A268413
a(n) = Sum_{k = 0..n} (-1)^k*14^k.
Original entry on oeis.org
1, -13, 183, -2561, 35855, -501969, 7027567, -98385937, 1377403119, -19283643665, 269971011311, -3779594158353, 52914318216943, -740800455037201, 10371206370520815, -145196889187291409, 2032756448622079727, -28458590280709116177, 398420263929927626479
Offset: 0
Cf. similar sequences of the type Sum_{k=0..n} (-1)^k*m^k:
A059841 (m=1),
A077925 (m=2),
A014983 (m=3),
A014985 (m=4),
A014986 (m=5),
A014987 (m=6),
A014989 (m=7),
A014990 (m=8),
A014991 (m=9),
A014992 (m=10),
A014993 (m=11),
A014994 (m=12),
A015000 (m=13), this sequence (m=14),
A239284 (m=15).
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I:=[1,-19]; [n le 2 select I[n] else -13*Self(n-1) +14*Self(n-2): n in [1..30]]; // G. C. Greubel, May 26 2018
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Table[((-1)^n 14^(n + 1) + 1)/15, {n, 0, 18}]
LinearRecurrence[{-13, 14}, {1, -13}, 19]
Table[Sum[(-1)^k*14^k, {k, 0, n}], {n, 0, 18}]
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x='x+O('x^30); Vec(1/(1 + 13*x - 14*x^2)) \\ G. C. Greubel, May 26 2018
A018084
Powers of fourth root of 14 rounded down.
Original entry on oeis.org
1, 1, 3, 7, 14, 27, 52, 101, 196, 379, 733, 1418, 2744, 5307, 10267, 19860, 38416, 74309, 143739, 278040, 537824, 1040332, 2012353, 3892567, 7529536, 14564655, 28172943, 54495951, 105413504, 203905179, 394421215, 762943322, 1475789056, 2854672519, 5521897022
Offset: 0
A038218
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j)*12^j (with i, j >= 0).
Original entry on oeis.org
1, 2, 12, 4, 48, 144, 8, 144, 864, 1728, 16, 384, 3456, 13824, 20736, 32, 960, 11520, 69120, 207360, 248832, 64, 2304, 34560, 276480, 1244160, 2985984, 2985984, 128, 5376, 96768, 967680, 5806080, 20901888, 41803776, 35831808
Offset: 0
From _Petros Hadjicostas_, Jul 23 2019: (Start)
Triangle T(i,j) (with rows i >= 0 and columns j >= 0) begins as follows:
1;
2, 12;
4, 48, 144;
8, 144, 864, 1728;
16, 384, 3456, 13824, 20736;
32, 960, 11520, 69120, 207360, 248832;
64, 2304, 34560, 276480, 1244160, 2985984, 2985984;
128, 5376, 96768, 967680, 5806080, 20901888, 41803776, 35831808;
... (End)
- B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
- Gábor Kallós, A generalization of Pascal's triangle using powers of base numbers, Ann. Math. Blaise Pascal 13(1) (2006), 1-15. [See Section 2 of the paper with title "ab-based triangles". Apparently, this is a 2(12)-based triangle; i.e., a = 2 and b = 12 even though b = 12 > 9. - _Petros Hadjicostas_, Jul 30 2019]
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/* As triangle */ [[Binomial(i,j)*2^(i-j)*12^j: j in [0..i]]: i in [0.. 15]]; // Vincenzo Librandi, Jul 24 2019
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Flatten[Table[Binomial[i, j] 2^(i - j) 12^j, {i, 0, 8}, {j, 0, i}]] (* Vincenzo Librandi, Jul 24 2019 *)
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