cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233
Offset: 1

Views

Author

Reinhard Zumkeller, May 20 2006

Keywords

Examples

			Triangle begins as:
   1;
   2,  2;
   3,  4,  3;
   4,  6,  6,  5;
   5,  8,  9, 10,  8;
   6, 10, 12, 15, 16, 13;
   7, 12, 15, 20, 24, 26,  21;
   8, 14, 18, 25, 32, 39,  42,  34;
   9, 16, 21, 30, 40, 52,  63,  68,  55;
  10, 18, 24, 35, 48, 65,  84, 102, 110,  89;
  11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
  12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;

Links

Crossrefs

Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Cf. A023652, A112469.

Programs

  • Magma
    A119457:= func< n,k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025
  • Mathematica
    (* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Second program *)
A119457[n_,k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n,k], {n,13}, {k,n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)
  • SageMath
    def A119457(n,k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Apr 16 2025

Formula

T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)

A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 1, 10, 5, 2, 21, 14, 9, 3, 40, 31, 24, 14, 5, 72, 61, 52, 38, 23, 8, 125, 112, 101, 83, 62, 37, 13, 212, 197, 184, 162, 135, 100, 60, 21, 354, 337, 322, 296, 263, 218, 162, 97, 34, 585, 566, 549, 519, 480, 425, 353, 262, 157, 55, 960, 939, 920, 886
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Comments

Principal diagonal: A213766.
Antidiagonal sums: A213767.
Row 1, (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.
Row 2, (1,3,5,7,9,...)**(1,2,3,5,8,...): A023652.
Row 3, (1,3,5,7,9,...)**(2,3,5,8,13,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....4....10....21....40....72
1....5....14....31....61....112
2....9....24....52....101...184
3....14...38....83....162...296
5....23...62....135...263...480
8....37...100...218...425...776
13...60...162...353...688...1256

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := 2 n - 1; c[n_] := Fibonacci[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}]  (* A213765 *)
Table[t[n, n], {n, 1, 40}] (* A213766 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213767 *)

Formula

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(F(n) + F(n+1)*x - F(n-1)*x^2) and g(x) = (1 - x - x^2)(1 - x )^2.
T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - Ehren Metcalfe, Jul 10 2019

A213774 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 14, 11, 5, 31, 26, 17, 7, 61, 53, 38, 23, 9, 112, 99, 75, 50, 29, 11, 197, 176, 137, 97, 62, 35, 13, 337, 303, 240, 175, 119, 74, 41, 15, 566, 511, 409, 304, 213, 141, 86, 47, 17, 939, 850, 685, 515, 368, 251, 163, 98, 53, 19, 1545, 1401, 1134
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Comments

Principal diagonal: A213775.
Antidiagonal sums: A213776.
Row 1, (1,2,3,5,8,...)**(1,3,5,7,9,...): A023652.
Row 2, (1,2,3,5,8,...)**(3,5,7,9,11,...).
Row 3, (1,2,3,5,8,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...31....61....112
3....11...26...53....99....176
5....17...38...75....137...240
7....23...50...97....175...304
9....29...62...119...213...368
11...35...74...141...251...432

Links

Crossrefs

Cf. A023652, A213500, A213768, A213775, A213776.

Programs

  • Mathematica
    b[n_] := Fibonacci[n + 1]; c[n_] := 2 n - 1;
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}]  (* A213774 *)
Table[t[n, n], {n, 1, 40}] (* A213775 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213776 *)

Formula

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 + 2*x - (2*n - 3)*x^2) and g(x) = (1 - x - x^2)*(1 - x )^2.
T(n,k) = 2*n*Fibonacci(k+3) + Lucas(k+3) - 4*(k+n+1). - Ehren Metcalfe, Jul 08 2019
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