A081434 -id:A081434 - cp's OEIS frontend

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

1, 7, 24, 58, 115, 201, 322, 484, 693, 955, 1276, 1662, 2119, 2653, 3270, 3976, 4777, 5679, 6688, 7810, 9051, 10417, 11914, 13548, 15325, 17251, 19332, 21574, 23983, 26565, 29326, 32272, 35409, 38743, 42280, 46026, 49987, 54169, 58578, 63220

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009
Row 1 of the convolution arrays A213831 and A213833. - Clark Kimberling, Jul 04 2012
Partial sums of A056109. - J. M. Bergot, Jun 22 2013
Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014
Row sums of A244418. - L. Edson Jeffery, Jan 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Christian Barrientos, The number of spanning trees of cyclic snakes, Indones. J. Comb. (2025) Vol. 9, No. 1, 21-30. See p. 29.
J. A. Dias da Silva and Pedro J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081435, A081437, A156933, A157705, A244418.

List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

A081436 := proc(n)
    (n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n),n=0..60) ; # R. J. Mathar, Jun 26 2013

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013

[(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.
G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)
a(n) = A110449(n, n-1), for n>1.
a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021

G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009

A081435 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 5, 18, 46, 95, 171, 280, 428, 621, 865, 1166, 1530, 1963, 2471, 3060, 3736, 4505, 5373, 6346, 7430, 8631, 9955, 11408, 12996, 14725, 16601, 18630, 20818, 23171, 25695, 28396, 31280, 34353, 37621, 41090, 44766, 48655, 52763, 57096, 61660

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081436, A081437.

List([0..40], n-> (n+1)*(2*(n+1)^2-3*n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+3*n^2+3*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013

a := n-> (n+1)*(2*(n+1)^2-3*n)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

Table[(n^3 +(n+1)^3 -1)/2 +1, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
CoefficientList[Series[(1 +3x^2 -4x^3)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,5,18,46},40] (* Harvey P. Dale, Dec 28 2024 *)

vector(40, n, n--; (n+1)*(2*(n+1)^2-3*n)/2) \\ G. C. Greubel, Aug 14 2019

[(n+1)*(2*(n+1)^2-3*n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3 +3*n^2 +3*n +2)/2.
G.f.: (1 +3*x^2 -4*x^3)/(1-x)^5.
E.g.f.: (2 +8*x +9*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A081433 Numbers n such that RevBinary(RevDecimal(n))=RevDecimal(RevBinary(n)), where RevDecimal(n) is the decimal reversal of n (A004086) and RevBinary(n) is the binary reversal of n (A030101).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 33, 51, 66, 99, 165, 313, 516, 535, 561, 585, 615, 626, 717, 759, 858, 929, 957, 1241, 1421, 2112, 2482, 2552, 2842, 3579, 4224, 5485, 5845, 6336, 7447, 8448, 9009, 9753, 11051, 12631, 13621, 15011, 15351, 15375

Offset: 1

Reinhard Zumkeller, Mar 20 2003

A030101(A004086(a(n)))=A004086(A030101(a(n))), A081431(a(n))=A081432(a(n)).

Cf. A081432, A081434, A007088.

def ok(n): return int(bin(int(str(n)[::-1]))[:1:-1], 2) == int(str(int(bin(n)[:1:-1], 2))[::-1])
print([k for k in range(2**14) if ok(k)]) # Michael S. Branicky, Jan 30 2023

A347209 Emirps in both base 2 and base 10.

Original entry on oeis.org

13, 37, 71, 97, 113, 167, 199, 337, 359, 701, 709, 739, 907, 937, 941, 953, 967, 1033, 1091, 1109, 1153, 1181, 1201, 1217, 1229, 1259, 1439, 1471, 1487, 1669, 1733, 1789, 1811, 1933, 1949, 3019, 3067, 3083, 3089, 3121, 3163, 3221, 3299, 3343, 3389, 3433, 3469, 3511, 3527, 3571, 3583, 3643, 3719

Offset: 1

J. M. Bergot and Robert Israel, Aug 23 2021

Primes p such that A030101(p) and A004086(p) are primes other than p.

