A064788 -id:A064788 - 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)

A060736 Array of square numbers read by antidiagonals in up direction.

Original entry on oeis.org

1, 2, 4, 5, 3, 9, 10, 6, 8, 16, 17, 11, 7, 15, 25, 26, 18, 12, 14, 24, 36, 37, 27, 19, 13, 23, 35, 49, 50, 38, 28, 20, 22, 34, 48, 64, 65, 51, 39, 29, 21, 33, 47, 63, 81, 82, 66, 52, 40, 30, 32, 46, 62, 80, 100

Offset: 1

Frank Ellermann, Apr 23 2001

A simple permutation of natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Jan 09 2013

			1 4 9 16 .. => a(1)= 1
2 3 8 15 .. => a(2)= 2, a(3)=4
5 6 7 14 .. => a(4)= 5, a(5)=3, a(6)=9
10 11 12 13 .. => a(7)=10, a(8)=6, a(9)=8, a(10)=16

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734. Inverse permutation: A064788, the first inverse function (numbers of rows) A194258, the second inverse function (numbers of columns) A194195.

Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2013 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i>=j:
   result=i**2-j+1
else:
   result=(j-1)**2+i
# Boris Putievskiy, Jan 09 2013

T(n+1, k)=n*n+k, T(k, n+1)=(n+1)*(n+1)+1-k, 1 <= k <= n+1.

a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i if i < j, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 09 2013

A064790 Inverse permutation to A060734.

Original entry on oeis.org

1, 3, 5, 2, 6, 9, 13, 8, 4, 10, 14, 19, 25, 18, 12, 7, 15, 20, 26, 33, 41, 32, 24, 17, 11, 21, 27, 34, 42, 51, 61, 50, 40, 31, 23, 16, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 36, 44, 53, 63, 74, 86, 99, 113, 98, 84, 71, 59, 48, 38, 29, 45, 54, 64, 75, 87, 100, 114

Offset: 1

N. J. A. Sloane, Oct 20 2001

From Boris Putievskiy, Mar 14 2013: (Start)
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and  from T(n,n) to T(n,1). This sequence is A188568 as table read  by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so. The same table A188568 read layer by layer clockwise is A194280. (End)

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  3,5,2;
  6,9,13,8,4;
  10,14,19,25,18,12,7;
  15,20,26,33,41,32,24,17,11;
  21,27,34,42,51,61,50,40,31,23,16;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's MathWorld, Pairing Function
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A064788, A188568, A194280.

a(n) = (i+j-1)*(i+j-2)/2+i, where i=min(t; t^2-n+1), j=min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1. - Boris Putievskiy, Dec 24 2012

cp's OEIS Frontend

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

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A060736 Array of square numbers read by antidiagonals in up direction.

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A064790 Inverse permutation to A060734.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A060736 Array of square numbers read by antidiagonals in up direction.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A064790 Inverse permutation to A060734.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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