A054569 -id:A054569 - cp's OEIS frontend

A357745 Numbers on the 8 main spokes of a square spiral with 1 in the center.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 53, 57, 61, 65, 69, 73, 77, 81, 86, 91, 96, 101, 106, 111, 116, 121, 127, 133, 139, 145, 151, 157, 163, 169, 176, 183, 190, 197, 204, 211, 218, 225, 233, 241, 249, 257, 265, 273

Offset: 1

Karl-Heinz Hofmann, Dec 22 2022

The 8 main spokes are (with 1 in the center, 2 to the east, 3 to the northeast): east: A054552; northeast: A054554; north: A054556; northwest: A053755; west: A054567; southwest: A054569; south: A033951; southeast: A016754.
Alternatively the 8 main spokes are pairwise part of the 4 main axes: horizontal: A317186; vertical: A267682; diagonal: A002061; antidiagonal: A080335.
And lastly the 4 main axes are giving two main crosses: Horizontal-vertical cross: A039823; Diagonal-antidiagonal cross: A200975.

			See visualization in links.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, Visualization of the first few terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.

Cf. A317186, A267682, A002061, A080335, A039823, A200975.

Rest@ CoefficientList[Series[x (1 - x^8 + x^9)/((1 - x)^3*(1 + x) (1 + x^2) (1 + x^4)), {x, 0, 63}], x] (* Michael De Vlieger, Dec 29 2022 *)
a[n_] := BitShiftRight[(n + 3)^2, 4] + Boole[BitAnd[n, 7] != 1]; Array[a, 65] (* Amiram Eldar, Dec 30 2022, after the PARI code *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11},70] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = sqr(n+3)>>4 + (bitand(n,7)!=1); \\ Kevin Ryde, Dec 30 2022

def A357745(n): return ((n+3)**2 >> 4) + 1 if n % 8 != 1 else (n+3)**2 >> 4

G.f.: x*(1-x^8+x^9)/((1-x)^3*(1+x)*(1+x^2)*(1+x^4)). - Joerg Arndt, Dec 29 2022

a(n) = floor((n+3)^2 / 16) + (1 if n != 1 mod 8). - Kevin Ryde, Dec 30 2022

A082041 a(n) = 16n^2 + 4n + 1.

Original entry on oeis.org

1, 21, 73, 157, 273, 421, 601, 813, 1057, 1333, 1641, 1981, 2353, 2757, 3193, 3661, 4161, 4693, 5257, 5853, 6481, 7141, 7833, 8557, 9313, 10101, 10921, 11773, 12657, 13573, 14521, 15501, 16513, 17557, 18633, 19741, 20881, 22053, 23257, 24493

Offset: 0

Paul Barry, Apr 02 2003

Also sequence found by reading the segment (1,21) together with the line from 21, in the direction 21, 73, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Column k=4 of A082039.

Cf. A054569, A082040, A074377.

Table[16n^2+4n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,73},50] (* Harvey P. Dale, Sep 28 2024 *)

a(n)=16*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (-1-18*x-13*x^2)/(x-1)^3 . - R. J. Mathar, Dec 03 2014
From Elmo R. Oliveira, Oct 28 2024: (Start)
E.g.f.: exp(x)*(1 + 20*x + 16*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A102094 a(n) = (2n-1)(2*n+1)^2.

Original entry on oeis.org

9, 75, 245, 567, 1089, 1859, 2925, 4335, 6137, 8379, 11109, 14375, 18225, 22707, 27869, 33759, 40425, 47915, 56277, 65559, 75809, 87075, 99405, 112847, 127449, 143259, 160325, 178695, 198417, 219539, 242109, 266175, 291785, 318987, 347829, 378359, 410625

Offset: 1

Gerald McGarvey, Feb 13 2005

Numbers which are both the sum of 2n+1 consecutive odd integers and, after skipping one odd integer, the sum of the 2n-1 immediately higher consecutive odd integers. See A082108(n-1) for the smallest of the 2n+1 odd integers, and A054569(n+1) for the skipped number. Odd integer counterpart to A059270. - Charlie Marion, Apr 30 2020

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, p. 123.
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.

