A088136 -id:A088136 - cp's OEIS frontend

A045873 a(n) = A006496(n)/2.

0, 1, 2, -1, -12, -19, 22, 139, 168, -359, -1558, -1321, 5148, 16901, 8062, -68381, -177072, -12239, 860882, 1782959, -738492, -10391779, -17091098, 17776699, 121008888, 153134281, -298775878, -1363223161, -1232566932

Offset: 0

N. J. A. Sloane

Partial sums of A006495. - Paul Barry, Mar 16 2006
This is the Lucas U(P=2,Q=5) sequence. - R. J. Mathar, Oct 24 2012
With different signs, 0, 1, -2, -1, 12, -19, -22, 139, -168, -359, 1558, ... we obtain the Lucas U(-2,5) sequence. - R. J. Mathar, Jan 08 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-5).
Index entries for Lucas sequences

Cf. A006495, A006496, A084102, A088136, A088137, A088139, A094423.

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-5*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 5*Self(n-2): n in [1..50]]; // G. C. Greubel, Oct 22 2018

seq(coeff(series(x/(1-2*x+5*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018

LinearRecurrence[{2,-5}, {0,1}, 40] (* G. C. Greubel, Jan 11 2024 *)

concat(0,Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ Charles R Greathouse IV, Dec 22 2011

[lucas_number1(n,2,5) for n in range(0, 29)] # Zerinvary Lajos, Apr 23 2009

A045873=BinaryRecurrenceSequence(2,-5,0,1)
[A045873(n) for n in range(41)] # G. C. Greubel, Jan 11 2024

a(n)^2 = A094423(n).
From Paul Barry, Sep 20 2003: (Start)
O.g.f.: x/(1 - 2*x + 5*x^2).
E.g.f.: exp(x)*sin(2*x)/2.
a(n) = 2*a(n-1) - 5*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1 + 2*i)^n - (1 - 2*i)^n)/(4*i), where i=sqrt(-1).
a(n) = Im{(1 + 2*i)^n/2}.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)*(-4)^k. (End)
a(n+1) = Sum_{k=0..n} binomial(k,n-k)*2^k*(-5/2)^(n-k). - Paul Barry, Mar 16 2006
G.f.: 1/(4*x - 1/G(0)) where G(k) = 1 - (k+1)/(1 - x/(x - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 5*x)/( x*(4*k+4 - 5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 1/sqrt(5)). - G. C. Greubel, Jan 11 2024

More terms from Paul Barry, Sep 20 2003

A088133 Sum of first and last digits of n. Different from A115299.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Zak Seidov, Sep 20 2003

Cf. A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088134, A088135, A088136.

Total[{First[IntegerDigits[#]],Last[IntegerDigits[#]]}]&/@Range[90] (* Harvey P. Dale, Aug 21 2018 *)

apply( {A088133(n)=n\10^logint(n+!n, 10)+n%10}, [0..99]) \\ M. F. Hasler, Apr 22 2024

list(map(A088133 := lambda n: int(str(n)[0])+n%10, range(99))) # M. F. Hasler, Apr 22 2024

a(n) = A000030(n) + A010879(n). - M. F. Hasler, Apr 22 2024

Extended to a(0) = 0 by M. F. Hasler, Apr 22 2024

A088134 Numbers n such that sum of first and last digits is prime.

Original entry on oeis.org

1, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 102, 104, 106, 111, 112, 114, 116, 121, 122, 124, 126, 131, 132, 134, 136, 141, 142, 144, 146, 151, 152, 154, 156, 161, 162, 164

Offset: 1

Zak Seidov, Sep 20 2003

Cf. A046704, A007953, A088133, A088135, A088136.

A088139 a(n) = 2a(n-1) - 6a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, -2, -16, -20, 56, 232, 128, -1136, -3040, 736, 19712, 35008, -48256, -306560, -323584, 1192192, 4325888, 1498624, -22958080, -54907904, 27932672, 385312768, 603029504, -1105817600, -5829812224, -5024718848, 24929435648, 80007184384

Offset: 0

Paul Barry, Sep 20 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-6).

