A079809 -id:A079809 - cp's OEIS frontend

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

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

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

A079808 Consider a triangle in which the 2n-th row contains first 2n positive integers in decreasing order and the (2n+1)-st row contains first 2n+1 positive integers in increasing order; sequence contains concatenation of numbers read upward at a 45-degree angle.

Original entry on oeis.org

1, 2, 11, 42, 133, 622, 1531, 8244, 17335, 102642, 193551, 1228446, 11137537, 142104662, 113395571, 1621248648, 11531157739, 182144106682, 117313597591, 20216412688410, 119315511779311, 2221841461086102, 1213175137995111

Offset: 1

Amarnath Murthy, Feb 10 2003

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

G. C. Greubel, Table of n, a(n) for n = 1..650

Cf. A079809.

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

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

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

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Extensions

A079808 Consider a triangle in which the 2n-th row contains first 2n positive integers in decreasing order and the (2n+1)-st row contains first 2n+1 positive integers in increasing order; sequence contains concatenation of numbers read upward at a 45-degree angle.

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Programs

Mathematica