A054569 -id:A054569 - cp's OEIS frontend

A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 21, 13, 1, 1, 21, 43, 43, 21, 1, 1, 31, 73, 91, 73, 31, 1, 1, 43, 111, 157, 157, 111, 43, 1, 1, 57, 157, 241, 273, 241, 157, 57, 1, 1, 73, 211, 343, 421, 421, 343, 211, 73, 1, 1, 91, 273, 463, 601, 651, 601, 463, 273, 91, 1, 1, 111, 343, 601, 813, 931, 931, 813, 601, 343, 111, 1

Offset: 0

Paul Barry, Apr 02 2003

			Square array T(n,k) begins:
  1  1  1   1   1   1 ...
  1  3  7  13  21  31 ...
  1  7 21  43  73 111 ...
  1 13 43  91 157 241 ...
  1 21 73 157 273 421 ...
  ...

Rows include A054569, A002061, A082040, A082041.
Main diagonal is A059826.
Cf. A082038.

A114254 Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral.

Original entry on oeis.org

1, 25, 101, 261, 537, 961, 1565, 2381, 3441, 4777, 6421, 8405, 10761, 13521, 16717, 20381, 24545, 29241, 34501, 40357, 46841, 53985, 61821, 70381, 79697, 89801, 100725, 112501, 125161, 138737, 153261, 168765, 185281, 202841, 221477, 241221

Offset: 0

William A. Tedeschi, Feb 06 2008, Mar 01 2008

			For n = 1, the 3 X 3 spiral is
.
       7---8---9
       |
       6   1---2
       |       |
       5---4---3
.
so a(1) = 7 + 9 + 1 + 5 + 3 = 25.
.
For n = 2, the 5 X 5 spiral is
.
  21--22--23--24--25
   |
  20   7---8---9--10
   |   |           |
  19   6   1---2  11
   |   |       |   |
  18   5---4---3  12
   |               |
  17--16--15--14--13
.
so a(2) = 21 + 25 + 7 + 9 + 1 + 5 + 3 + 17 + 13 = 101.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Project Euler, Problem 28. Number Spiral Diagonals
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016754, A054569, A053755, A054554 for diagonals from origin.

Cf. A325958 (first differences).

Array[1 + 10 #^2 + (16 #^3 + 26 #)/3 &, 36, 0] (* Michael De Vlieger, Mar 01 2018 *)

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3; \\ Joerg Arndt, Mar 01 2018

O.g.f.: 3/(-1+x) + 16/(-1+x)^2 + 44/(-1+x)^3 + 32/(-1+x)^4 = (1 + 21*x + 7*x^2 + 3*x^3)/(-1+x)^4. - R. J. Mathar, Feb 10 2008

a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3. [Corrected by Arie Groeneveld, Aug 17 2008]

A168026 Noncomposite numbers in the southwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 7, 43, 73, 157, 211, 421, 601, 1483, 2551, 2971, 3907, 4423, 6163, 6481, 8191, 12211, 19183, 22651, 26407, 27061, 28393, 31153, 35533, 37057, 37831, 42643, 47743, 55933, 60763, 71023, 74257, 77563, 83233, 84391, 98911, 110557, 113233

Offset: 1

Alonso del Arte, Nov 16 2009

From Peter Munn, Mar 17 2018: (Start)
Noncomposites of the form k^2 + k + 1 with k even and nonnegative (and the same values occur with k odd and negative). Equivalently, noncomposites of the form 4k^2 + 2k + 1 with k >= 0, or 4k^2 - 6k + 3 with k > 0.
A073337 lists those of the form k^2 + k + 1 with k odd and positive, and this is equivalently those of the form 4k^2 - 2k + 1 with k > 0.
(End)
Numbers that are the sum of A000217(2*k-3) + A000217(2*k-1) that result in either unity or a prime, for k,n >= 1. For k,n >= 0, a(n+1) = 4*k*2 + 2*k + 1 will give the same results. - J. M. Bergot, May 07 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Alonso del Arte, Ulam spiral (2009). [Note that "East" and "West" in this video match the cover of Scientific American, but "North" and "South" are switched.]
BackIssues.com, Scientific American March 1964 back issue
MathWorld, Prime Spiral
Scientific American, March 1964 cover
Wikipedia, Ulam Spiral

Cf. A054569, all numbers of the form 4k^2 - 6k + 3 with k > 0. Noncomposites of the eastern ray are in A168022. Primes of the northeastern ray are in A073337. Noncomposites of the northern ray are in A168023. Primes of the northwestern ray are in A121326. Noncomposites of the western ray are in A168025. Noncomposites of the southern ray are in A168027.

