A100314 -id:A100314 - cp's OEIS frontend

A131831 Duplicate of A100314.

1, 4, 8, 14, 24, 42, 76, 142, 272, 530, 1044, 2070, 4120, 8218, 16412, 32798, 65568, 131106, 262180, 524326, 1048616, 2097194, 4194348, 8388654, 16777264, 33554482, 67108916

Offset: 0

dead

A052548 a(n) = 2^n + 2.

Original entry on oeis.org

3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The most "compact" sequence that satisfies Bertrand's Postulate. Begin with a(1) = 3 = n, then 2n - 2 = 4 = n_1, 2n_1 - 2 = 6 = n_2, 2n_2 - 2 = 10, etc. = a(n), hence there is guaranteed to be at least one prime between successive members of the sequence. - Andrew S. Plewe, Dec 11 2007

Number of 2-sided prudent polygons of area n, for n>0, see Beaton, p. 5. - Jonathan Vos Post, Nov 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..240
Nicholas R. Beaton, Philippe Flajolet, and Anthony J. Guttmann, The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics, arXiv:1011.6195 [math.CO], Nov 29, 2010.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 485
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=2. [Annotated and scanned copy]
Eric Weisstein's World of Mathematics, Bertrand's Postulate
Index entries for sequences generated by sieves
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from initial term, same as A056469.
Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004, A058481, A100585, A100586, A058896, A000918, A173786.
Cf. also A000079, A000051, A100314.

a052548 = (+ 2) . a000079
a052548_list = iterate ((subtract 2) . (* 2)) 3
-- Reinhard Zumkeller, Sep 05 2015

[2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011

spec := [S,{S=Union(Sequence(Union(Z,Z)),Sequence(Z),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

2^Range[0,40]+2 (* Harvey P. Dale, Jun 26 2012 *)

a(n)=1<Charles R Greathouse IV, Nov 20 2011

G.f.: (3-5*x)/((1-2*x)*(1-x)) = (3-5*x)/(1 - 3*x + 2*x^2) = 2/(1-x) + 1/(1-2*x).
a(0)=3, a(1)=4, a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A058896(n)/A000918(n), for n>0. - Reinhard Zumkeller, Feb 14 2009
a(n) = A173786(n,1), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(n)*A000918(n) = A028399(2*n), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(0)=3, a(n) = 2*a(n-1) - 2. - Vincenzo Librandi, Aug 06 2010
E.g.f.: (2 + exp(x))*exp(x). - Ilya Gutkovskiy, Aug 16 2016

More terms from James Sellers, Jun 06 2000

A131830 Triangle read by rows: T(n,0) = T(n,n) = n + 1 for n >= 0, and T(n,k) = binomial(n,k) for 1 <= k <= n - 1, n >= 2.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 6, 4, 5, 6, 5, 10, 10, 5, 6, 7, 6, 15, 20, 15, 6, 7, 8, 7, 21, 35, 35, 21, 7, 8, 9, 8, 28, 56, 70, 56, 28, 8, 9, 10, 9, 36, 84, 126, 126, 84, 36, 9, 10, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12

Offset: 0

Gary W. Adamson, Jul 20 2007

Given Pascal's triangle, replace the two (1, 1, 1, ...) borders with (1, 2, 3, ...).

			First few rows of the triangle are:
  1;
  2, 2;
  3, 2,  3;
  4, 3,  3,  4;
  5, 4,  6,  4,  5;
  6, 5, 10, 10,  5, 6;
  7, 6, 15, 20, 15, 6, 7;
  ...

Georg Fischer, Table of n, a(n) for n = 0..10152 [Rows 0..141] (older version by B. D. Swan)

Row sums: A100314.

Cf. A007318, A131821, A131821.

Flatten[Table[If[Or[k==n,k==0], n+1, Binomial[n, k]], {n, 0, 11}, {k, 0, n}]] (* Georg Fischer, Feb 18 2020 *)

T(n, k) := if k = 0 or k = n then n + 1 else binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 19 2018 */

T(n,k) = A131821(n,k) + A007318(n,k) - 1.
From Franck Maminirina Ramaharo, Dec 19 2018: (Start)
G.f.: (1 - (1 + x)*y - 2*x*y^2 + (3*x + 3*x^2)*y^3 - (x + x^2 + x^3)*y^4)/((1 - y)^2*(1 - x*y)^2*(1 - y - x*y)).
E.g.f.: y*exp(y) + (x*y + exp(y))*exp(x*y). (End)

Edited by Franck Maminirina Ramaharo, Dec 19 2018

B-file corrected from a(1678) onwards by Georg Fischer, Feb 18 2020

A173740 Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 8, 6, 1, 1, 7, 12, 12, 7, 1, 1, 8, 17, 22, 17, 8, 1, 1, 9, 23, 37, 37, 23, 9, 1, 1, 10, 30, 58, 72, 58, 30, 10, 1, 1, 11, 38, 86, 128, 128, 86, 38, 11, 1, 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1, 1, 13, 57, 167, 332, 464, 464, 332, 167, 57, 13, 1

Offset: 0

Roger L. Bagula, Feb 23 2010

For n >= 1, row n sums to A131520(n).