Are there any cases where A030101(p) = A004086(p), i.e. emirps in A081434?

			a(3) = 71 is a term because 71 is prime, its base-10 reversal 17 is a prime other than 71, and its base-2 reversal 113 is a prime other than 71.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A006567 and A080790.
Subset of A136634.
Cf. A004086, A030101, A081434.

filter:= proc(n) local L,nL,i,r,s;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  nL:= nops(L);
  r:= add(10^(nL-i)*L[i],i=1..nL);
  if r=n or not isprime(r) then return false fi;
  L:= convert(n,base,2);
  nL:= nops(L);
  s:=add(2^(nL-i)*L[i],i=1..nL);
  s <> n and isprime(s)
end proc:
select(filter, [seq(i,i=3..10000,2)]);

Select[Range[4000], (ir = IntegerReverse[#]) != # && PrimeQ[#] && PrimeQ[ir] && (ir2 = IntegerReverse[#, 2]) != # && PrimeQ[ir2] &] (* Amiram Eldar, Aug 23 2021 *)
Select[Prime[Range[600]],!PalindromeQ[#]&&FromDigits[Reverse[IntegerDigits[#,2]],2]!=#&&AllTrue[{IntegerReverse[#],FromDigits[Reverse[IntegerDigits[#,2]],2]},PrimeQ]&] (* Harvey P. Dale, Oct 13 2022 *)

from sympy import isprime, primerange
def ok(p):
    s, b = str(p), bin(p)[2:]
    if s == s[::-1] or b == b[::-1]: return False
    return isprime(int(s[::-1])) and isprime(int(b[::-1], 2))
print(list(filter(ok, primerange(1, 3720)))) # Michael S. Branicky, Aug 24 2021

A176630 Nonpalindromic numbers whose binary representation when reversed is the same as binary representation of the number reversed in decimal.

Original entry on oeis.org

92, 732, 759, 957, 5485, 5845, 71869, 77360, 96817, 319773, 377913, 13162800, 39781062, 79497594, 94729789, 98792749, 144579540, 1231493321, 1233941321, 7075293947, 7493925707, 32817543720, 71461803829, 92830816417, 169709554740, 1432254694771, 1774964522341

Offset: 1

Gil Broussard, Apr 22 2010

The binary representation of a decimal number, when reversed, is also the reverse of the decimal number.

			92 = 1011100 mirrors 0011101 = 29.
732 = 1011011100 mirrors 0011101101 = 237.

Cf. A029742, A081434.

Select[Range[10^6], And[! PalindromeQ@ #, Drop[#, LengthWhile[#, # == 0 &]] &@ Reverse@ IntegerDigits[#, 2] === IntegerDigits[IntegerReverse[#], 2]] &] (* Michael De Vlieger, Dec 29 2020 *)

is(k)={my(t=fromdigits(Vecrev(digits(k,10)),10)); t<>k && t == fromdigits(Vecrev(digits(k,2)),2)} \\ Andrew Howroyd, Jan 14 2020

def agen():
  k = 0
  while True:
    strk = str(k)
    revstrk = strk[::-1]
    if revstrk != strk:
      if int(revstrk) == int((bin(k)[2:])[::-1], 2):
        yield k
    k += 1
g = agen()
print([next(g) for i in range(11)]) # Michael S. Branicky, Dec 29 2020

Intersection of A029742 and A081434. - Andrew Howroyd, Jan 14 2020

Name clarified and a(12)-a(17) from Andrew Howroyd, Jan 14 2020
a(18)-a(24) from Michael S. Branicky, Dec 29 2020
a(25)-a(27) from Jinyuan Wang, Apr 07 2025

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A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

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A081435 Diagonal in array of n-gonal numbers A081422.

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A081433 Numbers n such that RevBinary(RevDecimal(n))=RevDecimal(RevBinary(n)), where RevDecimal(n) is the decimal reversal of n (A004086) and RevBinary(n) is the binary reversal of n (A030101).

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A347209 Emirps in both base 2 and base 10.

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A176630 Nonpalindromic numbers whose binary representation when reversed is the same as binary representation of the number reversed in decimal.

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