Select[Table[4 n^2 - 6 n + 3, {n, 200}], Length[Divisors[ # ]] < 3 &]

lista(nn) = {print1(1, ", "); for(k=1, nn, if(isprime(p=4*k^2-6*k+3), print1(p, ", ")));} \\ Altug Alkan, Mar 22 2018

Numbers of the form 4k^2 - 6k + 3 with k > 0 and no more than two divisors. [corrected by Peter Munn, Mar 17 2018]

A200975 Numbers on the diagonals in Ulam's spiral.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49, 57, 65, 73, 81, 91, 101, 111, 121, 133, 145, 157, 169, 183, 197, 211, 225, 241, 257, 273, 289, 307, 325, 343, 361, 381, 401, 421, 441, 463, 485, 507, 529, 553, 577, 601, 625, 651, 677, 703, 729, 757, 785, 813, 841, 871, 901

Offset: 1

Ismael Bouya, Nov 25 2011

All entries are odd.
From Bob Selcoe, Oct 22 2014: (Start)
The following hold:
1. a(n) = (2k + 1)^2  when n = 4k + 1, k >= 0
2. a(n) = 4*k^2 + 1   when n = 4k - 1, k > 0
3  a(n) = k^2 + k + 1 when n = 2k, k > 0.
Conjecture 1: there must be at least one prime in [a(n), a(n+1)] inclusive.
Conjecture 2: generally, when j is in [(2m-1)^2+1, (2m+1)^2] inclusive, there must be at least one prime in [j-2m-1, j] inclusive. If true, then Conjecture 1 is true; also suggests A248623, A248835 and Oppermann's conjecture (see A002620) likely are true. (End)

			The numbers between ** are in this sequence.
.
  *21*--22---23---24--*25*
    |
    |
   20   *7*---8---*9*--10
    |    |              |
    |    |              |
   19    6   *1*---2   11
    |    |         |    |
    |    |         |    |
   18   *5*---4---*3*  12
    |                   |
    |                   |
  *17*--16---15---14--*13*

Todd Silvestri, Table of n, a(n) for n = 1..1000
Project Euler, Problem 28: Number spiral diagonals
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A016754, A054554, A053755, and A054569 interleaved, A002620,

Cf. A121658 (complementary)

Sort@ Flatten@ Table[4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 16}] (* Robert G. Wilson v, Jul 10 2014 *)
a[n_Integer/;n>0]:=Quotient[2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]+7,8] (* Todd Silvestri, Oct 25 2014 *)

al(n)=local(r=vector(n),j);r[1]=1;for(k=2,n,r[k]=r[k-1]+(k+2)\4*2);r /* Franklin T. Adams-Watters, Nov 26 2011 */

# prints all numbers on the diagonals of a sq*sq spiral
sq = 5
d = 1
while 2*d - 1 < sq:
    print(4*d*d - 4*d +1)
    print(4*d*d - 4*d +1 + 1* 2* d)
    print(4*d*d - 4*d +1 + 2* 2* d)
    print(4*d*d - 4*d +1 + 3* 2* d)
    d += 1
print(sq*sq)

a(4n) = 4n^2 + 2n + 1; a(4n+1) = 4n^2 + 4n + 1; a(4n+2) = 4n^2 + 6n + 3; a(4n+3) = 4n^2 + 8n + 5. [corrected by James Mitchell, Dec 31 2017]
G.f.: -x*(1+x+x^5-x^4) / ( (1+x)*(x^2+1)*(x-1)^3 ). - R. J. Mathar, Nov 28 2011
a(n) = (2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2)+7)/8 = (A249356(n)+7)/8. - Todd Silvestri, Oct 25 2014
a(n) = floor_(n*(n+2)/4) + floor_(n(mod 4)/3) + 1. - Bob Selcoe, Oct 27 2014

Edited with more terms by Franklin T. Adams-Watters, Nov 26 2011

A081348 First row in maze arrangement of natural numbers.

Original entry on oeis.org

1, 6, 7, 20, 21, 42, 43, 72, 73, 110, 111, 156, 157, 210, 211, 272, 273, 342, 343, 420, 421, 506, 507, 600, 601, 702, 703, 812, 813, 930, 931, 1056, 1057, 1190, 1191, 1332, 1333, 1482, 1483, 1640, 1641, 1806, 1807, 1980, 1981, 2162, 2163, 2352, 2353, 2550

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081347, A080335.

[(2*n^2+4*n+3-(2*n+1)*(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 + 5 x - x^2 + 3 x^3) / ((1 - x)^3 (1 + x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n) = (2*n^2+4*n+3-(2*n+1)(-1)^n)/2.
a(2*n) = A054569(n).
a(2*n+1) = 2*A014105(n+1).
G.f.: (1+5*x-x^2+3*x^3)/((1-x)^3*(1+x)^2). - Colin Barker, Apr 17 2012

A168547 a(n) = 1 - 2n^2 + 4n(1 + 2n^2)/3.

