A213759 -id:A213759 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A047838 a(n) = floor(n^2/2) - 1.

Original entry on oeis.org

1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403

Offset: 2

Michael Somos, May 07 1999

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006
From Dmitry Kamenetsky, Nov 29 2006: (Start)
This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:
  1 0 1 1 1
  1 0 1 0 1
  1 0 1 0 1
  1 0 0 0 1
  1 1 1 1 1 (End)
It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008
a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012
The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014
Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018
a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

			x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
Graham Cormode, Notes on the organization number of a permutation.
Eric Weisstein's World of Mathematics, Longest Path Problem.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Complement of A047839. First difference is A052928.
Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.
Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

[Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)
Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PARI
```
a(n) = n^2\2 - 1
```

a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003
a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006
Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007
G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008
a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010
a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011
a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011
a(n) = a(-n) = A007590(n) - 1.
a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015
a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017
From Guenther Schrack, May 12 2018: (Start)
Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
a(n) = A000982(n-1) + n - 2 for n > 1.
a(n) = 2*A033683(n) - 3 for n > 1.
a(n) = A061925(n-1) + n - 3 for n > 1.
a(n) = A074148(n) - n - 1 for n > 1.
a(n) = A105343(n-1) + n - 4 for n > 1.
a(n) = A116940(n-1) - n for n > 1.
a(n) = A179207(n) - n + 1 for n > 1.
a(n) = A183575(n-2) + 1 for n > 2.
a(n) = A265284(n-1) - 2*n + 1 for n > 1.
a(n) = 2*A290743(n) - 5 for n > 1. (End)
E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021
Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

Edited by Charles R Greathouse IV, Apr 23 2010

A213783 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 3, 1, 6, 4, 2, 11, 8, 6, 2, 17, 14, 11, 7, 3, 26, 22, 19, 13, 9, 3, 36, 32, 28, 22, 16, 10, 4, 50, 45, 41, 33, 27, 18, 12, 4, 65, 60, 55, 47, 39, 30, 21, 13, 5, 85, 79, 74, 64, 56, 44, 35, 23, 15, 5, 106, 100, 94, 84, 74, 62, 50, 38, 26, 16, 6, 133, 126, 120, 108

Offset: 1

Clark Kimberling, Jun 22 2012

Principal diagonal: A213759.
Antidiagonal sums: A213760.
Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744.
Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...).
Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...3....6....11...17...26...36....50
1...4....8....14...22...32...45....60
2...6....11...19...28...41...55....74
2...7....13...22...33...47...64....84
3...9....16...27...39...56...74....98
3...10...18...30...44...62...83....108
4...12...21...35...50...71...93....122
4...13...23...38...55...77...102...132

Clark Kimberling, Antidiagonals n = 1..80, flattened

Cf. A213500.

b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213783 *)
Table[t[n, n], {n, 1, 40}] (* A213759 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213760 *)

T(n,k) = 2*T(n,k-1) -T(n,k-2) - 4*T(n,k-3) +T(n,k-4) +2*T(n,k-5) -T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = [(n+1)/2] + [(n+2)/2]*x + ([(n-1)/2] + [(n+1)/2])*x^2 - (1+[n/2]-(n mod 2))*x^3 + [n/2]*x^4 and g(x) = (1 + x)^2 *(1 - x)^4, where [ ] = floor.

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

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A047838 a(n) = floor(n^2/2) - 1.

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A213783 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

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cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A047838 a(n) = floor(n^2/2) - 1.

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Magma

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A213783 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A213783 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.