			Triangle begins:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  5,   1;
  1,  6,  8,   6,   1;
  1,  7, 12,  12,   7,   1;
  1,  8, 17,  22,  17,   8,   1;
  1,  9, 23,  37,  37,  23,   9,   1;
  1, 10, 30,  58,  72,  58,  30,  10,  1;
  1, 11, 38,  86, 128, 128,  86,  38, 11,  1;
  1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A100314, A103451, A131520, A156050.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), this sequence (q=2), A173741 (q=4), A173742 (q=6).

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 2 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, m_] = Binomial[n, m] + 2*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 2$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 08 2018 */

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 2
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

From Franck Maminirina Ramaharo, Dec 08 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).
n-th row polynomial is 1 - (-1)^(2^n) + (1 + x)^n + 2*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 3*x*y^2 - 2*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 2*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 08 2018

A100315 Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 8, 14, 22, 34, 54, 90, 158, 290, 550, 1066, 2094, 4146, 8246, 16442, 32830, 65602, 131142, 262218, 524366, 1048658, 2097238, 4194394, 8388702, 16777314, 33554534, 67108970, 134217838, 268435570, 536871030, 1073741946, 2147483774, 4294967426, 8589934726

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A100314 (m=2), this sequence (m=3), A100316 (m=4).

[2^n+4*n+2*(1-0^n): n in [0..40]]; // G. C. Greubel, Feb 01 2023

Table[If[n==0, 1, 2^n+4*n+2], {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

[2^n+4*n+2*(1-0^n) for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 4*n + 2 for n>0, a(0)=1.
From Chai Wah Wu, Aug 26 2016: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3.
G.f.: 1 + 2*x*(4 - 9*x + 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*(1+2*x)*exp(x) - 2. - G. C. Greubel, Feb 01 2023

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 7, 7, 1, 1, 8, 10, 8, 1, 1, 9, 14, 14, 9, 1, 1, 10, 19, 24, 19, 10, 1, 1, 11, 25, 39, 39, 25, 11, 1, 1, 12, 32, 60, 74, 60, 32, 12, 1, 1, 13, 40, 88, 130, 130, 88, 40, 13, 1, 1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1, 1, 15, 59, 169, 334, 466, 466, 334, 169, 59, 15, 1

Offset: 0

Roger L. Bagula, Feb 23 2010

For n >= 1, row n sums to 2*A100314(n).

			Triangle begins:
  1;
  1,  1;
  1,  6,  1;
  1,  7,  7,   1;
  1,  8, 10,   8,   1;
  1,  9, 14,  14,   9,   1;
  1, 10, 19,  24,  19,  10,   1;
  1, 11, 25,  39,  39,  25,  11,   1;
  1, 12, 32,  60,  74,  60,  32,  12,  1;
  1, 13, 40,  88, 130, 130,  88,  40, 13,  1;
  1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A100314, A103451, A156050.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), A173740 (q=2), this sequence (q=4), A173742 (q=6).

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 4 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, m_] = Binomial[n, m] + 4*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 4$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 4
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

From Franck Maminirina Ramaharo, Dec 09 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 5*A007318(n,k) - 4*A132044(n,k).
n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

A100316 Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 16, 24, 34, 48, 70, 108, 178, 312, 574, 1092, 2122, 4176, 8278, 16476, 32866, 65640, 131182, 262260, 524410, 1048704, 2097286, 4194444, 8388754, 16777368, 33554590, 67109028, 134217898, 268435632, 536871094, 1073742012, 2147483842, 4294967496, 8589934798

Offset: 0

Sergey Kitaev, Nov 13 2004

nonn

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A100314 (m=2), A100315 (m=3), this sequence (m=4).