Cf. A045873, A084102, A088136, A088137.

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-6*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 22 2018

seq(coeff(series(x/(1-2*x+6*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018

Join[{a=0,b=1},Table[c=2*b-6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
TrigExpand@Table[(6^(n/2) Sin[n ArcTan[Sqrt[5]]])/Sqrt[5], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k + 1], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)
LinearRecurrence[{2,-6},{0,1},40] (* Harvey P. Dale, Nov 22 2024 *)

x='x+O('x^30); concat([0], Vec(x/(1-2*x+6*x^2))) \\ G. C. Greubel, Oct 22 2018

[lucas_number1(n,2,6) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1-2*x+6*x^2).
E.g.f.: exp(x)*sin(sqrt(5)*x)/sqrt(5).
a(n) = ((1+i*sqrt(5))^n-(1-i*sqrt(5))^n)/(2*i*sqrt(5)).
a(n) = Im{(1+i*sqrt(5))^n/sqrt(5)}.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)(-5)^k.
a(n+1) = (-1)^n*Sum_{k, 0<=k<=n} A172250(n,k)*(-2)^k. - Philippe Deléham, Feb 15 2012

A088135 Sum of first and last digits of n-th prime.

Original entry on oeis.org

4, 6, 10, 14, 2, 4, 8, 10, 5, 11, 4, 10, 5, 7, 11, 8, 14, 7, 13, 8, 10, 16, 11, 17, 16, 2, 4, 8, 10, 4, 8, 2, 8, 10, 10, 2, 8, 4, 8, 4, 10, 2, 2, 4, 8, 10, 3, 5, 9, 11, 5, 11, 3, 3, 9, 5, 11, 3, 9, 3, 5, 5, 10, 4, 6, 10, 4, 10, 10, 12, 6, 12, 10, 6, 12, 6, 12, 10, 5, 13, 13, 5, 5, 7, 13, 7, 13

Offset: 1

Zak Seidov, Sep 20 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046704, A007953, A088133, A088134, A088136.

sfl[p_]:=Module[{idn=IntegerDigits[p]},idn[[1]]+idn[[-1]]]; sfl/@Prime[Range[90]] (* Harvey P. Dale, Jan 31 2023 *)

A108660 Square-loop primes.

Original entry on oeis.org

2, 13, 31, 79, 97, 227, 881, 1013, 2797, 3181, 3631, 8101, 22727, 81001, 101363, 109013, 131363, 181813, 272227, 310181, 310901, 318181, 318881, 631013, 636313, 810401, 818101, 901097, 904097, 972227, 1018813, 1090013, 1810013, 2272727

Offset: 1

Zak Seidov, Jun 16 2005

Primes such that each pair of adjacent digits (and also the first and the last ones) sums up to a square. First term is arguable since there is 'no pair of adjacent digits', but there are the "first" and "last" digits.

Harvey P. Dale, Table of n, a(n) for n = 1..120

Cf. A046704, A007953, A088133, A088134, A088135, A088136, A108659.

Select[Prime[Range[200000]],And@@(IntegerQ[Sqrt[#]]&/@(Total/@Partition[ IntegerDigits[#],2,1,1]))&] (* Harvey P. Dale, Mar 03 2014 *)

A108659 Square-chain primes (including square-loop primes).

Original entry on oeis.org

2, 13, 31, 79, 97, 101, 109, 131, 181, 227, 313, 401, 409, 631, 727, 797, 881, 1009, 1013, 1097, 2797, 3109, 3181, 3631, 4001, 4013, 7901, 8101, 9001, 9013, 10009, 10181, 10909, 10979, 13109, 18131, 18181, 22279, 22727, 27901, 31013, 36313

Offset: 1

Zak Seidov, Jun 16 2005

Primes such that each pair of adjacent digits sums up to a square. First term is a square-loop prime, cf. A108660.

