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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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A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 7, 6, 4, 13, 18, 10, 5, 21, 36, 31, 15, 6, 31, 60, 64, 50, 21, 7, 43, 90, 109, 105, 71, 28, 8, 57, 126, 166, 180, 151, 98, 36, 9, 73, 168, 235, 275, 261, 210, 127, 45, 10, 91, 216, 316, 390, 401, 364, 274, 162, 55, 11, 111, 270, 409, 525, 571, 560, 477, 351, 199, 66
Offset: 1

Views

Author

Lamine Ngom, Sep 05 2021

Keywords

Comments

A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).
In column k, every term is the arithmetic mean of its neighbors minus A000217(k).

Examples

			Array, A(n, k), begins:
  1  3   6  10  15   21   28   36   45 ... A000217;
  2  7  18  31  50   71   98  127  162 ... A195605;
  3 13  36  64 105  151  210  274  351 ...
  4 21  60 109 180  261  364  477  612 ...
  5 31  90 166 275  401  560  736  945 ...
  6 43 126 235 390  571  798 1051 1350 ...
  7 57 168 316 525  771 1078 1422 1827 ...
  8 73 216 409 680 1001 1400 1849 2376 ...
  9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
   1;
   2,  3;
   3,  7,   6;
   4, 13,  18,  10;
   5, 21,  36,  31,  15;
   6, 31,  60,  64,  50,  21;
   7, 43,  90, 109, 105,  71,  28;
   8, 57, 126, 166, 180, 151,  98,  36;
   9, 73, 168, 235, 275, 261, 210, 127,  45;
  10, 91, 216, 316, 390, 401, 364, 274, 162,  55;

Links

Crossrefs

Family of sequences (k*n + 1)^2: A016754 (k=2), A016778 (k=3), A016814 (k=4), A016862 (k=5), A016922 (k=6), A016994 (k=7), A017078 (k=8), A017174 (k=9), A017282 (k=10), A017402 (k=11), A017534 (k=12), A134934 (k=14).
Cf. A000027, A000217, A002061, A028896, A085473, A195605.

Programs

  • Magma
    A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n,k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022
  • Mathematica
    A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n,k-1]; Table[Function[n, A[n, k]][m-k+1], {m,0,10}, {k,0,m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)
  • SageMath
    def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # G. C. Greubel, Dec 25 2022

Formula

A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.
A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.
A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.
A(n, 0) = A000027(n).
A(n, 1) = A002061(n+1).
A(n, 2) = A028896(n).
A(n, 3) = A085473(n).
From G. C. Greubel, Dec 25 2022: (Start)
A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).
T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).
Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)

A174679 a(4n) = n^2. a(4n+1) = (4n-1)^2. a(4n+2) = (2n)^2. a(4n+3) = (4n+1)^2.

Original entry on oeis.org

0, 1, 0, 1, 1, 9, 4, 25, 4, 49, 16, 81, 9, 121, 36, 169, 16, 225, 64, 289, 25, 361, 100, 441, 36, 529, 144, 625, 49, 729, 196, 841, 64, 961, 256, 1089, 81, 1225, 324, 1369, 100, 1521, 400, 1681, 121, 1849, 484, 2025, 144, 2209, 576
Offset: 0

Views

Author

Paul Curtz, Nov 30 2010

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,1,0,1,1,9,4,25,4,49,16,81},80] (* Harvey P. Dale, Apr 01 2018 *)

Formula

a(2n) = A174595(n).
a(2n+1) = A016754(n-1) = (2n-1)^2, n>0.
a(4n+1) = A016838(n-1).
a(4n+2) = A016742(n).
a(4n+3) = A016814(n).
a(n)= +3*a(n-4) -3*a(n-8) +a(n-12).
G.f.: -x*(1+x^2+x^3+6*x^4+4*x^5+22*x^6+x^7+25*x^8+4*x^9+9*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 01 2010
a(n) = ((16-(1+(-1)^n)*(5+i^n))*n-4*(8-(1+(-1)^n)*(3+i^n)))^2/256, where i=sqrt(-1). - Bruno Berselli, Jan 27 2011 - Apr 09 2011

A381196 Stellated octagon numbers: a(n) = 20*n^2 + 8*n + 1.

Original entry on oeis.org

1, 29, 97, 205, 353, 541, 769, 1037, 1345, 1693, 2081, 2509, 2977, 3485, 4033, 4621, 5249, 5917, 6625, 7373, 8161, 8989, 9857, 10765, 11713, 12701, 13729, 14797, 15905, 17053, 18241, 19469, 20737, 22045, 23393, 24781, 26209, 27677, 29185, 30733, 32321, 33949
Offset: 0

Views

Author

Aaron David Fairbanks, Feb 16 2025

Keywords

Comments

Sequence found by reading the line from 1, in the direction 1, 29, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Feb 17 2025

Examples

			Illustration of initial terms:
.
.                                     o
.                                   o o o
.                             o o o o o o o o o
.              o              o o o o o o o o o
.          o o o o o          o o o o o o o o o
.          o o o o o        o o o o o o o o o o o
.   o    o o o o o o o    o o o o o o o o o o o o o
.          o o o o o        o o o o o o o o o o o
.          o o o o o          o o o o o o o o o
.              o              o o o o o o o o o
.                             o o o o o o o o o
.                                   o o o
.                                     o
.
.   1          29                     97

Links

Crossrefs

Cf. A000217, A001844, A016742, A016814, A168668, A195162.

Programs

  • Mathematica
    a[n_] := 20*n^2 + 8*n + 1; Array[a, 42, 0] (* Amiram Eldar, Feb 17 2025 *)

Formula

a(n) = (4*n + 1)^2 + 4*n^2.
a(n) = A001844(3*n - 2) + 4*A000217(n - 1).
a(n) = 4 * A168668(n) + 1.
a(n) = A016814(n) + A016742(n).
G.f.: (13*x^2+26*x+1) / (1-x)^3.
E.g.f.: exp(x) * (1 + 28*x + 20*x^2). - Stefano Spezia, Feb 23 2025
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