Original entry on oeis.org

1, 3, 17, 59, 145, 291, 513, 827, 1249, 1795, 2481, 3323, 4337, 5539, 6945, 8571, 10433, 12547, 14929, 17595, 20561, 23843, 27457, 31419, 35745, 40451, 45553, 51067, 57009, 63395, 70241, 77563, 85377, 93699, 102545, 111931, 121873, 132387, 143489, 155195

Offset: 0

Paul Curtz, Nov 29 2009

Binomial transform of the quasi-finite sequence 1,2,12,16,0,... (0 continued).

A bisection of A168582.

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

[1-2*n^2+4*n*(1+2*n^2)/3: n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

Table[1-2*n^2+4*n*(1+2*n^2)/3, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,3,17,59},60] (* Harvey P. Dale, May 21 2023 *)

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (1 - x + 11*x^2 + 5*x^3)/(x-1)^4.
First differences: a(n+1) - a(n) = 2*A054569(n+1).
Second differences: a(n+2) - 2*a(n+1) + a(n) = 4*A004767(n).
Third differences: a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = 16.
a(n+1) = A166464(n) + A035597(n+1).
a(n) = 1 - 2*n^2 + 4*A005900(n). - R. J. Mathar, Dec 05 2009
E.g.f.: (1/3)*(3 + 6*x + 18*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Jul 26 2016

Edited and extended by R. J. Mathar, Dec 05 2009

A143861 Ulam's spiral (NNE spoke).

Original entry on oeis.org

1, 14, 59, 136, 245, 386, 559, 764, 1001, 1270, 1571, 1904, 2269, 2666, 3095, 3556, 4049, 4574, 5131, 5720, 6341, 6994, 7679, 8396, 9145, 9926, 10739, 11584, 12461, 13370, 14311, 15284, 16289, 17326, 18395, 19496, 20629, 21794, 22991, 24220

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Stanislaw M. Ulam was doodling during the presentation of a "long and very boring paper" at a scientific meeting in 1963. The spiral is its result. Note that conforming to trigonometric conventions, the spiral begins on the abscissa and rotates counterclockwise. Other spirals, orientations, direction of rotation and initial values exist, even in the OEIS.

Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 59, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Gardner, Mathematical Recreations: The Remarkable Lore of the Prime Number, Scientific American 210 3: 120 - 128.
Hermetic Systems, Prime Number Spiral
OEIS wiki, Ulam spiral
Ivars Peterson's MathTrek, Prime Spirals, Science News, May 3 2002.
Robert Sacks, Number Spiral
Scientific American, Cover page of the March 1964
Eric Weisstein's World of Mathematics, Prime Spiral
Wikipedia, Ulam spiral
Wikipedia, Boxing the compass
Robert G. Wilson v, Ulam's spiral
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.

List([1..40], n-> ((32*n-35)^2 +55)/64); # G. C. Greubel, Nov 09 2019

[((32*n-35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019

seq( ((32*n-35)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

(* From Robert G. Wilson v, Oct 29 2011 *)
f[n_]:= 16n^2 -35n +20; Array[f, 40]
LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
((32*Range[40] -35)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

[((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 35*n + 20. - R. J. Mathar, Sep 08 2008
G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A172979 Primes with locations of right angle turns in Ulam square spiral (primes in A033638).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 43, 73, 101, 157, 197, 211, 241, 257, 307, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191, 8837, 9901

Offset: 1

Michel Lagneau, Nov 21 2010

nonn

Except for the first term, 2, these are the primes on the main diagonals of the Ulam spiral. - Robert G. Wilson v, Jul 10 2014

Primes p for which floor(sqrt(p)) | (p-1). - Davide Rotondo, Jun 06 2025

			Ulam square spiral = 7 8 9 / 6 1 2 / 5 4 3 /...; changes of direction (right-angle)
  for the primes at 2 3 5 7 ...

Cf. A033638, A054554, A053755, A054569, A016754, A078784.

with(numtheory): a0:=1:for n from 1 to 200 do : a1:=a0+floor(n/2):a0:=a1:if
  type(a1,prime)=true then printf(`%d, `,a1):else fi:od:

Select[ Sort@ Flatten@ Table[ 4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 55}], PrimeQ]

for(n=0,10^3, my(t=n^2\4+1); if(isprime(t),print1(t,", "))); \\ Joerg Arndt, Jul 12 2014

A337641 One-quarter of the number of regions in the central square of an equal-armed cross with arms of length n (as in A331456).