Cf. A046704, A007953, A088133, A088134, A088135, A088136, A108660.

Join[{2},Select[Prime[Range[5,4000]],PrimeQ[#]&&AllTrue[Sqrt[#]&/@(Total/@Partition[ IntegerDigits[ #],2,1]),IntegerQ]&]] (* Harvey P. Dale, Jun 02 2024 *)

A109981 Primes such that both the sum of digits and the number of digits are primes.

Original entry on oeis.org

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Zak Seidov, Jul 06 2005

Cf. A046704 Additive primes: sum of digits is a prime, A088136 Primes such that sum of first and last digits is prime.

			a(86) = 10037 because both the sum (=11) and number (=5) of digits are primes.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A046704, A088136.

Cf. A010051, A055642.

a109981 n = a109981_list !! (n-1)
a109981_list = filter ((== 1) . a010051' . a055642) a046704_list
-- Reinhard Zumkeller, Nov 16 2012

Select[Prime[Range[200]], PrimeQ[Length[IntegerDigits[ # ]]]&&PrimeQ[Plus@@IntegerDigits[ # ]]&]

A109982 Primes p such that index of p, the sum of p's digits and the number of p's digits are all primes.

Original entry on oeis.org

11, 41, 67, 83, 157, 179, 191, 241, 283, 331, 353, 401, 461, 599, 739, 773, 797, 919, 991, 10079, 10169, 10433, 10457, 10589, 10631, 10723, 10853, 10909, 11311, 11447, 11867, 11953, 12097, 12143, 12301, 12457, 12479, 12503, 12547, 12763, 13003

Offset: 1

Zak Seidov, Jul 06 2005

			a(414) = 99551 because its index, 9551, the sum, 29 and number, 5, of digits are all primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046704 Additive primes: sum of digits is a prime, A088136 Primes such that sum of first and last digits is prime, A109981 Primes such that the sum of digits and the number of digits are primes.

Select[Prime[Range[200]], PrimeQ[Length[IntegerDigits[ # ]]]&&PrimeQ[Plus@@IntegerDigits[ # ]]&]
Select[Prime[Range[1600]],AllTrue[{PrimePi[#],Total[IntegerDigits[#]], IntegerLength[ #]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 15 2019 *)

A086924 Primes such that sum of the first and last digits is a square.

Original entry on oeis.org

2, 13, 31, 79, 97, 103, 113, 163, 173, 193, 227, 257, 277, 311, 331, 613, 643, 653, 673, 683, 709, 719, 739, 769, 811, 821, 881, 907, 937, 947, 967, 977, 997, 1013, 1033, 1063, 1093, 1103, 1123, 1153, 1163, 1193, 1213, 1223

Offset: 1

Zak Seidov, Sep 20 2003

Cf. A088133, A088134, A088135, A088136.

fldsQ[n_]:=Module[{idn=IntegerDigits[n]},IntegerQ[Sqrt[ idn[[1]] + idn[[-1]]]]]; Select[Prime[Range[200]],fldsQ] (* Harvey P. Dale, Aug 12 2017 *)

okdigs(n) = digs = digits(n); issquare(digs[1]+digs[#digs]);
isok(n) = isprime(n) && okdigs(n); \\ Michel Marcus, Oct 05 2013

cp's OEIS Frontend

A045873 a(n) = A006496(n)/2.

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A088133 Sum of first and last digits of n. Different from A115299.

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A088134 Numbers n such that sum of first and last digits is prime.

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A088139 a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.

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A088135 Sum of first and last digits of n-th prime.

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Mathematica

A108660 Square-loop primes.

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A108659 Square-chain primes (including square-loop primes).

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Mathematica

A109981 Primes such that both the sum of digits and the number of digits are primes.

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A088139 a(n) = 2a(n-1) - 6a(n-2), a(0)=0, a(1)=1.