[2^n+6*n+8*(1-0^n): n in [0..40]]; // G. C. Greubel, Feb 01 2023

Table[If[n==0, 1, 2^n+6*n+8], {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)

[2^n+6*n+8*(1-0^n) for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 6*n + 8 for n>0, a(0) = 1.
G.f.: (1+12*x-35*x^2+16*x^3)/((1-2*x)*(1-x)^2). - Alois P. Heinz, Dec 21 2018
E.g.f.: exp(2*x) + 2*(4+3*x)*exp(x) - 8. - G. C. Greubel, Feb 01 2023

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A321123 a(n) = 2^n + 2n^2 + 2n + 1.

Original entry on oeis.org

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

For n >= 2, a(n) is the number of evaluation points on the n-dimensional cube in Genz and Malik's degree 7 cubature rule.

Marius A. Burtea, Table of n, a(n) for n = 0..201
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics Vol. 6 (1980), 295-302.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001787, A001844, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A322593.

[2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 2*n + 1, {n, 0, 50}]

makelist(2^n + 2*n^2 + 2*n + 1, n, 0, 50);

a(n) = A000079(n) + A001844(n).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), n >= 4.
G.f.: (2 - 3*x - 3*x^3)/((1 - 2*x)*(1 - x)^3).
E.g.f.: exp(2*x) + (1 + 4*x + 2*x^2)*exp(x).

A322593 a(n) = 2^n + 2*n^2 + 1.

Original entry on oeis.org

2, 5, 13, 27, 49, 83, 137, 227, 385, 675, 1225, 2291, 4385, 8531, 16777, 33219, 66049, 131651, 262793, 525011, 1049377, 2098035, 4195273, 8389667, 16778369, 33555683, 67110217, 134219187, 268437025, 536872595, 1073743625, 2147485571, 4294969345, 8589936771

Offset: 0

Franck Maminirina Ramaharo, Dec 18 2018

For n = 3..7, a(n) is the number of evaluating points on the n-dimensional sphere (also n-space with weight function exp(-r^2) or exp(-r)) in a degree 7 cubature rule.

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Marius A. Burtea, Table of n, a(n) for n = 0..200
Ronald Cools, Encyclopaedia of Cubature Formulas
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A000051, A000079, A001580, A001787, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A321123.

[2^n + 2*n^2 + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Table[2^n + 2*n^2 + 1, {n, 0, 50}]
LinearRecurrence[{5,-9,7,-2},{2,5,13,27},50] (* Harvey P. Dale, Mar 23 2021 *)

Maxima
```
makelist(2^n + 2*n^2 + 1, n, 0, 50);
```

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = a(n-1) + A100315(n-1), n >= 2.
G.f.: (2 - 5*x + 6*x^2 - 7*x^3)/((1 - 2*x)*(1 - x)^3)
E.g.f.: exp(2*x) + (1 + 2*x + 2*x^2)*exp(x).

A340849 a(n) = A001045(n) + A052928(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 27, 49, 93, 179, 351, 693, 1377, 2743, 5475, 10937, 21861, 43707, 87399, 174781, 349545, 699071, 1398123, 2796225, 5592429, 11184835, 22369647, 44739269, 89478513, 178956999, 357913971, 715827913

Offset: 0

Paul Curtz, Jan 24 2021

nonn

a(2*n) is divisible by 3.
a(3*n+2) is divisible by 3.
a(n) is the minimum number of moves to solve a Towers of Hanoi puzzle with 4 pegs and n disks where a disk cannot move away from the destination peg (or symmetrically, a disk cannot return to the initial peg). - Woosuk Kwak, Jan 25 2024

Woosuk Kwak, 1+3 Towers of Hanoi, Puzzling Stack Exchange.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A001045, A052928, A100314, A109613, A128209.

LinearRecurrence[{3, -1, -3, 2}, {0, 1, 3, 5}, 32] (* Robert P. P. McKone, Jan 28 2021 *)

a(n+1) - 2*a(n) = -A109613(n-2), for a(0)=0, a(1)=1. a(n) + a(n+1) = A100314(n).
a(n+1) - a(n) = A128209(n) for n >= 0.
a(n+2) = 1 + 2*A086445(n). - Hugo Pfoertner, Jan 24 2021
From Woosuk Kwak, Jan 25 2024: (Start)
a(n) = n + floor(2^n/3).
a(n) = n + A000975(n-1) for n >= 1. (End)

cp's OEIS Frontend

A131831 Duplicate of A100314.

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A052548 a(n) = 2^n + 2.

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A131830 Triangle read by rows: T(n,0) = T(n,n) = n + 1 for n >= 0, and T(n,k) = binomial(n,k) for 1 <= k <= n - 1, n >= 2.

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A173740 Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

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A100315 Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

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A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

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A100316 Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

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A321123 a(n) = 2^n + 2n^2 + 2n + 1.