A081585 -id:A081585 - cp's OEIS frontend

A275543 A081585 and A069129 interleaved.

1, 1, 9, 17, 33, 49, 73, 97, 129, 161, 201, 241, 289, 337, 393, 449, 513, 577, 649, 721, 801, 881, 969, 1057, 1153, 1249, 1353, 1457, 1569, 1681, 1801, 1921, 2049, 2177, 2313, 2449, 2593, 2737, 2889, 3041, 3201, 3361, 3529, 3697, 3873, 4049, 4233, 4417, 4609

Offset: 0

Daniel Poveda Parrilla, Aug 01 2016

a(A000129(n)) is a square.
(n^2)*a(n) = A275496(n) which is a triangular number.
(A000129(n)^2)*a(A000129(n)) = A275496(A000129(n)) = A001110(n) which is a square triangular number.
a(2n+1)/a(2n) is convergent to 1.

			a(1) = A275496(1) = 1.
a(5) = A275496(5)/25 = 1225/25 = 49.
a(7) = A275496(7)/49 = 4753/49 = 97.
a(12) = A275496(12)/144 = 41616/144 = 289.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..500000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081585(n) = a(2n), A069129(n) = a(2n + 1).

Cf. A000129, A000217, A000290, A001110, A077221, A093954, A275496.

CoefficientList[Series[(1 - x + 7 x^2 + x^3)/((1 - x)^3 (1 + x)), {x, 0, 48}], x] (* or as defined *)
Riffle[LinearRecurrence[{3, -3, 1}, {1, 9, 33}, #], FoldList[#1 + #2 &, 1, 16 Range@ #]] &@ 25 (* Michael De Vlieger, Aug 01 2016, after Vincenzo Librandi at A081585 and Robert G. Wilson v at A069129 *)

a(n)=(-1)^n + 2*n^2 \\ Charles R Greathouse IV, Aug 03 2016

Vec((1-x+7*x^2+x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Aug 21 2016

a(0) = 1; a(n) = A275496(n)/(n^2) for n > 0.
From Colin Barker, Aug 01 2016: (Start)
a(n) = (2*n^2 + (-1)^n).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 -x +7*x^2 +x^3) / ((1 - x)^3*(1 + x)).
(End)
From Daniel Poveda Parrilla, Aug 18 2016: (Start)
a(2n) = A077221(2n) + 1.
a(2n + 1) = A077221(2n + 1). (End)
Sum_{n>=0} 1/a(n) = (1 + (tan(c) + coth(c))*c)/2, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022

A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

Original entry on oeis.org

1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898

Offset: 0

N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Milan Janjic, Two Enumerative Functions
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513.

[1] cat [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

Join[{1},Table[16n^2+2,{n,50}]] (* Harvey P. Dale, Oct 15 2012 *)

A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.
G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
E.g.f.: (x*(x+1)*16+2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - Harvey P. Dale, Oct 15 2012
G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - Peter Bala, Apr 09 2017
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth( Pi*sqrt(2)/4) = 1.095237238050... - R. J. Mathar, May 07 2024
a(n) = 2*A081585(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069129(n)+A069129(n+1). - R. J. Mathar, May 07 2024

A081578 Pascal-(1,3,1) array.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 33, 13, 1, 1, 17, 73, 73, 17, 1, 1, 21, 129, 245, 129, 21, 1, 1, 25, 201, 593, 593, 201, 25, 1, 1, 29, 289, 1181, 1921, 1181, 289, 29, 1, 1, 33, 393, 2073, 4881, 4881, 2073, 393, 33, 1, 1, 37, 513, 3333, 10497, 15525, 10497, 3333, 513, 37, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016813, A081585, A081586. Coefficients of the row polynomials in the Newton basis are given by A013611.

As a number triangle, this is the Riordan array (1/(1-x), x*(1+3*x)/(1-x)). It has row sums A015518(n+1) and diagonal sums A103143. - Paul Barry, Jan 24 2005

			Square array begins as:
  1,  1,   1,   1,    1, ... A000012;
  1,  5,   9,  13,   17, ... A016813;
  1,  9,  33,  73,  129, ... A081585;
  1, 13,  73, 245,  593, ... A081586;
  1, 17, 129, 593, 1921, ...
As a triangle this begins:
  1;
  1,  1;
  1,  5,   1;
  1,  9,   9,    1;
  1, 13,  33,   13,     1;
  1, 17,  73,   73,    17,     1;
  1, 21, 129,  245,   129,    21,     1;
  1, 25, 201,  593,   593,   201,    25,    1;
  1, 29, 289, 1181,  1921,  1181,   289,   29,   1;
  1, 33, 393, 2073,  4881,  4881,  2073,  393,  33,  1;
  1, 37, 513, 3333, 10497, 15525, 10497, 3333, 513, 37, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
P. Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

