A178482 Phi-antipalindromic numbers.
1, 3, 4, 7, 8, 10, 11, 18, 19, 21, 22, 25, 26, 28, 29, 47, 48, 50, 51, 54, 55, 57, 58, 65, 66, 68, 69, 72, 73, 75, 76, 123, 124, 126, 127, 130, 131, 133, 134, 141, 142, 144, 145, 148, 149, 151, 152, 170, 171, 173, 174
Offset: 1
Examples
The vectors of exponents of 4 and 5 are (-2,0,2) and (-4,-1,3) correspondingly (cf.A104605). Therefore by definition 4 is a phi-antipalindromic number, while 5 is not. Let n=38. Then k=5. Thus a(38)=A005248(5)+a(6)=123+10=133. The vector of exponents of phi in the base-phi expansion of 133 is (-10,-4,-2,2,4,10).
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3071 from R. J. Mathar)
- Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.
Crossrefs
Programs
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Mathematica
phiAPQ[1] = True; phiAPQ[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; Reverse[e] == -e]; Select[Range[200], phiAPQ] (* Amiram Eldar, Apr 23 2020 *)
Formula
For k>=1, a(2^k)=A005248(k); if 2^k
A104626 Numbers having three 1's in their base-phi representation.
4, 5, 6, 8, 19, 48, 124, 323, 844, 2208, 5779, 15128, 39604, 103683, 271444, 710648, 1860499, 4870848, 12752044, 33385283, 87403804, 228826128, 599074579, 1568397608, 4106118244, 10749957123, 28143753124, 73681302248
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.
- Eric Weisstein's World of Mathematics, Phi Number System
- Wikipedia, Golden ratio base
Programs
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Magma
[4,5,6] cat [1 + Fibonacci(2*n-3) + Fibonacci(2*n-5): n in [4..50]]; // G. C. Greubel, Aug 13 2018
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Mathematica
Join[{4, 5, 6}, Table[LucasL[2*n-4] + 1, {n, 4, 50}]] (* G. C. Greubel, Aug 13 2018 *) -
PARI
for(n=1,50, print1(if(n==1,4, if(n==2, 5, if(n==3, 6, 1 + fibonacci(2*n-3) + fibonacci(2*n-5)))), ", ")) \\ G. C. Greubel, Aug 13 2018
Formula
{n: A055778(n) = 3}. - R. J. Mathar, Sep 05 2010
a(n) = Lucas(2*n-4) + 1, for n>3. - Ralf Stephan, Nov 13 2010
Extensions
Terms 5 and 6 added by Jaroslav Krizek, May 25 2010
Edited by R. J. Mathar, Sep 05 2010
A104628 Numbers having 5 1's in their base-phi representation.
11, 15, 22, 23, 24, 26, 30, 31, 32, 34, 35, 37, 41, 42, 43, 45, 46, 51, 52, 53, 55, 66, 83, 95, 112, 127, 128, 129, 131, 142, 171, 217, 247, 293, 326, 327, 328, 330, 341, 370, 446, 568, 645, 767, 847, 848, 849, 851, 862, 891, 967, 1166, 1487, 1687, 2008, 2211
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..300
- Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.
- Eric Weisstein's World of Mathematics, Phi Number System.
Programs
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Mathematica
q[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]] == 5; Select[Range[2000], q] (* Amiram Eldar, Jan 20 2022 *)
Formula
There is a 46-state automaton accepting precisely the Zeckendorf representation of members of this sequence. - Jeffrey Shallit, May 03 2023
A178493 Numbers of powers of phi in base-phi expansion of phi-antipalindromic numbers (A178482).
1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 5, 6, 7, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 4, 5, 6, 7, 6, 7, 8
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
- Eric Weisstein's World of Mathematics, Phi Number System
Programs
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Mathematica
powNum[1] = 1; powNum[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]] - Flatten @ Position[d[[1]], 1]; If[Reverse[e] == -e, Length[e], 0]]; Select[Array[powNum, 400], # > 0 &] (* Amiram Eldar, Apr 23 2020 *)
Extensions
Extended by T. D. Noe, May 20 2011
Comments