The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A053088

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A053088

a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.
(history; published version)

#100 by Michael De Vlieger at Sat Mar 30 21:19:39 EDT 2024

STATUS

proposed

approved

#99 by Robert C. Lyons at Sat Mar 30 20:03:11 EDT 2024

STATUS

editing

proposed

#98 by Robert C. Lyons at Sat Mar 30 20:03:09 EDT 2024

COMMENTS

Growth of happy bug population in GCSE maths math course work assignment.

STATUS

approved

editing

#97 by N. J. A. Sloane at Thu Nov 04 22:31:00 EDT 2021

STATUS

editing

approved

#96 by N. J. A. Sloane at Thu Nov 04 22:30:56 EDT 2021

COMMENTS

From Paul Curtz, Nov 02 2021 (Start)

a(n-2) difference table (from 0, 0, a(n)):

0 0 1 0 3 2 9 12 31 54 ...

0 1 -1 3 -1 7 3 19 23 63 ...

1 -2 4 -4 8 -4 16 4 40 44 ...

-3 6 -8 12 -12 20 -12 36 4 84 ...

9 -14 20 -24 32 -32 48 -32 80 0 ...

-23 34 -44 56 -64 80 -80 112 -80 176 ...

57 -78 100 -120 144 -160 192 -192 256 -192 ...

... .

The signature is valid for every row.

a(n-2) + a(n-1) = A001045(n).

a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).

a(n-2) + a(n+3) = see A144472(n+1).

Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).

First subdiagonal: -A036895(n) = -2*A001787(n).

Main diagonal: A001787(n) = -first and -third upper diagonals.

Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

FORMULA

From Paul Curtz, Nov 02 2021 (Start)

a(n-2) + a(n-1) = A001045(n), for a(-1) = a(-2) = 0 here and below too.

a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).

a(n-2) + a(n+3) = A144472(n+1).

b(n) = A091919(n-4), if b >= 4 else b(n) = 0.

a(n-2) = b(n+2) - 2*b(n+1) + b(n).

a(n) - a(n-2) = A078008(n) (End).

From Paul Curtz and Thomas Scheuerle, Nov 02 2021 (Start)

A014480(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(n, k)*(a(2*n-k) - 2*a(2*n-k-1) - a(2*n-k-2)), if a(-1) = a(-2) = 0.

a(n) = Sum_{k=0..floor((n+2)/2)}*binomial(n-k+2, k)*A001787(k)

- Sum_{k=0..floor((n+1)/2)}*binomial(n-k+1, k)*A036289(k).

a(n+1) = Sum_{k=0..floor((n+1)/2)}*binomial(n-k+1, k)*A001792(k)

- Sum_{k=0..floor(n/2)}*binomial(n-k, k)*A001787(k+1).

a(n+3) = Sum_{k=0..floor((n+1)/2)}*binomial(n-k+1, k)*A045891(k)

+ Sum_{k=0..floor(n/2)}*binomial(n-k, k)*A001787(k+1).

a(n+4) = Sum_{k=0..floor((n+1)/2)}*binomial(n-k+1, k)*A172160(k+1)

+ Sum_{k=0..floor(n/2)}*binomial(n-k, k)*A045891(k+3).

(End)

CROSSREFS

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036289, A036895, A062510, A045891, A172160.

Cf. A045891, A144472, A159697, A172160, A232603, A232604.

Cf. A034299, A078008, A091919.

STATUS

approved

editing

#95 by N. J. A. Sloane at Thu Nov 04 22:29:31 EDT 2021

STATUS

proposed

approved

#94 by Paul Curtz at Thu Nov 04 09:19:39 EDT 2021

STATUS

editing

proposed

#93 by Paul Curtz at Thu Nov 04 09:12:53 EDT 2021

FORMULA

a(n-2) = b(n+2) - 2*b(n+1) + b(n)0.

STATUS

proposed

editing

Discussion

Thu Nov 04

09:19

Paul Curtz: Please do as you like.

#92 by Thomas Scheuerle at Thu Nov 04 02:49:43 EDT 2021

STATUS

editing

proposed

Discussion

Thu Nov 04

03:09

Thomas Scheuerle: Idea:  What if we put the original difference table + all the binomials into a LaTex-PDF document and attach this to this sequence. In the sequence it self then only a comment that the attached file shows how other sequences appear there ?

04:27

Kevin Ryde: Uploads usually have a lower (if that's the word!) standard of critique.  It'd have to be pretty awful before upload notes were rejected.  Obviously the aim is still good interesting relevant work, but freedom of expression opened up.

#91 by Thomas Scheuerle at Thu Nov 04 02:43:09 EDT 2021

FORMULA

a(n-2) + a(n-1) = A001045(n), if for a(-1) = a(-2) = 0 here and below too.

a(n-2) + a(n+1) = A062510(n) = 3*A001045(n), if a(-1) = a(-2) = 0.

a(n-2) + a(n+3) = A144472(n+1), if a(-1) = a(-2) = 0.

a(n-2) = b(n+2) - 2*b(n+1) + b(n), for a(-1) = a(-2) = 0.

STATUS

proposed

editing

Discussion

Thu Nov 04

02:49

Thomas Scheuerle: Dear Paul, I stop editing here, because I have hopelessly messed up your work. I am sorry. Now I see here at least three options:
1. Revert back to comment with difference table.
2. Start new sequence from difference table as proposed by Kevin.
3. leave it up to the editors.
4. Remove something ...
5. .....