G. C. Greubel, Table of n, a(n) for n = 1..1000
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002388.

List([1..40], n-> (2*n-1)*(2*n+1)^2); # G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2: n in [1..40]]; // G. C. Greubel, Oct 27 2019

seq((2*n-1)*(2*n+1)^2, n=1..40); # G. C. Greubel, Oct 27 2019

Table[(2n-1)(2n+1)^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{9,75,245,567},40] (* Harvey P. Dale, Jul 24 2012 *)

vector(40, n, (2*n-1)*(2*n+1)^2) \\ G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2 for n in (1..40)] # G. C. Greubel, Oct 27 2019

Sum_{n>=1} 1/a(n) = (12 - Pi^2)/16.
Sum_{n>=1} n/a(n) = (Pi^2 - 4)/32. - Sign flipped by Bernard Schott, May 06 2020
From Harvey P. Dale, Jul 24 2012: (Start)
a(1)=9, a(2)=75, a(3)=245, a(4)=567, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (9 + 39*x - x^2 + x^3)/(1-x)^4. (End)
E.g.f.: 1 + (-1 + 10*x + 28*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Oct 27 2019

More terms from Harvey P. Dale, Jul 24 2012

A244687 The spiral of Champernowne read by the Southwest ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

			See A244677 for the spiral of David Gawen Champernowne.

Cf. A033307, A054569, A033952, A244677-A244688, A244690-A244692.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 4n^2 - 6n + 3 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

See A244677 formula section.

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 36, 43, 64, 73, 100, 111, 144, 157, 196, 211, 256, 273, 324, 343, 400, 421, 484, 507, 576, 601, 676, 703, 784, 813, 900, 931, 1024, 1057, 1156, 1191, 1296, 1333, 1444, 1483, 1600, 1641, 1764, 1807, 1936, 1981, 2116, 2163, 2304, 2353, 2500, 2551

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th element of the diagonal of the triangle A325655. Equivalently, a(n) is the element M_{n,1} of the matrix M(n) whose permanent is A322277(n).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A054569, A317614, A322277, A325655.

Flat(List([0..55], n->(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2));

[(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2: n in [0..55]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..55);

Table[(1/2)*(- 1+(-1)^n)*(n-1)+n^2,{n,0,55}]

a(n) = (1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2;

O.g.f.: (-1 - 3*x - x^2 - 3*x^3)/((-1 + x)^3*(1+x)^2).
E.g.f.: (1/2)*exp(-x)*(-1 - x + exp(2*x)*(1 + x + 2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4
a(n) = n^2 if n is even.
a(n) = n^2 - n + 1 if n is odd.

A357744 a(n) is the least k such that prime(n) * k occurs in one of the eight main spokes of a square spiral with 1 in the center.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 25, 1, 17, 1, 59, 1, 13, 37, 1, 4, 3, 13, 5, 1, 21, 8, 2, 4, 1, 131, 3, 1, 2, 1, 1, 1, 2, 37, 4, 13, 58, 7, 1, 34, 1, 7, 23, 4, 1, 29, 1, 251, 1, 5, 25, 3, 13, 1, 7, 30, 1, 311, 31, 38, 3, 49, 3, 6, 5, 37, 19, 16, 7, 5, 149, 3, 1, 7, 419, 1, 1, 91, 10, 2

Offset: 1

Karl-Heinz Hofmann, Dec 01 2022

nonn

Numbers on the spokes of the spiral are A357745.
a(n) = 1 when prime(n) is directly on a main spoke.
a(n) <= prime(n) since odd squares are on the southeast spoke (A016754).

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, Graphic example.
Karl-Heinz Hofmann, Graphic together with the first 10000 primes.

Cf. A000040, A054552, A054554, A054556, A053755, A054567.

Cf. A054569, A033951, A016754, A357745.

from sympy import sieve
A357744, A357745, aupto = [], [], 82
for n in range (1, sieve[aupto]**2):
    A357745.append(((n+3)**2 >> 4) + 1 if n % 8 != 1 else (n+3)**2 >> 4)
for p in sieve[1:aupto + 1]:
    k = 1
    while (p*k) not in A357745: k += 1
    A357744.append(k)
print(A357744)

A098966 Number of (k+1)-tuples of integers modulo n (x_1,...,x_k,s) such that at least one subset of the x_i sums to s mod n. In other words, n^k times the expected number of distinct subset sums mod n of k integers mod n chosen uniformly at random. Read by antidiagonals, i.e., with entries in the order (n,k)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 1, 3, 1, 7, 5, 1, 15, 21, 7, 1, 31, 73, 43, 9, 1, 63, 233, 215, 73, 11, 1, 127, 717, 951, 497, 111, 13, 1, 255, 2173, 3971, 2865, 959, 157, 15, 1, 511, 6545, 16171, 15161, 6863, 1657, 211, 17, 1, 1023, 19665, 65167, 77369, 44391, 14521, 2631, 273, 19

Offset: 1

Andrew Childs (amchilds(AT)caltech.edu) and Wim van Dam (vandam(AT)cs.ucsb.edu), Oct 13 2004

a(n,k) <= n^(k+1).

