A139271 -id:A139271 - cp's OEIS frontend

A139277 a(n) = n(8n+5).

0, 13, 42, 87, 148, 225, 318, 427, 552, 693, 850, 1023, 1212, 1417, 1638, 1875, 2128, 2397, 2682, 2983, 3300, 3633, 3982, 4347, 4728, 5125, 5538, 5967, 6412, 6873, 7350, 7843, 8352, 8877, 9418, 9975, 10548, 11137, 11742, 12363, 13000

Offset: 0

Omar E. Pol, Apr 26 2008

Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139273 in the same spiral.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250.

Cf. A139271, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Table[n (8 n + 5), {n, 0, 50}] (* Bruno Berselli, Aug 22 2018 *)
LinearRecurrence[{3,-3,1},{0,13,42},50] (* Harvey P. Dale, Dec 04 2018 *)

a(n)=n*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 8*n^2 + 5*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x*{c+8 + (8-c)*x}/(1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - R. J. Mathar, May 12 2008
a(n) = 16*n + a(n-1) - 3 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
Sum_{n>=1} 1/a(n) = (sqrt(2)-1)*Pi/10 - 4*log(2)/5 + sqrt(2)*log(sqrt(2)+1)/5 + 8/25. - Amiram Eldar, Mar 18 2022
E.g.f.: exp(x)*x*(13 + 8*x). - Elmo R. Oliveira, Dec 15 2024

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

Original entry on oeis.org

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 20, 4, 0, 1, 8, 54, 176, 10, 0, 1, 10, 104, 996, 1876, 28, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0

Offset: 0

Alois P. Heinz, Mar 10 2023

Column k is INVERTi transform of k-th row of A287318.

			Square array A(n,k) begins:
  1,  1,     1,      1,       1,        1,        1, ...
  0,  2,     4,      6,       8,       10,       12, ...
  0,  2,    20,     54,     104,      170,      252, ...
  0,  4,   176,    996,    2944,     6500,    12144, ...
  0, 10,  1876,  22734,  108136,   332050,   796860, ...
  0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364.
Rows n=0-2 give: A000012, A005843, A139271.
Main diagonal gives A361297.
Cf. A000108, A287318.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
g:= proc(n, k) option remember; `if` (n<1, -1,
      -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
    end:
A:= (n,k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *)

A(n,1)/2 = A000108(n-1) for n >= 1.

G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - Shel Kaphan, Mar 19 2023

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 2, 3, 6, 9, 10, 13, 14, 7, 8, 11, 12, 15, 18, 19, 22, 23, 26, 27, 16, 17, 20, 21, 24, 25, 28, 31, 32, 35, 36, 39, 40, 43, 44, 29, 30, 33, 34, 37, 38, 41, 42, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 69

Offset: 1

Boris Putievskiy, Feb 14 2013

Permutation of the 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.
Enumeration table T(n,k). The order of the list:
T(1,1) = 1;
T(1,3), T(2,2), T(1,2), T(2,1), T(3,1);
. . .
T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.
Table contains the following:
row 1 is alternation of elements A130883 and A100037,
row 2 accommodates elements A033816 in even places;
column 1 is alternation of elements A000384 and A014106,
column 2 is alternation of elements A091823 and A071355,
column 4 accommodates elements A130861 in odd places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,
diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,
diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

			The start of the sequence as a table:
   1   4   2   9   7   8  16 ...
   5   3  10   8  19  17  32 ...
   6  13  11  22  20  35  33 ...
  14  12  23  21  36  34  53 ...
  15  26  24  39  37  56  54 ...
  27  25  40  38  57  55  78 ...
  28  43  41  60  58  81  79 ...
  ...
The start of the sequence as a triangle array read by rows:
   1
   4  5
   2  3  6
   9 10 13 14
   7  8 11 12 15
  18 19 22 23 26 27
  16 17 20 21 24 25 28
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
   1
   4  5  2  3  6
   9 10 13 14  7  8 11 12 15
  18 19 22 23 26 27 16 17 20 21 24 25 28
  ...
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+6, 2*r*r-5*r+7, ..., 2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
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. A000384, A002260, A003056, A003057, A004736, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135, A194431, A211377.

T:=(n,k)->((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 - (-1)^n - (-1)^(n+k)(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3-(-1)**i-(t+2)*(-1)**t)/2

As a table:
T(n,k) = ((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2.
As a linear sequence:
a(n) = (A003057(n)^2-4*A004736(n)+3-(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3-(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 4, 2, 3, 6, 10, 9, 14, 13, 7, 8, 11, 12, 15, 19, 18, 23, 22, 27, 26, 16, 17, 20, 21, 24, 25, 28, 32, 31, 36, 35, 40, 39, 44, 43, 29, 30, 33, 34, 37, 38, 41, 42, 45, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 70

Offset: 1

Boris Putievskiy, Feb 15 2013

Permutation of the 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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);
. . .
T(1,2*n+1), T(2,2*n), T(2,2*n-1), T(1,2*n), ...T(2*n-1,3), T(2*n,2), T(2*n,1), T(2*n-1,2), T(2*n+1,1);
. . .
Movement along two adjacent antidiagonals - step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.
Table contains:
row  1 accommodates elements A130883 in odd places,
row  2 is alternation of elements A100037 and A033816;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A014106 and A071355,
column 3 accommodates elements A130861 in even places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal accommodates elements A033566 in even places,
diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,
diagonal 3, located above the main diagonal  accommodates of elements A033585.

			The start of the sequence as table:
1....5...2..10...7..19..16...
4....3...9...8..18..17..31...
6...14..11..23..20..36..33...
13..12..22..21..35..34..52...
15..27..24..40..37..57..54...
26..25..39..38..56..55..77...
28..44..41..61..58..82..79...
. . .
The start of the sequence as triangle array read by rows:
1;
5,4;
2,3,6;
10,9,14,13;
7,8,11,12,15;
19,18,23,22,27,26;
16,17,20,21,24,25,28;
. . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
1;
5,4,2,3,6;
10,9,14,13,7,8,11,12,15;
19,18,23,22,27,26,16,17,20,21,24,25,28;
. . .
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+7, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
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. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.

T:=(n,k)->((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 + (-1)^k - 2(-1)^n - (n+k)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3+(-1)**j-2*(-1)**i-(t+2)*(-1)**t)/2

As table
T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2.
As linear sequence
a(n) = (A003057(n)^2-4*A004736(n)+3+(-1)^A004736(n)-2*(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3+(-1)^j-2*(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

cp's OEIS Frontend

A139277 a(n) = n*(8*n+5).

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A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

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A213171 T(n,k) = ((k+n)^2 - 4*k + 3 - (-1)^n - (k+n)*(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

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A213205 T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

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A139277 a(n) = n(8n+5).

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.