a081578 n k = a081578_tabl !! n !! k
a081578_row n = a081578_tabl !! n
a081578_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 3) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A081578:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081578(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 4], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 4).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 3*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+3*x)^k/(1-x)^(k+1).
T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*3^(j-k). - Paul Barry, Oct 23 2006
E.g.f. for the n-th subdiagonal of the triangle, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(4*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 8*x + 16*x^2/2) = 1 + 9*x + 33*x^2/2! + 73*x^3/3! + 129*x^4/4! + 201*x^5/5! + .... - Peter Bala, Mar 05 2017
From G. C. Greubel, May 26 2021: (Start)
T(n, k, m) = Hypergeometric2F1([-k, k-n], [1], m+1), for m = 3.
T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 3.
Sum_{k=0..n} T(n, k, 3) = A015518(n+1). (End)

A158575 a(n) = 32*n^2 + 1.

Original entry on oeis.org

1, 33, 129, 289, 513, 801, 1153, 1569, 2049, 2593, 3201, 3873, 4609, 5409, 6273, 7201, 8193, 9249, 10369, 11553, 12801, 14113, 15489, 16929, 18433, 20001, 21633, 23329, 25089, 26913, 28801, 30753, 32769, 34849, 36993, 39201, 41473, 43809, 46209, 48673, 51201, 53793

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (32*n^2 + 1)^2 - (256*n^2 + 16)*(2*n)^2 = 1 can be written as a(n)^2-A158574(n)*A005843(n)^2 = 1. - Comment rewritten by R. J. Mathar, Oct 16 2009

Sequence found by reading the line segment from 1 to 33 together with the line from 33, in the direction 33, 129, ..., in the square spiral whose vertices are the generalized 18-gonal numbers A274979. - Omar E. Pol, Apr 21 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005843, A010021, A081585, A158563, A158574, A244082.

Cf. A274979 (generalized 18-gonal numbers).

I:=[1, 33, 129]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

LinearRecurrence[{3, -3, 1}, {1, 33, 129}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
32*Range[0,40]^2+1 (* Harvey P. Dale, Jul 20 2021 *)

for(n=0, 50, print1(32*n^2+1", ")); \\ Vincenzo Librandi, Feb 15 2012

G.f.: (1+30*x+33*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
For n > 0 a(n) = sqrt(8*(A000217(4*n-1)^2 + A000217(4*n)^2) + 1). - J. M. Bergot, Sep 03 2015
a(n) = A244082(n) + 1. - Omar E. Pol, Apr 21 2021
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + coth(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 32*x + 32*x^2).
a(n) = A081585(2*n). (End)

a(0) added by R. J. Mathar, Oct 16 2009

A081586 Fourth row of Pascal-(1,3,1) array A081578.

Original entry on oeis.org

1, 13, 73, 245, 593, 1181, 2073, 3333, 5025, 7213, 9961, 13333, 17393, 22205, 27833, 34341, 41793, 50253, 59785, 70453, 82321, 95453, 109913, 125765, 143073, 161901, 182313, 204373, 228145, 253693, 281081, 310373, 341633, 374925, 410313, 447861

Offset: 0

Paul Barry, Mar 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081578, A081585.

[(3+28*n-24*n^2+32*n^3)/3: n in [0..40]]; // Vincenzo Librandi, Nov 16 2011

Table[(3+28n-24n^2+32n^3)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,13,73,245},40] (* Harvey P. Dale, Nov 06 2011 *)

[(3+28*n-24*n^2+32*n^3)/3 for n in (0..40)] # G. C. Greubel, May 26 2021

From Harvey P. Dale, Nov 06 2011: (Start)
a(n) = (3 + 28*n - 24*n^2 + 32*n^3)/3.
G.f.: (1+3*x)^3/(1-x)^4.
a(0)=1, a(1)=13, a(2)=73, a(3)=245, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
E.g.f.: (1/3)*(3 + 36*x + 72*x^2 + 32*x^3)*exp(x). - G. C. Greubel, May 26 2021

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A118465 a(n) = 8*n^3 + n.