			Table begins
  1,  1,   1,    1,     1, ...
  3,  7,  15,   31,    63, ...
  5, 21,  73,  233,   717, ...
  7, 43, 215,  951,  3971, ...
  9, 73, 497, 2865, 15161, ...
  ...

First column is A005408; second column is A054569; second row is A000225.

Mathematica
```
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a(n, 1) = 2*n - 1;
a(n, 2) = 4*n^2 - 6*n + 3;
a(n, 3) = 8*n^3 - 28*n^2 + 44*n - 23, n odd;
a(n, 3) = 8*n^3 - 28*n^2 + 44*n - 25, n even;
a(1, k) = 1;
a(2, k) = 2^(k+1) - 1;
a(3, k) = 3^(k+1) - 2*k - 2.

A161372 In Ulam's spiral starting at 101, take the elements not used so far from the two spokes SW, NE, SW, NE, SW, NE ...

Original entry on oeis.org

107, 101, 121, 103, 143, 113, 173, 131, 211, 157, 257, 191, 311, 233, 373, 283, 443, 341, 521, 407, 607, 481, 701, 563, 803, 653, 913, 751, 1031, 857, 1157, 971, 1291, 1093, 1433, 1223, 1583, 1361, 1741, 1507, 1907, 1661, 2081, 1823, 2263, 1993, 2453

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Jun 08 2009

NE to SW Diagonal of Ulam's Spiral, with 101 at center.

The sequence did not match the original definition. A working definition might be a(2n)=4n^2-10n+107, a(2n-1)=4n^2+2n+101, a(n)=n^2-n+105+2*(-1)^n*(1-2*n), but this seems to be unrelated to Ulam spirals. [R. J. Mathar, Jun 11 2009]

			SW-NE diagonal in:
  137 136 135 134 133 132 131
  138 117 116 115 114 113 130
  139 118 105 104 103 112 129
  140 119 106 101 102 111 128
  141 120 107 108 109 110 127
  142 121 122 123 124 125 126
  143 144 145 146 147 148 149

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054569 (SW spoke), A054554 (NE spoke).

CoefficientList[Series[(107*x^4-6*x^3-194*x^2-6*x+107)/((1+x)^2*(1-x)^3), {x,0,32}], x] (* Georg Fischer, Dec 03 2024 *)

Edited by Georg Fischer, Dec 03 2024

A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

Original entry on oeis.org

0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42, -30, 43, 57, 72, -56, 73, 91, 110, -90, 111, 133, 156, -132, 157, 183, 210, -182, 211, 241, 272, -240, 273, 307, 342, -306, 343, 381, 420, -380, 421, 463, 506, -462, 507, 553, 600, -552, 601, 651, 702, -650, 703, 757

Offset: 0

Enrico Santilli, Jun 23 2013

Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A081348 (second bisection); A002939, A054554, A054569, A068377.

[IsZero(n) select 0 else IsOdd(Self(n)) select n+Self(n) else n-Self(n): n in [0..60]]; // Bruno Berselli, Jul 01 2013

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42}, 60] (* Bruno Berselli, Jul 01 2013 *)

makelist(coeff(taylor(x*(1+3*x+6*x^2-2*x^3+4*x^4+4*x^5+2*x^6-6*x^7+3*x^8+x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3), x, 0, n), x, n), n, 0, 60); /* Bruno Berselli, Jul 01 2013 */

G.f.: x*(1 +3*x +6*x^2 -2*x^3 +4*x^4 +4*x^5 +2*x^6 -6*x^7 +3*x^8 +x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3). [Bruno Berselli, Jul 01 2013]
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12). [Bruno Berselli, Jul 01 2013]
a(4n) = -A002939(n), a(4n+1) = A054569(n+1), a(4n+2) = A054554(n+2), a(4n+3) = A068377(n+2). [Bruno Berselli, Jul 02 2013]

More terms from Bruno Berselli, Jul 01 2013

cp's OEIS Frontend

A357745 Numbers on the 8 main spokes of a square spiral with 1 in the center.

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A082041 a(n) = 16*n^2 + 4*n + 1.

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A102094 a(n) = (2*n-1)*(2*n+1)^2.

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A244687 The spiral of Champernowne read by the Southwest ray.

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.

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A357744 a(n) is the least k such that prime(n) * k occurs in one of the eight main spokes of a square spiral with 1 in the center.

A082041 a(n) = 16n^2 + 4n + 1.

A102094 a(n) = (2n-1)(2*n+1)^2.

A325657 a(n) = (1/2)(-1 + (-1)^n)(n-1) + n^2.