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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.

User: David Z Crookes

David Z Crookes's wiki page.

David Z Crookes has authored 3 sequences.

A341043 a(n) = 16*n^3 - 36*n^2 + 30*n - 9.

Original entry on oeis.org

1, 35, 189, 559, 1241, 2331, 3925, 6119, 9009, 12691, 17261, 22815, 29449, 37259, 46341, 56791, 68705, 82179, 97309, 114191, 132921, 153595, 176309, 201159, 228241, 257651, 289485, 323839, 360809, 400491, 442981, 488375, 536769, 588259, 642941, 700911, 762265
Offset: 1

Views

Author

David Z Crookes, Feb 03 2021

Keywords

Comments

The n-th term of A155883 (hexagonal bifrustum numbers) has a hexagonal pyramid of [n - 1] set on each of its two hexagonal faces.
The digital roots run recursively 1, 8, 9.
The sum of the first n consecutive terms is the square of the n-th hexagonal number.

Examples

			For n = 3 the solution is 173 + 8 + 8 = 189.

Links

Crossrefs

Cf. A000384, A000578, A155883.

Programs

  • PARI
    Vec(x*(9*x^3+55*x^2+31*x+1)/(x-1)^4 + O(x^38)) \\ Elmo R. Oliveira, Sep 01 2025

Formula

a(n) = 16*n^3 - 36*n^2 + 30*n - 9.
a(n) = A155883(n) + 2*A000578(n-1).
G.f.: x*(1 + 31*x + 55*x^2 + 9*x^3)/(1 - x)^4. - Stefano Spezia, Feb 04 2021
From Elmo R. Oliveira, Sep 01 2025: (Start)
E.g.f.: 9 + exp(x)*(-9 + 10*x + 12*x^2 + 16*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Extensions

More terms from Elmo R. Oliveira, Sep 01 2025

A338795 Each term of A003215 (centered hexagonal numbers) is multiplied by the corresponding term of A003154 (centered dodecagonal numbers).

Original entry on oeis.org

1, 91, 703, 2701, 7381, 16471, 32131, 56953, 93961, 146611, 218791, 314821, 439453, 597871, 795691, 1038961, 1334161, 1688203, 2108431, 2602621, 3178981, 3846151, 4613203, 5489641, 6485401, 7610851, 8876791, 10294453, 11875501, 13632031, 15576571, 17722081, 20081953
Offset: 1

Views

Author

David Z Crookes, Nov 09 2020

Keywords

Comments

The digital root (A010888) of each term is 1.

Examples

			The centered hexagonal number of 4 is 37, and the centered dodecagonal number of 4 is 73, so the fourth term of the series is 37*73 = 2701.

Links

Crossrefs

Cf. A003154, A003215, A010888.

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{1,91,703,2701,7381},40] (* Harvey P. Dale, May 13 2022 *)

Formula

a(n) = A003215(n)*A003154(n).
a(n) = 18*n^4 - 36*n^3 + 27*n^2 - 9*n + 1.
From Elmo R. Oliveira, Sep 01 2025: (Start)
G.f.: -x*(1 + 86*x + 258*x^2 + 86*x^3 + x^4)/(x - 1)^5.
E.g.f.: -1 + exp(x)*(1 + 45*x^2 + 72*x^3 + 18*x^4).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

A155883 a(n) = 14*n^3 - 30*n^2 + 24*n - 7.

Original entry on oeis.org

1, 33, 173, 505, 1113, 2081, 3493, 5433, 7985, 11233, 15261, 20153, 25993, 32865, 40853, 50041, 60513, 72353, 85645, 100473, 116921, 135073, 155013, 176825, 200593, 226401, 254333, 284473, 316905, 351713, 388981, 428793, 471233, 516385, 564333, 615161, 668953
Offset: 1

Views

Author

David Z Crookes, Jan 29 2009

Keywords

Comments

A three-dimensional version of the centered hexagonal numbers (A003215). Two examples: the third term 173 is built up as 19 + 37 + 61 + 37 + 19 and the fourth term 505 is built up as 37 + 61 + 91 + 127 + 91 + 61 + 37.
The sequence's digital roots are 1, 6, 2 (repeat).

Links

Crossrefs

Cf. A003215.

Programs

  • Magma
    I:=[1, 33, 173, 505]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]];  // Vincenzo Librandi, Jun 30 2012
  • Mathematica
    CoefficientList[Series[(1+29*x+47*x^2+7*x^3)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)

Formula

From N. J. A. Sloane, Feb 16 2025: (Start)
Let h(i) denote the centered hexagonal number A003215(i). Then for n >= 1,
a(n) = h(2*n-2) + 2*Sum_{i=n-1..2*n-3} h(i).
E.g. a(3) = h(2) + h(3) + h(4) + h(3) + h(2), as in the COMMENTS.
This sequence should really have had offset 0, not 1, which would have given a simpler formula. (End)
G.f.: x*(1+29*x+47*x^2+7*x^3)/(1-x)^4. - Colin Barker, Jun 16 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012
E.g.f.: 7 + exp(x)*(-7 + 8*x + 12*x^2 + 14*x^3). - Elmo R. Oliveira, Sep 02 2025

Extensions

More terms from Colin Barker, Jun 16 2012
New name using explicit formula from Joerg Arndt, Jan 15 2021
Edited by N. J. A. Sloane, Feb 16 2025

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