A277699 -id:A277699 - cp's OEIS frontend

A284273 Main diagonal of A284270.

0, 0, 0, 0, 2, 0, 0, 0, 2, 8, 8, 0, 12, 0, 0, 0, 2, 8, 11, 12, 16, 10, 3, 0, 9, 22, 18, 0, 12, 0, 0, 0, 2, 8, 30, 32, 11, 6, 26, 8, 19, 22, 0, 40, 1, 12, 37, 0, 17, 36, 18, 36, 7, 18, 16, 0, 27, 48, 13, 0, 12, 0, 0, 0, 2, 8, 30, 32, 53, 50, 40, 56, 32, 44, 2, 24, 73, 26, 61, 32, 14, 76, 28, 4, 34, 0, 64, 72, 86, 4, 73, 48, 16, 54, 11, 0, 33

Offset: 1

Antti Karttunen, Apr 10 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A048720, A065621, A277699, A284270.

(define (A284273 n) (A284270bi n n)) ;; A284270bi, see A284270.
(define (A284273 n) (modulo (A277699 n) n))

a(n) = A277699(n) mod n = A048720(n, A065621(n)) mod n.

A277199 Transpose of square array A277320.

Original entry on oeis.org

1, 2, 2, 7, 4, 3, 4, 14, 6, 4, 13, 8, 9, 8, 5, 14, 26, 12, 28, 10, 6, 11, 28, 23, 16, 27, 12, 7, 8, 22, 18, 52, 20, 18, 14, 8, 25, 16, 29, 56, 57, 24, 21, 16, 9, 26, 50, 24, 44, 54, 46, 28, 56, 18, 10, 31, 52, 43, 32, 39, 36, 35, 32, 63, 20, 11, 28, 62, 46, 100, 40, 58, 42, 104, 36, 54, 22, 12, 21, 56, 33, 104, 125, 48, 49, 112, 101, 40, 49, 24, 13

Offset: 1

Antti Karttunen, Nov 01 2016

See A277320.

			The top left corner of the array:
   1,  2,  7,  4,  13,  14,  11,   8,  25,  26,  31,  28
   2,  4, 14,  8,  26,  28,  22,  16,  50,  52,  62,  56
   3,  6,  9, 12,  23,  18,  29,  24,  43,  46,  33,  36
   4,  8, 28, 16,  52,  56,  44,  32, 100, 104, 124, 112
   5, 10, 27, 20,  57,  54,  39,  40, 125, 114,  99, 108
   6, 12, 18, 24,  46,  36,  58,  48,  86,  92,  66,  72
   7, 14, 21, 28,  35,  42,  49,  56,  79,  70,  93,  84
   8, 16, 56, 32, 104, 112,  88,  64, 200, 208, 248, 224
   9, 18, 63, 36, 101, 126,  83,  72, 209, 202, 231, 252
  10, 20, 54, 40, 114, 108,  78,  80, 250, 228, 198, 216
  11, 22, 49, 44, 127,  98,  69,  88, 227, 254, 217, 196
  12, 24, 36, 48,  92,  72, 116,  96, 172, 184, 132, 144

Transpose: A277320.
Main diagonal: A277699.
Row 1: A065621, Row 3: 3: A277823.
Column 1: A000027, Column 3: A048727.
Cf. A048720.

(define (A277199 n) (A277320bi (A004736 n) (A002260 n))) ;; Code for A277320bi given in A277320.

A(r,c) = A048720(r, A065621(c)).

A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.

Original entry on oeis.org

0, 16, 64, 144, 256, 912, 576, 784, 1024, 1296, 3648, 1936, 2304, 7312, 3136, 8208, 4096, 11824, 5184, 5776, 14592, 7056, 7744, 8464, 9216, 41232, 29248, 11664, 12544, 27568, 32832, 15376, 16384, 17424, 47296, 44688, 20736, 61104, 23104, 65808, 58368, 78096, 28224, 29584, 30976, 73872

Offset: 0

Torlach Rush, Aug 08 2019

nonn

For this sequence the square spiral begins with 0 and is the second illustration in the comments of A317186, where 0 is the origin of our circles.
a(n) >= A001107(n) + A033991(n) + A007742(n) + A033954(n).
a(n) = A016802(n) iff A046109(n) = 4.
a(n) = A016802(n) iff n <> k * A002144(m), k,m >= 1.
a(n) is congruent to 0 mod 16 and is the sum of one or more terms of A016802.
Conjecture: a(n) is a term of A277699 iff a(n)/16 = A277699(n).

			16 is a term because 16 = 16*(1)^2.
912 is a term because 912 = 16*(5)^2 + (2*(16*(4)^2)).
41232 is a term because 41232 = 16*(25)^2 + (2*((16*(24)^2) + (16*(20)^2))).

Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A001107, A002144, A007742, A016802, A033954, A033991, A046109, A277699, A317186.

Tb(n) = {return(16 * n * n)}
llsum(n) = {my(x=0); for (i = 1, n - 2, for (ii = i+1, n - 1, if(n*n == (ii*ii) + (i*i), x+=(2 * Tb(ii))))); return(x)}
Tx(n) = {my(x=0); forprimestep(x = 5, n, 4, if(n%x==0, return(llsum(n))))}
Tn(n) = {for (i = 0, n, print1(Tb(i) + Tx(i), ", "))}
Tn(45)

cp's OEIS Frontend

A284273 Main diagonal of A284270.

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A277199 Transpose of square array A277320.

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A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.

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