A010731 -id:A010731 - cp's OEIS frontend

A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

3, 5, 5, 5, 8, 5, 5, 8, 8, 5, 5, 8, 8, 8, 5, 5, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the first row and the first column, the rest of the elements are 8's.

For the same idea but for a right triangle see A278480; for an isosceles triangle see A278481; for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The corner of the square array begins:
3,5,5,5,5,5,5,5,5,5,...
5,8,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,...
5,8,8,8,8,8,...
5,8,8,8,8,...
5,8,8,8,...
5,8,8,...
5,8,...
5,...
...

Robert Israel, Table of n, a(n) for n = 1..10000

Antidiagonal sums give 3 together with the elements > 2 of A017089.

Cf. A010722, A010731, A274912, A274913, A278317, A278290, A278480, A278481.

3, seq(op([5,8$i,5]),i=0..20); # Robert Israel, Dec 04 2016

G.f. 3+x+8*x/(1-x)-3*(1+x)*Theta_2(0,sqrt(x))/(2*x^(1/8)) where Theta_2 is a Jacobi Theta function. - Robert Israel, Dec 04 2016

A349118 Row sums of a triangle based on A261327.

Original entry on oeis.org

1, 5, 3, 18, 8, 47, 18, 100, 35, 185, 61, 310, 98, 483, 148, 712, 213, 1005, 295, 1370, 396, 1815, 518, 2348, 663, 2977, 833, 3710, 1030, 4555, 1256, 5520, 1513, 6613, 1803, 7842, 2128, 9215, 2490, 10740, 2891, 12425, 3333, 14278, 3818, 16307, 4348, 18520, 4925

Offset: 2

Paul Curtz, Nov 08 2021

The following triangle has A261327 as its diagonals:
  1
      5
  1       2
      5       13
  1       2        5
      5       13       29
  1       2        5        10
      5       13       29        53
  1       2        5        10        17
      5       13       29        53        85
  ...
a(0) = a(1) = 0.
a(n)'s final digit: neither 4 nor 9.
First full bisection difference table:
  0,  1,  3,  8,  18,  35,  61,  98, ...    = 0, A081489 = b(n)
  1,  2,  5, 10,  17,  26,  37,  50, ...    = A002522
  1,  3,  5,  7,   9,  11,  13,  15, ...    = A005408
  2,  2,  2,  2,   2,   2,   2,   2, ...    = A007395
  0,  0,  0,  0,   0,   0,   0,   0, ...    = A000004
Second full bisection difference table:
  0,   5,  18,  47, 100, 185, 310, 483, ...    = c(n)
  5,  13,  29,  53,  85, 125, 173, 229, ...    = A078370
  8,  16,  24,  32,  40,  48,  56,  64, ...    = A008590(n+1)
  8,   8,   8,   8,   8,   8,   8,   8, ...    = A010731
  0,   0,   0,   0,   0,   0,   0,   0, ...    = A000004
Both bisections are cubic polynomials.
c(-n) = -c(n).

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A002522, A005408, A007395, A078370, A081489 (first bisection).

Cf. also A008590, A010731, A261327.

LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {1, 5, 3, 18, 8, 47, 18, 100}, 50] (* Amiram Eldar, Nov 08 2021 *)

G.f.: (5*x^5+2*x^4-2*x^3-x^2+5*x+1)/((x-1)^4*(x+1)^4).

cp's OEIS Frontend

A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

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