A164359 -id:A164359 - cp's OEIS frontend

A008577 Crystal ball sequence for planar net 4.8.8.

1, 4, 9, 17, 28, 41, 57, 76, 97, 121, 148, 177, 209, 244, 281, 321, 364, 409, 457, 508, 561, 617, 676, 737, 801, 868, 937, 1009, 1084, 1161, 1241, 1324, 1409, 1497, 1588, 1681, 1777, 1876, 1977, 2081, 2188, 2297, 2409, 2524, 2641, 2761, 2884, 3009, 3137, 3268

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 4*x + 9*x^2 + 17*x^3 + 28*x^4 + 41*x^5 + 67*x^6 + ... - _Michael Somos_, May 02 2020

T. D. Noe, Table of n, a(n) for n = 0..1000
Brian Galebach, Uniform Tiling 2 of 11
W. M. Meier and H. J. Moeck, Topology of 3-D 4-connected nets ..., J. Solid State Chem 27 1979 349-355, esp. p. 351.
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Partial sums of A008576.

Cf. A099837, A164359.

A099837[0] = 1; A099837[n_] := Mod[n+2, 3] - Mod[n, 3]; a[n_] := 4*n/3*(n+1) + 10/9 + A099837[n+2]/9; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Feb 15 2012, after R. J. Mathar *)
CoefficientList[Series[((1 + x)^2 (1 + x^2))/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{1,4,9,17,28},40] (* Harvey P. Dale, Dec 17 2017 *)
a[ n_] := Quotient[(n^2 + n + 1)*4, 3]; (* Michael Somos, May 02 2020 *)

a(n) = (n^2 + n + 1)*4\3; /* Michael Somos, May 02 2020 */

G.f.: ((1+x)^2*(1+x^2)) / ((1-x)^3*(1+x+x^2)). - Ralf Stephan, Apr 24 2004
a(n) = 4*(n/3)*(n+1)+10/9+A099837(n+2)/9. - R. J. Mathar, Nov 20 2010
The above g.f. and formula were originally stated as conjectures, but I now have a proof. This also justifies the b-file. Details will be added later. - N. J. A. Sloane, Dec 29 2015
From Michael Somos, May 02 2020: (Start)
Euler transform of length 3 sequence [4, -1, 1, -1].
a(n) = a(-1-n) = floor((n^2+n+1)*4/3) for all n in Z.
a(n) - 2*a(n+1) + a(n+2) = A164359(n) unless n=0.
(End)

A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, -1, -2, -3, 0, 0, 1, 3, 6, -1, -1, -1, -2, -5, -11, 0, 1, 2, 3, 5, 10, 21, 0, 0, -1, -3, -6, -11, -21, -42, 1, 1, 1, 2, 5, 11, 22, 43, 85, 0, -1, -2, -3, -5, -10, -21, -43, -86, -171, 0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, -1, -1, -1, -2, -5, -11, -22, -43, -85, -170, -341, -683, 0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365

Offset: 0

Paul Curtz, Nov 07 2009

The array contains A131708(0) in diagonal 0, then -A024495(0..1) in diagonal 1, then A024493(0..2) in diagonal 2, then -A131708(0..3), then A024495(0..4), then -A024493(0..5).

			The table starts in row n=0 with columns k >= 0 as:
0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0 A131531
0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1 A092220
1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2 A131556
-3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3 A164359
6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5
-11, 10, -11, 11, -10, 11, -11, 10, -11, 11, -10, 11, -11, 10, -11
21, -21, 22, -21, 21, -22, 21, -21, 22, -21, 21, -22, 21, -21, 22

Cf. A167617 (antidiagonal sums).

A131531 := proc(n) op((n mod 6)+1,[0,0,1,0,0,-1]) ; end proc:
A167613 := proc(n,k) option remember; if n= 0 then A131531(k); else procname(n-1,k+1)-procname(n-1,k) ; end if;end proc: # R. J. Mathar, Dec 17 2010

nmax = 13;
A131531 = Table[{0, 0, 1, 0, 0, -1}, {nmax}] // Flatten;
T[n_] := T[n] = Differences[A131531, n];
T[n_, k_] := T[n][[k]];
Table[T[n-k, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 20 2023 *)

T(0,k) = A131531(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.
T(n,n) = A001045(n). T(n,n+1) = -A001045(n). T(n,n+2) = A078008(n).
T(n,0) = -T(n,3) = (-1)^(n+1)*A024495(n).
T(n,1) = (-1)^(n+1)*A131708(n).
T(n,2) = (-1)^n*A024493(n).
T(n,k+6) = T(n,k).
a(n) = A131708(0), -A024495(0,1), A024493(0,1,2), -A131708(0,1,2,3), A024495(0,1,2,3,4), -A024493(0,1,2,3,4,5).

A244893 a(n) = a(n-a(n-1)) with initial values 2,3,2.

Original entry on oeis.org

2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3

Offset: 1

N. J. A. Sloane, Jul 17 2014

nonn

After first term, periodic with period {3,2,3}.

S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

f:=proc(n,o,S) option remember; # o = offset, S = init. terms
L:=nops(S);
if n < 0 then 0
elif n <= o+L-1 then S[n-o+1]
else f(n-f(n-1,o,S),o,S); fi; end;
[seq(f(n,1,[2,3,2]),n=1..100)];

Join[{2},LinearRecurrence[{0, 0, 1},{3, 2, 3},75]] (* Ray Chandler, Aug 25 2015 *)

a(n) = A164359(n), n>1. - R. J. Mathar, Jul 25 2014

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A008577 Crystal ball sequence for planar net 4.8.8.

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A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

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A244893 a(n) = a(n-a(n-1)) with initial values 2,3,2.

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