A185505 -id:A185505 - cp's OEIS frontend

A063490 a(n) = (2n - 1)(7n^2 - 7n + 6)/6.

1, 10, 40, 105, 219, 396, 650, 995, 1445, 2014, 2716, 3565, 4575, 5760, 7134, 8711, 10505, 12530, 14800, 17329, 20131, 23220, 26610, 30315, 34349, 38726, 43460, 48565, 54055, 59944, 66246, 72975, 80145, 87770, 95864, 104441, 113515, 123100, 133210, 143859, 155061

Offset: 1

N. J. A. Sloane, Aug 01 2001

From Omar E. Pol, Oct 23 2019: (Start)
a(n) is also the sum of terms that are in the n-th finite row and in the n-th finite column of the square [1,n]x[1,n] of the natural number array A000027; e.g., the [1,3]x[1,3] square is
1..3..6
2..5..9
4..8..13,
so that a(1) = 1, a(2) = 2 + 3 + 5 = 10, a(3) = 4 + 6 + 8 + 9 + 13 = 40.
Hence the partial sums give A185505. (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000
Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022. (See p. 14).
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A064788, A185505.

[(2*n-1)*(7*n^2-7*n+6)/6: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Table[(2*n-1)*(7*n^2-7*n+6)/6, {n,1,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,10,40,105}, 50] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(7*n^2 - 7*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-6 + 12*x + 21*x^2 + 14*x^3 )*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017

G.f.: x*(1+x)*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
a(n) = Sum_{k = n^2-2*n+2..n^2} A064788(k). - Lior Manor, Jan 13 2013
From G. C. Greubel, Dec 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (-6 + 12*x + 21*x^2 + 14*x^3)*exp(x)/6 + 1. (End)

A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 7, 11, 10, 14, 23, 26, 20, 25, 42, 51, 50, 35, 41, 70, 88, 94, 85, 56, 63, 109, 140, 156, 155, 133, 84, 92, 161, 210, 240, 250, 237, 196, 120, 129, 228, 301, 350, 375, 374, 343, 276, 165, 175, 312, 416, 490, 535, 550, 532, 476, 375, 220, 231, 415, 558, 664, 735, 771, 770, 728, 639, 495, 286

Offset: 1

Clark Kimberling, Jan 29 2011

Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array. The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.)

The formula for the integer T(n,k) has denominator 12. The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509.

			The natural number array A000027 starts with
  1, 2,  4,  7, ...
  3, 5,  8, 12, ...
  6, 9, 13, 18, ...
  ...
T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:
   1,  3,   7,  14,  25,  41
   4, 11,  23,  42,  70, 109
  10, 26,  51,  88, 140, 210
  20, 50,  94, 156, 240, 350
  35, 85, 155, 250, 375, 535

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Cf. A000027, A185507, A185508, A185509.

Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal).

f[n_,k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12.

A336186 Side length of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board such that the sum of the integers in the square is a perfect square.

Original entry on oeis.org

1, 17, 127, 1871, 13969, 205793, 1536463, 22635359, 168996961, 2489683697

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 11 2020

Consider a diagonally numbered 2D board shown in the example below. Draw a square, including the 1 at the top-left corner, around a block of integers and sum the integers within the square. This sequence gives the number of integers on the side of that square such that the resulting sum of integers is a perfect square.
The corresponding perfect square sum is given in A336189.
Integers m such that A185505(m) is a square. - Michel Marcus, Jul 11 2020

			Board is numbered as follows:
.
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
a(1) = 1 is a term as 1 = 1^2 is a perfect square.
a(2) = 17 is a term as the block of integers, with the seventeen numbers {1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137} along the top edge and the seventeen numbers {1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153} along the left edge, sum to 48841 = 221^2 which is a perfect square.

Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]

Cf. A336189, A185505, A000290, A000124, A000217.

isok(m) = issquare((7*m^4 + 5*m^2)/12); \\ Michel Marcus, Jul 11 2020

Conjectures from Colin Barker, Jul 11 2020: (Start)
G.f.: x*(1 + x)*(1 + 16*x + x^2) / (1 - 110*x^2 + x^4).
a(n) = 110*a(n-2) - a(n-4) for n>4. (End)
Empirical from Bill McEachen, Aug 08 2025: (Start)
a(n) = floor(k1*B^(n+1)) for odd n, and floor(k2*B^n) for even n, where k1 =(26*sqrt(21)-119)/14, k2 = (2*sqrt(21)-7)/14, and B = sqrt(55 + 12*sqrt(21)).
Above closed-forms via Amiram Eldar equate to Barker's recurrence. (End)

a(10) from Michel Marcus, Jul 11 2020

A336189 The perfect square integer sum of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board.

Original entry on oeis.org

1, 48841, 151757761, 7148452448281, 22211509021338121, 1046258952151234702321, 3250912043200499426917081, 153132136343696050161247674961, 475808694603918281112156880430641

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 11 2020

See A336186 for the corresponding square edge length and an explanation of the sequence. Note that the currently known terms all end in 1.

			a(1) = 1 = 1^2.
a(2) = 48841 = 221^2.
a(3) = 151757761 = 12319^2.
a(4) = 7148452448281 = 2673659^2.
a(5) = 22211509021338121 = 149035261^2.
a(6) = 1046258952151234702321 = 32345926361^2.
a(7) = 3250912043200499426917081 = 1803028575259^2.
a(8) = 153132136343696050161247674961 = 391321014441719^2.
a(9) = 475808694603918281112156880430641 = 21813039554448121^2.
See A336186 for a diagram of the 2D board and examples.

Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]

Cf. A336186, A185505, A000290, A000124, A000217.

A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 25, 31, 37, 39, 51, 61, 63, 69, 81, 87, 97, 99, 109, 117, 135, 145, 147, 151, 153, 163, 165, 171, 183, 189, 195, 201, 207, 213, 219, 223, 229, 235, 241, 249, 253, 267, 271, 273, 277, 297, 307, 319, 325, 337, 343, 345, 355, 373, 381, 387, 391, 393, 409, 435, 447, 451, 457

Offset: 1

Eric Angelini and Scott R. Shannon, Jul 13 2020

nonn

			The board is numbered as follows:
.
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
a(1) = 1 as the four numbers {1,2,5,3} form the corners of a square of size 1, and the sum of these number is 11, a prime number.
a(2) = 3 as the four numbers {1,7,25,10} form the corners of a square of size 3, and the sum of these number is 43, a prime number.
a(3) = 7 as the four numbers {1,29,113,36} form the corners of a square of size 7, and the sum of these number is 179, a prime number.

Harvey P. Dale, Table of n, a(n) for n = 1..2000
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]

Cf. A000040, A185505, A000124, A000217, A282513.

Select[Range[1,501,2],PrimeQ[3#^2+4#+4]&] (* Harvey P. Dale, May 26 2022 *)

The sequence is the values of d where 3*d^2+4*d+4, the sum of the four numbers for a square of size d, is prime. For even d this sum will always be even, thus all terms are odd.

cp's OEIS Frontend

A063490 a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6.

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A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

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A336186 Side length of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board such that the sum of the integers in the square is a perfect square.

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A336189 The perfect square integer sum of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A063490 a(n) = (2n - 1)(7n^2 - 7n + 6)/6.