A079808 -id:A079808 - cp's OEIS frontend

A079809 Consider a triangle in which the 2n-th row contains first 2n positive integers in increasing order and the (2n+1)-st row contains first 2n+1 positive integers in decreasing order; sequence contains concatenation of numbers read upward at a 45-degree angle.

1, 1, 32, 12, 521, 143, 7234, 1632, 92541, 18345, 1127436, 1103652, 13294561, 11238547, 1521147638, 1143105672, 17213496581, 11631258749, 19215411678310, 1183145107692, 212174136985101, 120316512789411, 232194156118710312

Offset: 1

Amarnath Murthy, Feb 10 2003

			Triangle begins with:
  1;
  1 2;
  3 2 1;
  1 2 3 4;
  5 4 3 2 1;
  1 2 3 4 5 6;

G. C. Greubel, Table of n, a(n) for n = 1..715

Cf. A079808, A079810.

a[n_] := FromDigits[Join@@Table[IntegerDigits[If[OddQ[n+1-i], n+2-2i, i]], {i, 1, Ceiling[n/2]}]]

A079810 Sums of diagonals (upward from left to right) of the triangle shown in A079809.

Original entry on oeis.org

1, 1, 5, 3, 8, 8, 16, 12, 21, 21, 33, 27, 40, 40, 56, 48, 65, 65, 85, 75, 96, 96, 120, 108, 133, 133, 161, 147, 176, 176, 208, 192, 225, 225, 261, 243, 280, 280, 320, 300, 341, 341, 385, 363, 408, 408, 456, 432, 481, 481, 533, 507, 560, 560, 616, 588, 645, 645

Offset: 1

Amarnath Murthy, Feb 10 2003

nonn

			a(7) = T(7,1) + T(6,2) + T(5,3) + T(4,4) = 7 + 2 + 3 + 4 = 16.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A079808, A079809, A079811, A092542.

R:=PowerSeriesRing(Integers(), 71); Coefficients(R!( x*(1+4*x^2-2*x^3+3*x^4)/((1-x)*(1-x^4)^2) )); // G. C. Greubel, Dec 12 2023

LinearRecurrence[{1,0,0,2,-2,0,0,-1,1}, {1,1,5,3,8,8,16,12,21}, 70] (* G. C. Greubel, Dec 12 2023 *)

def A079810_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+4*x^2-2*x^3+3*x^4)/((1-x)*(1-x^4)^2) ).list()
a=A079810_list(71); a[1:] # G. C. Greubel, Dec 12 2023

a(4k) = 3k^2. a(4k+1) = a(4k+2) = 3k^2+4k+1. a(4k+3) = 3k^2+8k+5.
From Chai Wah Wu, Feb 03 2021: (Start)
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(1 + 4*x^2 - 2*x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2*(1 + x^2)^2). (End)
From G. C. Greubel, Dec 12 2023: (Start)
a(n) = (1/32)*( (6*n^2 + 14*n + 5) - (-1)^n*(10*n + 9) + 2*((3 - i)*(-i)^n + (3 + i)*i^n) - 8*(-1)^floor(n/2)*floor((n+2)/2) ).
E.g.f.: 4*(1-x)*cos(x) - 4*(2-x)*sin(x) + 2*(3*x^2 + 15*x - 2)*cosh(x) 2*(3*x^2 + 5*x + 7)*sinh(x). (End)

Edited by David Wasserman, May 11 2004

A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

Original entry on oeis.org

1, 2, 43, 75, 1186, 16129, 22171310, 29231814, 3730241915, 4638312520, 564739322621, 675748403327, 79685849413428, 92806959504235, 10693817060514336, 121107948271615244, 137122108958372625345, 1541381231099684736354, 1721551391241109785746455

Offset: 1

Amarnath Murthy, Feb 11 2003

   1
   2  3
   4  5  6
   7  8  9 10
  11 12 13 14 15
  16 17 18 19 20 21
  ...
a(n) also is the concatenation of the terms of the n-th row of A056536. - Michel Marcus, Dec 14 2023

G. C. Greubel, Table of n, a(n) for n = 1..395

Cf. A056536, A079808, A079824.

read("transforms"):
A079823aux := proc(n,k)
    A000124(n)+k ;
end proc:
A079823 := proc(n)
    local L,k,n0 ;
    n0 := n-1 ;
    L := [] ;
    for k from 0 do
        if k > n0-k then
            break;
        end if;
        L := [op(L),A079823aux(n0-k,k)] ;
    end do:
    digcatL(L) ;
end proc: # R. J. Mathar, Aug 23 2012
# second Maple program:
T:= (i, j)-> i*(i-1)/2+j:
a:= n-> parse(cat(seq(T(n-j,j+1), j=0..(n-1)/2))):
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 03 2022

Table[FromDigits[Join@@IntegerDigits[Table[Binomial[n-k+1,2] + k, {k, Ceiling[n/2]}]]], {n,30}] (* G. C. Greubel, Dec 13 2023 *)

More terms from Jason D. W. Taff (jtaff(AT)jburroughs.org), Oct 31 2003

Corrected by Philippe Deléham, Feb 16 2004

A079811 Sum of numbers read upward at a 45-degree angle in A079809.

Original entry on oeis.org

1, 2, 2, 6, 7, 10, 10, 18, 19, 24, 24, 36, 37, 44, 44, 60, 61, 70, 70, 90, 91, 102, 102, 126, 127, 140, 140, 168, 169, 184, 184, 216, 217, 234, 234, 270, 271, 290, 290, 330, 331, 352, 352, 396, 397, 420, 420, 468, 469, 494, 494, 546, 547, 574, 574, 630, 631

Offset: 1

Amarnath Murthy, Feb 10 2003

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A079808, A092542.

The two bisections are A309805 and twice A211538 (with leading zeros dropped).

Terms a(8) and beyond from Andrey Zabolotskiy, Jan 18 2024

cp's OEIS Frontend

A079809 Consider a triangle in which the 2n-th row contains first 2n positive integers in increasing order and the (2n+1)-st row contains first 2n+1 positive integers in decreasing order; sequence contains concatenation of numbers read upward at a 45-degree angle.

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A079810 Sums of diagonals (upward from left to right) of the triangle shown in A079809.

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A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

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A079811 Sum of numbers read upward at a 45-degree angle in A079809.

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