Original entry on oeis.org

0, 9, 66, 219, 516, 1005, 1734, 2751, 4104, 5841, 8010, 10659, 13836, 17589, 21966, 27015, 32784, 39321, 46674, 54891, 64020, 74109, 85206, 97359, 110616, 125025, 140634, 157491, 175644, 195141, 216030, 238359, 262176, 287529, 314466, 343035, 373284, 405261

Offset: 0

Mohamed Bouhamida, May 16 2006, Oct 02 2007

(8*n^3 + n, 8*n^3 - n) solves the Diophantine equation 2*(X-Y)^3-(X+Y)=0.
(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 and where m is a positive integer. Also ((m*n^k+n)/2, (m*n^k-n)/2) solves the Diophantine equation: m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 where m is an odd number.
24*a(n) = (4*n+1)^3 + (4*n)^3 + (4*n-1)^3. - Bruno Berselli, May 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006003, A081585, A143166.

[8*n^3 + n: n in [0..30]]; // Wesley Ivan Hurt, May 13 2014

A118465:=n->8*n^3 + n; seq(A118465(n), n=0..30); # Wesley Ivan Hurt, May 13 2014

Table[8 n^3 + n, {n, 0, 35}]
CoefficientList[Series[3 x (x + 3) (3 x + 1)/(-1 + x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,9,66,219},40] (* Harvey P. Dale, Feb 01 2023 *)

G.f.: 3*x*(x+3)*(3*x+1)/(-1+x)^4. - R. J. Mathar, Nov 14 2007
a(n) = n*A081585(n). - Vincenzo Librandi, May 13 2014
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: exp(x)*x*(9 + 24*x + 8*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*A143166(n). (End)

Edited by Stefan Steinerberger, Jul 24 2007

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

A157912 a(n) = 64*n^2 + 16.

Original entry on oeis.org

80, 272, 592, 1040, 1616, 2320, 3152, 4112, 5200, 6416, 7760, 9232, 10832, 12560, 14416, 16400, 18512, 20752, 23120, 25616, 28240, 30992, 33872, 36880, 40016, 43280, 46672, 50192, 53840, 57616, 61520, 65552, 69712, 74000, 78416, 82960, 87632, 92432, 97360, 102416

Offset: 1

Vincenzo Librandi, Mar 09 2009

The identity (8*n^2 + 1)^2 - (64*n^2 + 16)*n^2 = 1 can be written as A081585(n)^2 - a(n)*n^2 = 1. - Vincenzo Librandi, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A053755, A081585.

I:=[80, 272, 592]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 09 2012

A157912:=n->64*n^2 + 16: seq(A157912(n), n=1..50); # Wesley Ivan Hurt, Oct 27 2014

64Range[50]^2+16  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{3, -3, 1}, {80, 272, 592}, 40] (* Vincenzo Librandi, Feb 09 2012 *)

for(n=1, 40, print1(64*n^2 + 16", ")); \\ Vincenzo Librandi, Feb 09 2012

From Vincenzo Librandi, Feb 09 2012: (Start)
G.f.: x*(80+32*x+16*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/32.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/32. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 16*(exp(x)*(4*x^2 + 4*x + 1) - 1).
a(n) = 16*A053755(n). (End)

A158444 a(n) = 16*n^2 + 4.

Original entry on oeis.org

20, 68, 148, 260, 404, 580, 788, 1028, 1300, 1604, 1940, 2308, 2708, 3140, 3604, 4100, 4628, 5188, 5780, 6404, 7060, 7748, 8468, 9220, 10004, 10820, 11668, 12548, 13460, 14404, 15380, 16388, 17428, 18500, 19604, 20740, 21908, 23108, 24340, 25604, 26900, 28228

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (8*n^2 + 1)^2 - (16*n^2 + 4)*(2*n)^2 = 1 can be written as A081585(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by Bruno Berselli, Sep 06 2011]

Sequence found by reading the line from 20, in the direction 20, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A053755, A074377, A081585.

Magma
```
[16*n^2+4: n in [1..50]];
```

a[n_] := 16*n^2 + 4; Array[a, 50] (* Amiram Eldar, Mar 05 2023 *)

a(n)=16*n^2+4 \\ Charles R Greathouse IV, Jun 17 2017

From Bruno Berselli, Sep 06 2011: (Start)
G.f.: 4*x*(5 + 2*x + x^2)/(1-x)^3.
a(n) = 4*A053755(n). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/8. (End)
E.g.f.: 4*(exp(x)*(4*x^2 + 4*x + 1) - 1). - Elmo R. Oliveira, Jan 27 2025

cp's OEIS Frontend

A275543 A081585 and A069129 interleaved.

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A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

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A081578 Pascal-(1,3,1) array.

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A158575 a(n) = 32*n^2 + 1.

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A081586 Fourth row of Pascal-(1,3,1) array A081578.

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A185438 a(n) = 8*n^2 - 2*n + 1.

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A118465 a(n) = 8*n^3 + n.

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A185438 a(n) = 8n^2 - 2n + 1.