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.

User: John Molokach

John Molokach's wiki page.

John Molokach has authored 4 sequences.

A225800 Triangle of rising diagonals of A011973 (with rows displayed as centered text).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 5, 4, 1, 1, 7, 6, 1, 1, 1, 9, 8, 6, 3, 1, 1, 11, 10, 15, 10, 1, 1, 13, 12, 28, 21, 1, 1, 1, 15, 14, 45, 36, 10, 4, 1, 1, 17, 16, 66, 55, 35, 20, 1, 1, 19, 18, 91, 78, 84, 56, 1, 1, 1, 21, 20, 120, 105, 165, 120, 15, 5, 1
Offset: 1

Views

Author

John Molokach, Jul 27 2013

Keywords

Comments

Row sums are A227300.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  1;
  1,  1,  1;
  1,  1,  3,  2;
  1,  1,  5,  4;
  1,  1,  7,  6,  1;
  1,  1,  9,  8,  6,  3;
  1,  1, 11, 10, 15, 10;
  1,  1, 13, 12, 28, 21,  1;

Crossrefs

Cf. A011973, A000045, A227300, A005314, A224838.

Programs

  • Mathematica
    Table[Binomial[2 n - k - 2 - 3 Floor[k/2], Floor[k/2]], {n, 1, 25}, {k, 0, Floor[(2 n - 1)/3]}] (* John Molokach, Jul 29 2013 *)

Formula

r(n) = binomial(2n-k-2-3*floor(k/2), floor(k/2)), k = 0..floor((2n-1)/3). - John Molokach, Jul 29 2013

A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).

Original entry on oeis.org

0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152
Offset: 0

Views

Author

John Molokach, Jul 27 2013

Keywords

Comments

This sequence is part of a family of Fibonacci-like sequences, where:
Sum_{k=0..n} binomial(n,k)*m^(n-k)*Fibonacci(n+k) produces a sequence whose terms are divisible by (m+1); m>=1.
A recurrence relation for a(n) (m not equal to zero) is:
a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
Notable values of m include:
m = 1: Fibonacci(3n),
m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
m = -1: the zero sequence,
m = -2: (-1)*Fibonacci(n), or A152163(n+2).
For any value of m, the sequence gives a(n*k) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
Equivalent sequences are given by: Sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.

Links

Crossrefs

Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13,109},{0,11},30] (* Harvey P. Dale, Jul 31 2018 *)

Formula

a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by Georg Fischer, May 10 2019

A224838 Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 4, 6, 5, 1, 1, 10, 10, 6, 1, 1, 5, 20, 15, 7, 1, 6, 15, 35, 21, 8, 1, 1, 21, 35, 56, 28, 9, 1, 1, 7, 56, 70, 84, 36, 10, 1, 8, 28, 126, 126, 120, 45, 11, 1, 1, 36, 84, 252, 210, 165, 55, 12, 1, 1, 9, 120, 210, 462, 330, 220, 66, 13, 1
Offset: 1

Views

Author

John Molokach, Jul 21 2013

Keywords

Comments

The successive rows have lengths 1,2,2; 3,4,4; 5,6,6; 7,8,8; ...
Sum of row n is A005314(n).
Old definition was: "Triangle of falling diagonals of A011973 (with rows displayed as centered text)."

Examples

			First 11 rows of the triangle:
  1;
  1,  1;
  2,  1;
  1,  3,  1;
  1,  3,  4,  1;
  4,  6,  5,  1;
  1, 10, 10,  6,  1;
  1,  5, 20, 15,  7,  1;
  6, 15, 35, 21,  8,  1;
  1, 21, 35, 56, 28,  9,  1;
  1,  7, 56, 70, 84, 36, 10,  1;

Links

Crossrefs

Cf. A005314, A227300, A001973, A000045, A004396.

Programs

  • Mathematica
    Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* T. D. Noe, Jul 25 2013 *)
Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* John Molokach, Jul 25 2013 *)

Formula

r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), k = 1..floor((2n+2)/3).
R(n) = binomial(n-floor((k+1)/2), n-floor((3k-1)/2)), k = 1..floor((2n+2)/3), gives the terms of each row in reverse order.

Extensions

Entry revised by N. J. A. Sloane, Jul 07 2024

A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text).

Original entry on oeis.org

1, 2, 2, 3, 7, 11, 16, 28, 48, 77, 126, 211, 349, 573, 947, 1568, 2588, 4271, 7058, 11661, 19256, 31804, 52538, 86779, 143329, 236744, 391046, 645900, 1066850, 1762163, 2910634, 4807590, 7940870, 13116238, 21664568, 35784145, 59105987, 97627533, 161254953, 266350689
Offset: 1

Views

Author

John Molokach, Jul 09 2013

Keywords

Comments

Rising diagonal sums of triangle A011973, taken with rows as centered text.

Examples

			a(1)  = 1;
a(2)  = 1 +  1;
a(3)  = 1 +  1;
a(4)  = 1 +  1 +  1;
a(5)  = 1 +  1 +  3 +  2;
a(6)  = 1 +  1 +  5 +  4;
a(7)  = 1 +  1 +  7 +  6 +  1;
a(8)  = 1 +  1 +  9 +  8 +  6 +  3;
a(9)  = 1 +  1 + 11 + 10 + 15 + 10;
a(10) = 1 +  1 + 13 + 12 + 28 + 21 +  1.

Links

Crossrefs

Cf. A011973 (triangle), A000045 (row sums of triangle), A005314 (falling diagonal sums of triangle). Expansion of terms begin with A055624 at a(1) and adds A016813 at a(4), A016754 at a(7), and A100157 at a(10).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 2, 0, 0, -1}, {1, 2, 2, 3, 7, 11}, 40] (* T. D. Noe, Jul 11 2013 *)
  • PARI
    a(n) = if(n<=1, 1, sum(k=0, floor((n-1)/3), binomial(2*n-2-5*k,k)+binomial(2*n-1-5*k,k)) ); \\ Joerg Arndt, Jul 11 2013

Formula

a(n) = Sum_{k=0..floor((n-1)/3)} (binomial(2*n-2-5*k,k) + binomial(2*n-3-5*k,k)) for n >= 2; a(1)=1. - John Molokach, Jul 11 2013
a(n) = a(n-1) + 2*a(n-3) - a(n-6), starting with {1, 2, 2, 3, 7, 11}. - T. D. Noe, Jul 11 2013
G.f.: x*(1+x-x^3)/(1-x-2*x^3+x^6) - John Molokach, Jul 15 2013
a(n) = Sum_{k=0..floor((2n-1)/3)} binomial(2n-k-2-3*floor(k/2),floor(k/2)). - John Molokach, Jul 29 2013

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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