bruno Berselli, Aug 08 2013
Sum_{n>=1} 1/a(n) = Pi^2/42 + tan(sqrt(3/7)*Pi/2)*Pi/sqrt(21). - Amiram Eldar, Jan 16 2023

A195043 Concentric 11-gonal numbers.

Original entry on oeis.org

0, 1, 11, 23, 44, 67, 99, 133, 176, 221, 275, 331, 396, 463, 539, 617, 704, 793, 891, 991, 1100, 1211, 1331, 1453, 1584, 1717, 1859, 2003, 2156, 2311, 2475, 2641, 2816, 2993, 3179, 3367, 3564, 3763, 3971, 4181, 4400, 4621, 4851, 5083, 5324, 5567

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric hendecagonal numbers. A033584 and A069173 interleaved.

Partial sums of A175885. - 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).

Column 11 of A195040.

Cf. A032527, A032528, A033584, A069173, A175885, A195042, A195142, A195143, A195045.

a195043 n = a195043_list !! n
a195043_list = scanl (+) 0 a175885_list
-- Reinhard Zumkeller, Jan 07 2012

[11*n^2/4+7*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,11,23},50] (* Harvey P. Dale, May 20 2019 *)

Vec(-x*(x^2+9*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = 11*n^2/4 + 7*((-1)^n - 1)/8.
a(n) = -a(n-1) + A069125(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 15 2013: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(x^2+9*x+1) / ((x-1)^3*(x+1)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/66 + tan(sqrt(7/11)*Pi/2)*Pi/sqrt(77). - Amiram Eldar, Jan 16 2023

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

cp's OEIS Frontend

A056020 Numbers that are congruent to +-1 mod 9.

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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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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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A016766 a(n) = (3*n)^2.

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A195142 Concentric 10-gonal numbers.

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A069131 Centered 18-gonal numbers.

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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.