Original entry on oeis.org

1, 14, 70, 231, 576, 1207, 2255, 3883, 6267, 9588, 14088, 20021, 27667, 37306, 49240, 63859, 81517, 102603, 127545, 156769, 190739, 229932, 274898, 326181, 384332, 449878, 523472, 605766, 697380, 799053, 911449, 1035371, 1171471, 1320566, 1483374, 1660873, 1853819, 2063133, 2289607, 2534326, 2798159

Offset: 0

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Sep 17 2020

nonn

Without loss of generality, we may assume the central square has vertices (0,0), (1,0), (0,1), (1,1).
Suppose n >= 1. Then all nodes in the graph, whether or not in the central square, have rational coordinates with denominator at most 4*n^2 + 2*n + 1, which for n = 1, 2, 3, ... is 7, 21, 43, 73, 111, ... (cf. A054569).
This maximum is always attained, for example by the node at the intersection of the lines 2*n*x + y = n, joining (0,n) to (1, -n) and -x + (2*n+1)*y = n, joining (-n,0) to (n+1,1).
In the central square, the maximum number of sides in any region is (for n = 0, 1, 2, 3, ...) 3, 4, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 7, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, ...  We conjecture that 7 is the maximum. - Lars Blomberg, Sep 19 2020.
See A331456 for further illustrations.

Lars Blomberg, Table of n, a(n) for n = 0..74
Scott R. Shannon, Colored illustration for a(0): there are 4 regions, so a(0) = 1.
Scott R. Shannon, Colored illustration for a(1): there are 56 regions, so a(1) = 14.
Scott R. Shannon, Colored illustration for a(2): there are 280 regions, so a(2) = 70.
Scott R. Shannon, Colored illustration for a(3): there are 924 regions, so a(3) = 231.
Scott R. Shannon, Black and white illustration for a(1) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(2) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(3) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(4) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(5) (Shows vertices and regions for each square)
Scott R. Shannon, Black and white illustration for a(6) (Shows vertices and regions for each square)

Cf. A054569, A331456, A337640.

A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2m + 4m^2).

Original entry on oeis.org

1, 2, 7, 9, 7, 2, 8, 7, 4, 2, 2, 8, 1, 8, 9, 6, 8, 3, 3, 6, 4, 7, 2, 7, 5, 7, 0, 1, 5, 0, 7, 6, 3, 0, 6, 7, 2, 2, 6, 2, 6, 0, 3, 6, 7, 5, 0, 7, 5, 7, 8, 2, 6, 1, 9, 3, 0, 6, 8, 3, 0, 5, 8, 8, 1, 6, 9, 3, 0, 6, 6, 0, 7, 2, 2, 1, 3, 6, 4, 9, 0, 6, 6, 2, 1, 1, 5, 3, 2, 9, 9, 0, 5, 3, 5, 3, 2, 2, 7, 3, 7, 1, 9, 7, 1, 3, 2, 9, 2, 3

Offset: 1

Artur Jasinski, Nov 28 2020

This constant K and the constant J = A339135 allow the expression of the real and imaginary parts of:
Psi(1/4 + i*sqrt(3)/4) = - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*sqrt(3)*K;
Psi(-1/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3) - sqrt(3)*K + Pi*tanh(Pi*sqrt(3)/2));
Psi(3/4 + i*sqrt(3)/4)= - J - i*sqrt(3)*k - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/4) + i*Pi*tanh(Pi*sqrt(3)/2).
Psi(-3/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3)/3 + sqrt(3)*K).
where Psi is the digamma function and i=sqrt(-1).

			1.27972874228189683364727570150763067226260...

Cf. A054569 (terms), A339135.

K:= Re(sum(1/(1+2*n+4*n^2), n=0..infinity)):
evalf(K, 120);  # Alois P. Heinz, Dec 06 2020

RealDigits[N[Re[Sum[1/(1 + 2*n + 4*n^2), {n, 0, Infinity}]], 110]][[1]]

sumpos(n=0, 1/(1+2*n+4*n^2)) \\ Michel Marcus, Nov 28 2020

Equals -i/(2*sqrt(3)) * (Psi(1/4 + i*sqrt(3)/4) - Psi(1/4 - i*sqrt(3)/4)).

Equals Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/3 - Sum_{m>=0} 1/(3 + 6*m + 4*m^2).

cp's OEIS Frontend

A082039 Symmetric square array defined by T(n,k) = k^2*n^2 + k*n + 1, read by antidiagonals.

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A114254 Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral.

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A168026 Noncomposite numbers in the southwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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A200975 Numbers on the diagonals in Ulam's spiral.

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A081348 First row in maze arrangement of natural numbers.

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A168547 a(n) = 1 - 2*n^2 + 4*n*(1 + 2*n^2)/3.

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A143861 Ulam's spiral (NNE spoke).

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A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.

A168547 a(n) = 1 - 2n^2 + 4n(1 + 2n^2)/3.

A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2m + 4m^2).